text:
\section*{Preface (by Yuri Bilu)}

This text is based on the expanded notes of the mini-courses on the work of  Bombieri, Pila, Wilkie, Zannier and others that  I gave in Basel in April 2011, Yaroslavl in August 2011 and Chennai in February 2012. I thank  the organizers who made it possible:
David Masser and Evelina Viada in Basel;
Fedor Bogomolov, Misha Verbitski and Alexei Zykin in Moscow;
V. Balaji, Sanoli Gun and Purusottam Rath in Chennai.

I also thank all the listeners for their precious comments. I am especially indebted to Lyosha Beshenov and Purusottam Rath, who prepared excellent notes of my courses in Yaroslavl and Chennai, respectively, and to Margaret Bilu, who attended my courses in Basel and Yaroslavl, and who wrote the present text following the notes by Lyosha and Purusottam. I only had to re-read and slightly polish the exposition at a few points, correct some inaccuracies\footnote{probably creating some new ones\dots} and add some references.

I thank Fedor Petrov and Umberto Zannier for useful discussions, Martin Klazar for detecting an error in the previous version, and Elina Wojciechowska for inspiration.

Finally, I thank the referee for many helpful comments.

Please do not forget  that this text is based on very informal lecture notes,  and do not expect completeness and precision (in particular in the references). Please do not forget either that English is not my mother tongue. I assume full responsibility for all remaining inaccuracies, and no responsibility for the damage they may cause.

 %

This work was supported by Bogomolov's mega-grant, ALGANT Scholarship Program, University of Basel, Erwin Schroedinger Institute (Vienna)
and the ANR project ``HAMOT'' (ANR 2010 BLAN-0115-01).

{\footnotesize

\tableofcontents }

\section{Introduction: Jarn\'ik and Dörge}
Let $N\in\N$ be an integer. We want to study the number of intersection points of a curve with an $N\times N$ lattice. More precisely, we want to bound the number of points of the lattice $\Lm_N = \frac{1}{N}\Z^2$
 lying on a curve $\G \subset [0,1)^2$ . Obviously,
$$|\G\cap \Lm_N|\leq N^2,$$
and we can find a curve $\G$ that makes this an equality, going through all vertices of the lattice in $[0,1)^2$. Suppose now that $\G$ is a graph of a function, that is, a set of the form $\{(x,y)\in [0,1)^2 \mid y = f(x)\}$. Then again, there is an easy bound
$$|\G\cap \Lm_N|\leq N,$$
and there exist graphs $\G$ realizing it. To make the problem more interesting, we are going to assume that $f$ is strictly concave or convex. The following statement is essentially due to Jarn\'ik~\cite{Ja26}.

\begin{theorem}[Jarn\'ik, 1926]
\label{thja}
Let $\G$ be the graph of a strictly convex or concave monotonic function ${[0,1]\to[0,1]}$. Then
\begin{equation}
\label{eja}
|\G\cap \Lm_N|\leq 3\pi^{-2/3}N^{2/3} + O(N^{1/3}\log N).
\end{equation}
\end{theorem}

The proof of Jarn\'ik's theorem will not be needed further, but it is simple and beautiful, and we sketch it below. Skipping it should not in any way harm the reader's understanding of the main content of this text.
We thank Fedor Petrov who explained to us Jarnik's argument and his subsequent work on the topic, and Martin Klazar, who detected an error in the previous version of the proof.

It will be more convenient to prove a (formally) more general statement, by considering functions ${I\to[0,1]}$, defined on a closed sub-interval~$I$ of $[0,1]$. Thus, define
$$
\mu(N)=\max_f|\Gamma_f\cap\Lambda_N|,
$$
where~$f$ runs over the set of strictly increasing and strictly convex functions from a closed sub-interval of $[0,1]$ to $[0,1]$, and~$\Gamma_f$ denotes the graph of~$f$. Further, let~$I$ be a shortest sub-interval of $[0,1]$ such that there exists ${f:I\to[0,1]}$ as above with ${|\Gamma_f\cap\Lambda_N|=\mu(N)}$. We fix these~$I$ and~$f$ and write ${\Gamma=\Gamma_f}$.

Call a vector ${(x,y)\in \Z^2}$ \emph{primitive} if ${d=\gcd(x,y)=1}$; in general, we call $(x/d,y/d)$ the \emph{primitivization} of $(x,y)$. Let~$S_X$ denotes the set of primitive vectors with positive coordinates whose sum is  bounded by~$X$:
$$
S_X=\{(x,y)\in \Z^2:\ x,y>0,\ \gcd(x,y)=1,\ x+y\le X\}
$$
Then
\begin{equation}
\label{esx}
|S_X|=\frac3{\pi^2}X^2+O(X\log X), \qquad \sum_{(x,y)\in S_X}x=\frac1{\pi^2}X^3+O(X^2\log X).
\end{equation}
(These relations are pretty standard. See, for instance, \cite[Section~18.5]{HW08} for the proof of a similar statement; both relations~\eqref{esx} can be proved analogously.)

Let ${P_0, \ldots, P_n}$ be the lattice points on~$\Gamma$ (so that ${n+1=\mu(N)}$) ordered by the increasing first coordinates.
The vectors ${(x_i,y_i)=N(P_i-P_{i-1})}$ are pairwise distinct (because the function is convex) and have positive coordinates (because it is increasing). We may also assume them to be primitive; if one of them is not, then, replacing it by the primitivization, we may find a new (strictly increasing and strictly convex) function  with graph having  ${n+1}$ lattice points, but defined on a shorter interval,
contradicting our choice of~$I$.

Let $X$ be the largest real number such that ${n+1\ge |S_X|}$. Then
$$
\begin{gathered}
|\Gamma\cap\Lambda_N|=n+1=\frac3{\pi^2}X^2+O(X\log X), \\
2N\ge\sum_i(x_i+y_i)\ge 2\sum_{(x,y)\in S_X}x=\frac2{\pi^2}X^3+O(X^2\log X),
\end{gathered}
$$
whence the result.

\bigskip

It follows from the proof that bound~\eqref{eja} is optimal in the sense that for each~$N$ we can find a curve $\G_N$ (satisfying the assumptions of Theorem~\ref{thja})
such that
$$
|\G_N\cap\Lm_N| \ge 3\pi^{-2/3}N^{2/3}(1+o(1)).
$$
Here the curve~$\Gamma_N$ depends on~$N$. One may wonder whether there  exists a ``universal Jarn\'ik curve'' $\G$ satisfying ${|\G_N\cap\Lm_N| >\kappa N^{2/3}}$ for infinitely many  $N$, where~$\kappa$ is a positive real number, not necessarily equal to $3\pi^{-2/3}$. This question, which  is attributed to J.-M. Deshouillers and A.~Plagne~\cite{Pl99}, was  recently answered negatively by  F.~Petrov~\cite{Pe06}, who showed that for any fixed $\G$ we have
$$|\G\cap\Lm_N| = o(N^{2/3})\quad \text{as}\quad N\longrightarrow \infty.$$

Now we  give a totally different  proof of a weaker version of Jarn\'ik's theorem. This proof is based on an idea of Dörge~\cite{Doe27} (1927) (see also \cite[Section 9.2]{La83}) that will turn out to be very fruitful.

 \begin{theorem}(Weak Jarn\'ik) Let $I$ be a compact interval and $\G$ be the graph $\{(x,y) \mid y = f(x)\}$ of a strictly convex function $f\in\CC^2(I)$. Then
 $$|\G\cap\Lm_N|\leq c(I,f)\,N^{\frac23}.$$
 \end{theorem}
Note that this is weaker than Jarn\'ik's theorem because $f$ is assumed to be $\CC^2$ and because the constant in the bound depends on $f$.

The proof uses the following generalization of Lagrange's mean value theorem:

\begin{proposition}(Schwarz mean value theorem) Let~$I$ be an interval, ${f\in\CC^n(I)}$, and ${x_0, \ldots, x_n \in I}$. Then there is $\tau\in I$ such that
$$ \left|\begin{array}{ccccc}
1 & x_0 & \cdots & x_0^{n-1} & f(x_0) \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_n & \cdots & x_n^{n-1} & f(x_n) \\
\end{array}\right| = \frac{f^{(n)} (\tau)}{n!} \, V(x_0, \ldots, x_n).$$
\end{proposition}
Here $V(x_0,\ldots,x_n)$ denotes the Vandermonde determinant
$$ \left|\begin{array}{ccccc}
1 & x_0 & \cdots & x_0^{n-1} & x_0^n \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_n & \cdots & x_n^{n-1} & x_n^n \\
\end{array}\right|. $$

\noindent\textit{Proof of Schwarz mean value theorem.} Let $g(t) = a_nt^n + \cdots + a_1t + a_0$ be the Lagrange interpolation polynomial for $f$ at the points $x_0,\ldots, x_n$. Then $(f-g)(x_i) = 0$ for all $i = 0,\ldots n$, and therefore by a ``generalized Rolle theorem'' there is $\tau\in I$ such that $f^{(n)}(\tau) = g^{(n)}(\tau)$.  Moreover, $g^{(n)}(\tau) = n!\,a_n$. Noting that
$$V(x_0,\ldots,x_n)\,a_n  =  \left|\begin{array}{ccccc}
1 & x_0 & \cdots & x_0^{n-1} & f(x_0) \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
1 & x_n & \cdots & x_n^{n-1} & f(x_n) \\
\end{array}\right|$$
then yields the result.\qed

\bigskip

\noindent\textit{Proof of the weak Jarn\'ik theorem.} Let us pick three points ${P_i = (x_i,y_i)}$, ${i = 0,1,2}$ on $\G$. Because of the strict convexity of $f$, they cannot lie on one straight line, so
$$ \Delta = \left|\begin{array}{ccc}
1 & x_0 & y_0 \\
1 & x_1 & y_1 \\
1 & x_2 & y_2 \\
\end{array}\right| \ne 0. $$
By the Schwarz mean value theorem, there exists ${\tau\in I}$ such that
$$ \Delta = \left|\begin{array}{ccc}
1 & x_0 & f(x_0) \\
1 & x_1 & f(x_1) \\
1 & x_2 & f(x_2) \\
\end{array}\right| = \frac{f^{\prime\prime}(\tau)}{2!}\,V(x_0,x_1,x_2). $$
Since $I$ is compact, this gives us an upper bound
$$|\Delta|\leq c(I,f)\max_{1\leq i < j \leq 3} |x_i-x_j|^3.$$
Assume now that ${P_0,P_1,P_2\in \G \cap\Lm_N}$, that is, the $P_i$ are rational points whose coordinates have denominators dividing $N$. Then $\Delta$ is a non-zero rational number with denominator at most $N^2$, so $$|\Delta|\geq \frac{1}{N^2}.$$ This, combined with the above upper bound, gives us
$$\max_{1\leq i < j \leq 3} |x_i-x_j| \geq c\,N^{-2/3}.$$ We have thus proved that an interval containing the projections of three points lying in $\G \cap\Lm_N$ has length bounded from below by $c N^{-2/3}$. Cutting $I$ into small intervals of length $\frac{c}{2}N^{-{2}/{3}}$, we have at most two points projecting to each such small interval. Denoting by $\ell(I)$ the length of $I$, there are at most $ \ell (I)\frac{2}{c}N^{{2}/{3}}$ such intervals, so the result follows.\qed

In fact, we have proved the following
\begin{proposition}Let $I$ be a compact interval and $\G$ the graph $\{(x,y) \mid y = f(x)\}$ of a  function $f\in\CC^2(I)$. There is a constant $c$ depending only on~$I$ and~$f$, having the following property. If for the points
$$P_0,\ldots,P_k \in\G\cap\Lm_N,\ \ \ P_i = (x_i,y_i)$$
the coordinates $x_1,\ldots,x_k$ lie in an interval of length smaller than $c\,N^{-{2}/{3}}$, then $P_0,\ldots,P_k$ lie on a straight line.\end{proposition}

In particular, the set $\G\cap\Lm_N$ is covered by $O_f(N^{2/3})$ straight lines. Bombieri and Pila~\cite{BP89} came up with a crucial idea:  consider not only straight lines, but also more general algebraic curves. We discuss it in the next section.

\section{The theorem of Bombieri and Pila}
Generalizing Dörge's ideas, Bombieri and Pila~\cite{BP89} proved  that a plane analytic compact curve cannot contain many rational points with a given denominator.

In the sequel by an \emph{irreducible plane curve of degree~$d$} we mean a subset of $\R^2$ consisting of points $(x,y)$ satisfying $F(x,y)=0$, where $F(X,Y)$ is an irreducible real polynomial of degree~$d$. A \emph{segment} of such a curve is its connected compact subset.

\begin{theorem}\label{bombieri}(Bombieri, Pila, 1989) Let $\G\subset\R^2$ be a real analytic plane compact curve.
\begin{enumerate}\item Assume that $\G$ is transcendental.  Then for any $\eps>0$ there is a constant $C(\G,\eps)$ such that
$$|\G\cap\Lm_N|\leq C(\G,\eps)\,N^{\eps},$$
where  $\Lm_N = \frac{1}{N}\Z^2$.
\item Assume that $\G$ is a segment of an irreducible plane algebraic curve of degree $d$. Then for any $\eps>0$ there is a constant $C(\G,\eps)$ such that
$$|\G\cap\Lm_N|\leq C(\G,\eps)\,N^{\frac{1}{d}+\eps}.$$
\end{enumerate}
\end{theorem}
\paragraph{Remarks}\begin{enumerate} \item The example of a curve given by $y =x^d$ shows that the exponent in the second part of the theorem cannot be made smaller than ${1}/{d}$.
\item In the second part of the theorem one can replace $C(\G,\eps)$ by $C(d,\eps)$, if one assumes in addition that $\G\subset[0,1]^2$.
\end{enumerate}

We are going to use the following generalization of the mean value theorem:

\begin{proposition} (Generalized Schwarz mean value theorem) Let~$I$ be an interval, ${f_0,\ldots,f_n\in \CC^n(I)}$, and ${x_0,\ldots,x_n\in I}$. Then for all $i,j$ such that ${0\leq i,j \leq n}$ there is $\tau_{ij}\in I$ such that
$$ \det [f_i (x_j)]_{i,j} = \det \left[\frac{f_i^{(j)} (\tau_{ij})}{j!}\right]_{ij} \cdot V (x_0, \ldots, x_n). $$
\end{proposition}
 Here is the proof  for $n=1$: by the Lagrange mean value theorem, there are ${\tau_{01},\tau_{11}\in I}$ such that ${f_i(x_1) - f_i(x_0) = f'_i(\tau_{i1})\,(x_1 - x_0)}$ for $i=0,1$. Thus,
$$ \left|\begin{array}{cc}
f_0 (x_0) & f_0 (x_1) \\
f_1 (x_0) & f_1 (x_1) \\
\end{array}\right| =
\left|\begin{array}{cc}
f_0 (x_0) & f_0 (x_1) - f_0 (x_0) \\
f_1 (x_0) & f_1 (x_1) - f_1 (x_0) \\
\end{array}\right| =
\left|\begin{array}{cc}
f_0 (x_0) & f_0^\prime (\tau_{01}) \\
f_1 (x_0) & f_1^\prime (\tau_{11}) \\
\end{array}\right|\,(x_1-x_0), $$
and we can choose $\tau_{00} = \tau_{10} = x_0$.

The proof of the general case is very similar, but with Lagrange replaced by Schwarz, see \cite{vieux}.
\begin{corollary}\label{cor} If the interval $I$ is compact, there is a constant $c$ depending only on $f_0,\ldots,f_n$ such that
$$\left|\det [f_i (x_j)]_{i,j}\right|\leq c\,|V(x_0,\ldots,x_n)|.$$
\end{corollary}
\begin{definition} Let $d,e\in\N$. We call a \emph{$(d,e)$-curve} a (possibly reducible) plane algebraic curve with equation of the form $P(x,y) = 0$ where $P\in\R[x,y]$ is a non-zero polynomial such that $\deg_x P < d$ and $\deg_y P < e$.
\end{definition}
For example, a $(2,2)$-curve is of the form $a + bx + cy + hxy = 0$.

\paragraph{Facts about $(d,e)$-curves}\begin{enumerate}\item Any collection of $de-1$ points lies on a $(d,e)$-curve.
\item A collection of $de$ points $P_k = (x_k,y_k)$, $k = 1,\ldots,de$, lies on a $(d,e)$-curve if and only if
$$ \Delta := \det [x_k^i\,y_k^j]_{\substack{(i,j) = (0,0), \ldots, (d-1,e-1)\\ k = 1, \ldots, d\,e}} = 0. $$
Note that the rows in the above determinant are numbered by the bi-index $(i,j)$. For example, for $(2,2)$-curves this condition for four points $P_1,\ldots, P_4$ takes the form
$$ \left|\begin{array}{cccc}
1 & 1 & 1 & 1 \\
x_1 & x_2 & x_3 & x_4 \\
y_1 &  y_2& y_3 & y_4 \\
x_1\,y_1 & x_2\,y_2 & x_3\,y_3 & x_4\,y_4 \\
\end{array}\right| = 0. $$
\item If $P_1,\ldots,P_s$ do not lie on a $(d,e)$-curve (in particular $s\geq de$  by the above), then one can select $de$ points among them not lying on a $(d,e)$-curve.
\end{enumerate}
Let $\G$ be the graph $y = f(x)$ of a function $f\in \CC^{de-1}(I)$ where $I$ is an interval, and consider $P_1,\ldots,P_{de}\in\G\cap\Lm_N$ not lying on a $(d,e)$-curve. Then, by the above, writing $g_{i,j}(x) = x^if(x)^j$ we have
$$ \Delta = \det [g_{ij}(x_k)]_{\substack{(i,j) = (0,0), \ldots, (d-1,e-1)\\ k = 1, \ldots, d\,e}} \ne 0, $$
where $P_i = (x_i,f(x_i))$. Using Corollary \ref{cor} for the functions $g_{ij}$, we get
$$|\Delta|\leq c(f)\,|V(x_1,\ldots,x_{d\,e})|\leq c\left(\max_{1\leq k<l\leq d\,e}|x_k-x_l|\right)^{\frac{de(de-1)}{2}},$$
where the constant $c$ depends only on~$I$ and~$f$. On the other hand, the coordinates of $P_i$'s are rational numbers with denominator dividing $N$. By assumption, the determinant $\Delta$ evaluates to a non-zero rational number, and the entries in its $(i,j)$-th column have a common denominator dividing $N^{i+j}$. Thus the denominator of $\Delta$ is bounded by
$$\prod_{(i,j)} N^{i+j} = N^{\sum_{(i,j)} i+j} = N^{\frac{de(d+e-2)}{2}}$$
and we have
$$|\Delta|\geq N^{-\frac{de(d+e-2)}{2}}.$$
We can therefore conclude that if points $P_1, \ldots, P_{de}\in\G\cap\Lm_N$ do not lie on a $(d,e)$-curve, then
$$\max |x_k - x_l| \ge c(f) \cdot N^{- \frac{e+d-2}{ed-1}} = c(f) \cdot N^{-\delta},$$
where $\delta = \frac{d+e-2}{de-1}.$
This means that if $de$ points of $\G\cap\Lm_N$ project to an interval of length $\frac{c}{2}N^{\delta}$, they necessarily lie on a $(d,e)$-curve. Since we can cut our interval $I$ into $c(f,d,e,I)\,N^\delta$ such intervals, we have the following.

{\sloppy

\begin{lemma}(Main lemma of Bombieri--Pila) Let~$\G$ be the graph ${y = f(x)}$ of a function ${f\in \CC^{de-1}(I)}$. Then the set $\G\cap\Lm_N$ is covered by $cN^\delta$ \quad  $(d,e)$-curves, where~$c$ depends on~$I$,~$f$,~$d$,~$e$ and  ${\delta = \frac{d+e-2}{de-1}}$.
\end{lemma}
Note that the approach of Bombieri and Pila is an exact analogue of the proof of the weak Jarn\'ik theorem we gave above, with straight lines replaced by more general $(d,e)$-curves.

}

\bigskip
\noindent\textit{Proof of Theorem \ref{bombieri}.}\ We prove the two statements of the theorem separately:
 \begin{enumerate}\item Take $\G$ transcendental, analytic and compact as in the statement of the theorem, and $\eps>0$. Choose moreover $d,e$ such that ${\frac{d+e-2}{de-1}<\eps}$. We can break up $\G$ into a finite number of pieces of the form $y = f(x)$ or $x = f(y)$, to each of which we apply the main lemma. Thus $\G \cap\Lm_N$ is covered by $cN^\eps$ \quad $(d,e)$-curves, where ${c= c(\G,\eps)}$. On the other hand, since $\G$ is transcendental, the size of the intersection of $\G$ with a $(d,e)$-curve is bounded uniformly by a constant depending only on $\G$, $d$, $e$.

\item Assume now that $\G$ is a segment of an irreducible algebraic curve of degree $d$. By applying a linear change of variables, we may assume that it is defined by the polynomial equation ${F(x,y)=0}$ where ${\deg_xF=d}$.
Choose $e$ big enough such that
$\frac{d+e-2}{de-1}<\frac{1}{d} + \eps.$ Then, applying the main lemma, $\G\cap\Lm_N$ is covered by $cN^{1/d+\eps}$ \quad  $(d,e)$-curves, where ${c=c(\G,\eps)}$. On the other hand, since ${\deg_xF=d}$, our curve~$\Gamma$ cannot be contained in a $(d,e)$-curve. By the Bézout theorem, the size of the intersection of $\G$ with a $(d,e)$-curve is bounded by $c(d,e)$, and we are done. \qed
\end{enumerate}

\section{Counting rational points with bounded height}
In this section, instead of counting lattice points on $\G$, we are going to count points of bounded height.

\begin{definition} Let $\alpha = \frac{p}{q}\in\Q$ with $p$ and $q$ coprime. Define the height of $\alpha$ by
$$H(\alpha) = \max\{|p|,|q|\}.$$
\end{definition}
Denoting $\G(\Q,N) = \{(x,y)\in\G\cap\Q^2,\ H(x),H(y)\leq N\}$, we can state the following stronger version of Theorem \ref{bombieri}:

\begin{theorem}
(Bombieri--Pila theorem for rational points)
\label{tbprat}
Let $\G$ be a plane compact analytic curve.\begin{enumerate} \item If $\G$ is transcendental, then for any $\eps>0$ there is a constant $c(\G,\eps)$ such that
$$|\G(\Q,N)|\leq c(\G,\eps)\,N^{\eps}.$$
\item If $\G$ is algebraic of degree $d$, then for any $\eps>0$ there is a constant $c(\G,\eps)$ such that
$$|\G(\Q,N)|\leq c(\G,\eps)\,N^{\frac{2}{d}+\eps}.$$
\end{enumerate}
\end{theorem}
Again, the second part cannot be improved, as shown by the example $y = x^d$.

\bigskip

\noindent \textit{Proof.} To get this result, we just need to modify a little the proof of Theorem \ref{bombieri}, more precisely, the proof of the Main Lemma.  Consider points
$$P_1,\ldots,P_{de}\in \G(\Q,N),\ \ \ P_k = (x_k,y_k)$$
not lying on a $(d,e)$-curve, where $\G$ is a graph $y=f(x)$. We still have the bound
$$|\Delta| \le c(f)\left(\max_{1\leq k<l\leq d\,e}|x_k-x_l|\right)^{\frac{de(de-1)}{2}}$$
obtained directly from the general mean value theorem where
$$ \Delta = \det [x_k^iy_k^j]_{\substack{(i,j) = (0,0), \ldots, (d-1,e-1)\\ k = 1, \ldots, d\,e}}\ne 0. $$
In fact, the only thing that needs to be changed is the computation of the lower bound for $\Delta$.
The difference is that since now our condition is on the height of $x_k$'s and $y_k$'s, we have less information on their denominators, and in particular we can no longer use $N$ as a common denominator. Write $(x_k,y_k) = \left(\frac{\cdot}{M_k},\frac{\cdot}{N_k}\right)$, with $0<M_k,N_k \le N$. Then the common denominator in column $k$ is at most $M_k^{d-1}N_k^{e-1}\le N^{d+e-2}$, and $\Delta$ has denominator at most $N^{de(d+e-2)}$, which gives us
$$|\Delta|\ge N^{-de(d+e-2)}. $$
The rest of the proof works in exactly the same way. Note that in the proof of the first version of the Main Lemma, the exponent was $\frac{de(d+e-2)}{2}$, that is, we lost the factor $\frac{1}{2}$. Therefore, the new version of the Main Lemma is the following:
\begin{lemma}(Second Main Lemma) Let $I$ be an interval and $\G$ a graph $y = f(x)$ of a function $f\in\CC^{de-1}(I)$. Then $\G(\Q,N)$ is covered by $c\,N^{\delta}$ \quad $(d,e)$-curves, where $c$ depends only on $I$, $f$, $d$ and $e$, and $\delta = \frac{2(d+e-2)}{de-1}$.
\end{lemma}

\section{Generalization to higher dimensions}

\label{shigher}

Let $X\subset\R^m$ be a compact real-analytic transcendental manifold. We denote by $X(\Q,N)$ the set of rational points of $X$ whose coordinates are all of height smaller than $N$. To generalize the previous results, we would like to get something like
$$|X(\Q,N)|\le c(X,\eps) \, N^{\eps}.$$
However this is easily seen to be false, as the example of the surface $z = x^y$ where $1\leq x,y\leq 2$ shows. Indeed, the problem that arises in the higher dimensional case is the fact that a transcendental manifold can very well contain whole pieces of algebraic curves, which can have many rational points. Let us therefore define $\xalg$ to be the union of all segments of algebraic curves inside $X$, and $\xtr = X\setminus \xalg$. The above statement then turns out to be true when replacing $X$ by $\xtr$:

\begin{theorem}\label{higher}(Pila, Wilkie) For every $\eps>0$ there is a constant $c(X,\eps)$ such that
$$\xtr(\Q,N)\le c(X,\eps)N^{\eps}.$$
\end{theorem}
We  sketch now the proof in the case where $X$ is a (two-dimensional) surface inside $\R^3$. In this special case, Theorem \ref{higher} was proved by Pila in \cite{Pi05}.

 \begin{theorem}\label{pila05}(Pila, 2005) Let $X\subset\R^3$ be an analytic surface.  For every $\eps>0$, there is a constant $c(X,\eps)$ such that
$$\xtr(\Q,N)\le c(X,\eps)\,N^{\eps}.$$
 \end{theorem}

In fact, Pila proved this result for subanalytic surfaces, projections of analytic surfaces. See \cite{BM88} for a tutorial on these topics.

We start from an analog of the Main Lemma. Instead of algebraic curves of bounded degree, as in the one-dimensional case, we now use algebraic surfaces of bounded degree: for a positive integer $d$, a \textsl{$d$-surface}\footnote{To make it compatible with the $(d,e)$-curves, we should have probably said \textsl{$(d,d,d)$-surface}, but this is too lengthy.} is defined by $P(x,y,z) = 0$ where
$\deg_x P,\, \deg_y P,\, \deg_z P < d$. The Main Lemma asserts that rational points on a sufficiently smooth compact surface lie on a few $d$-surfaces.

\begin{proposition}(Main Lemma in dimension~$2$)
Let $\eps >0$. There are integers $d$ and $D$ depending on $\eps$, such that if $X$ is a compact $\CC^D$-surface in $\R^3$, then $X(\Q,N)$ is covered by $c\,N^\eps$ \quad $d$-surfaces, where ${c=c(X,\eps)}$.
\end{proposition}

The proof goes along the same lines as in dimension~$1$. Fix~$d$ and~$D$, to be specified later. Points $P_1,\ldots,P_s\in \R^3$ belong to a $d$-surface if a certain determinant $\Delta$ vanishes. We may assume that~$X$ is given by ${z = f(x,y)}$,  where $f$ is a function   defined on some compact domain in $\R^2$ having bounded continuous derivatives of order up to~$D$. If  $P_1,\ldots,P_s\in X$ project to a small square on the $x,y$-plane, and~$D$ is large enough (in terms of~$d$), then $\Delta$ can be bounded from above using some analogues of the Mean Value Theorem. On the other hand, if $P_1,\ldots,P_s$ are rational points of height at most~$N$ and $\Delta\ne 0$, then $\Delta$ is bounded from below by some negative power of~$N$. When the square is small enough, the upper bound contradicts the lower bound, which means that $\Delta$ must be $0$. In other words, if the points project to a sufficiently small square, then they must belong to a $d$-surface. Selecting~$d$ suitably large, we see that ``sufficiently small square'' means ``square with   side length $c(X,\eps)N^{-\eps}$\;''. Now subdividing the domain of definition of the function~$f$ into small squares, we complete the proof. For the details see \cite[Appendix A, Lemma A.3]{Za12}.

\bigskip

Having this, let us see what kind of intersection $X$ can have with a $d$-surface. Since $X$ is transcendental, such an intersection is of dimension~$1$; by compactness, it must be a finite union of irreducible analytic curves. The number of irreducible components in the intersection is controlled by the following result due to Gabrielov~\cite{Ga68}.

\begin{theorem}(Gabrielov) Let $V,W$ be complex semi-analytic manifolds with $W$ compact, and $\pi:V\to W$ an analytic map. There exists a constant $c(\pi)$ depending only on $\pi$ such that for all $\omega \in W$ the number of connected components of $\pi^{-1}(\omega)$ is bounded by $c(\pi)$.
\end{theorem}

Applying this theorem (more precisely, an analogous statement about irreducible components) with~$W$ being the space of all non-zero polynomials defining $d$-surfaces (which is $\P^{d^3-1}(\R)$, hence compact), we bound the number of components in the intersection of~$X$ with a $d$-surface. Now let us consider one such component. It is a curve, algebraic or transcendental. If it is algebraic, it is contained in $\xalg$ and we don't have to worry about it. If it is transcendental, then we reduced the problem to the one-dimensional case. Projecting it to one of the coordinate planes, we obtain a plane transcendental curve, and to the latter we may apply Theorem~\ref{tbprat}.

The argument above can be briefly summarized as follows: rational points of bounded height from~$X^\tr$ belong to ``few'' transcendental curves, and each of the latter has ``few'' rational points of bounded height by Theorem~\ref{tbprat}.

However, there is an important difficulty: Theorem~\ref{tbprat}  involves a constant depending on the curve. Thus we need to have a sort of ``uniform'' version of this theorem, which would follow if we get a similar uniform version of the one-dimensional Main Lemma.

Let us therefore for a moment go back to the one-dimensional case. Let~$\Gamma$ be the graph  ${y=f(x)}$, where~$f$ is defined on some interval~$I$ and has sufficiently many bounded derivatives therein. The Main Lemma asserts that
the set  $\G(\Q,N)$ is contained inside a union of $cN^{\eps}$ \quad $(d,d)$-curves, where $c$ depends on~$I$,~$f$,~$d$ and~$\eps$.

Now assume that~$f$ runs through a continuous family of functions ${\{f_\tau:I_\tau\to \R: \tau\in \Tau\}}$ with a compact base~$\Tau$. We will manage to complete the argument above if we  bound the constant~$c$ uniformly in~$\tau$. This would have been possible if the constant depended only on the length $|I|$ of the interval and the sup-norm $\|f\|$: both are bounded in a continuous family with compact base. Unfortunately, the constant depends also on the  sup-norms of the derivatives $\|f^{(k)}\|$ with ${k = 0,\ldots,d^2-1}$, and the derivatives are not bounded in a  continuous family with compact base (for example, take ${\Tau=[0,1]}$, ${I_\tau= [0,1]}$ and ${f_\tau(x)=\sqrt{x+\tau}}$).

One can take care of the first derivative $f'$ by noticing that on the part of the interval where $|f'|\ge1$ our curve can be written as ${x=g(y)}$ with ${|g'|\le 1}$. However, one cannot deal like this with higher order derivatives.

Instead, Pila uses the fact that derivatives can be large only on short intervals. The following lemma is easy to prove.

\begin{lemma} Assume that $f$ is of class $\mathcal{C}^2$ on  $I=[a,b]$, and $f,f',f''$ do not vanish on $I$.  Put $J = [a+\delta,b-\delta]$ where $0 < \delta < \frac{b-a}{2}$. Then
${\|f'\vert_J\|< \delta^{-1}\|f\|}$.
\end{lemma}

Indeed, assume, for instance, that $f,f',f''>0$ on $I$. Then for ${x\in J}$  we have ${f(x)> 0}$ and ${f'(u)> f'(x)>0}$ for ${u\in [x,b]}$. Hence
$$
f(b) = f(x)+\int_x^b f'(u)\,du > f'(x)\,(b-x)\ge f'(x)\,\delta,
$$
whence ${0<f'(x)< \delta^{-1}f(b)= \delta^{-1}\|f\|}$, proving the lemma in this case. The other cases are treated similarly.

By induction, one shows that if ${f\in \mathcal{C}^{n+1}(I)}$ and $f,f',\ldots,f^{(n+1)}$ do not vanish on $I$, then ${\|f^{(n)}\vert_J\|\le (n/\delta)^n\,\|f\|}$.

Thus high order derivatives may grow uncontrollably only near the points where one of them vanishes.

Now return to our curve $y=f(x)$. Put ${n=d^2-1}$ and assume that $f,f',\ldots, f^{n+1}$ have $k$ roots altogether. Fix a small positive number~$\delta$ and throw away from our interval $I$ all the $\delta$-neighborhoods  of these roots. On the remaining part of~$I$ the sup-norms of derivatives of order up to~$n$ are controlled by the sup-norm of~$f$. As for the part thrown away, it consists of~$k$ tiny intervals, and Pila applies to them an ingenious re-scaling argument, going back to the original article with Bombieri. The details are quite intricate and cannot appear here: the reader may consult Pila's article \cite{Pi05} or the exposition in \cite[Appendix A]{Za12}.

This reasoning implies a new version of the Main Lemma, with constant depending only on $|I|, \|f\|$ and the number of zeros of derivatives of order up to $n$. For the curves occurring in the proof of Theorem~\ref{pila05} the first two of these parameters are estimated immediately, just from the compactness of the space of $d$-surfaces. As for the number of zeros of derivatives, this can be estimated using Gabrielov's theorem.

This is, in general terms, how Theorem~\ref{pila05} was proved.

\section{The Manin--Mumford conjecture}

In this section we show how Pila's techniques apply to the famous problem of Manin--Mumford. We start with some generalities about complex abelian varieties.

Let $A$ be an abelian variety over $\C$ of dimension $g$. Recall that there is a complex analytic group isomorphism
$$A(\C)\cong \C^g/\Lm,$$
where $A(\C)$ is the group of complex points of $A$ and $\Lm$ is a lattice inside $\C^g$. It follows  that
$$A[N]\cong \left(\Z/N\Z\right)^{2g},$$
where $A[N] = \{x\in A \mid Nx = 0\}$  are the $N$-torsion points of $A$. We will denote the group of all torsion points of $A$ by $A^{\text{tors}} = \cup_{N\ge 1} A[N].$
 In its simplest form, the Manin--Mumford conjecture can be stated as follows:
\begin{theorem}
\label{tmmsi}
(Manin--Mumford conjecture, simplest form) Let $X$ be an algebraic curve on an abelian variety~$A$. If $X$ is not an elliptic curve, then $|X\cap A^{\text{tors}}|$ is finite.
\end{theorem}
If $X\subset A$ is an elliptic curve passing through a torsion point of~$A$, then the above set is equal to the set of torsion points of $X$, so is infinite: this explains why we need to exclude elliptic curves.

Theorem~\ref{tmmsi} was proved by Raynaud~\cite{Ra83} in 1983. To state a more general form of the Manin--Mumford conjecture, we need the following definition.

\begin{definition} A subvariety $B\subset A$ is called a \emph{torsion subvariety} if it is a translate of an abelian subvariety by a torsion point.  That is,  there exist  a torsion point $b\in A$ and an abelian subvariety $B_0\subset A$ such that $B = b + B_0$.
\end{definition}
\begin{theorem} (Manin--Mumford conjecture, general form) Let $X \subset A$ be a complex algebraic variety. Then $X$ has only finitely many maximal torsion subvarieties.
\end{theorem}
This was proved by Raynaud~\cite{Ra83a} in 1983 as well. After the work of Raynaud a number of other proofs emerged.

Recently a new proof was suggested by Pila and Zannier (2006). They do prove the general conjecture, but we will discuss their argument only  in the simplest case where $X$ is an algebraic curve.  Let us consider the inverse image of $X$ under the the complex analytic uniformization map for $A$
\begin{eqnarray*}
\pi:\mathbb{C}^g & \longrightarrow& A,\\
\frac{1}{N}\Lm & \longrightarrow &A[N],\\
\tilX = \pi^{-1} (X) & \longrightarrow & X.
\end{eqnarray*}
Let ${\Delta\subset \C^g}$ be a fundamental domain of the lattice $\Lm$. Then $\pi\vert_{\Delta}:\Delta\longrightarrow A$ is surjective, and any point inside $A[N]\cap X$ comes from a point in $\tilX\cap\frac{1}{N}\Lm\cap\Delta$, so we have
$$\bigl|X\cap A[N]\bigr|\le\left|\tilX\cap\frac{1}{N}\Lm\cap\Delta\right|.$$
Thus we have reduced the problem to rational points on the complex analytic curve (and real analytic surface) $\tilX$. Our goal is to apply Theorem \ref{higher}, for which we have to determine~$\tilX^{\mathrm{alg}}$. This will be done in two steps:

\paragraph{Claim 1:} Each irreducible component of~$\tilX$ is transcendental.

\bigskip

First of all, let us observe  the following property of plane algebraic curves.

\begin{lemma}
Let ${C\subset \R^2}$ be an irreducible plane algebraic curve and let $\Lambda$ be a lattice in $\mathbb{R}^2$. Then either $C$ is a straight line and the intersection ${C\cap\Lambda}$ is infinite, or ${C+\Lambda}$ is dense in $\R^2$ (in the real topology).
\end{lemma}

We do not prove this lemma, but it is not very difficult.  Since $\tilX$ is invariant under translations by elements of the lattice, but not dense, the lemma implies that its only possible algebraic components are straight lines having infinite intersection with the lattice. But having such a component would mean that $X$ has an elliptic curve as a component, a contradiction. This proves Claim 1.

\paragraph{Claim 2:} $\tilX$ cannot contain a segment of a real algebraic curve.

\bigskip

Indeed, by analytic continuation this would give us a complex algebraic curve inside $\tilX$, contradicting Claim 1.

\bigskip

Thus $\tilX^{\mathrm{alg}} = \varnothing$. We are therefore ready to apply the Pila--Wilkie theorem to get an upper bound
$$\bigl|X\cap A[N]\bigr|\le\left|\tilX\cap\frac{1}{N}\Lm\cap\Delta\right|\le c(X,\eps)\,N^{\eps}.$$
(Notice that while $\tilX$ can a priori have infinitely many components, only finitely many of them may intersect $\Delta$.)

For the lower bound, we may assume without loss of generality that $A$ and $X$ are defined over a number field $K$. Then if $P\in X\cap A[N]$, all conjugates of $P$ over~$K$ lie in $X\cap A[N]$ as well, which implies $|X\cap A[N]|\ge [K(P):K]$ where $K(P)$ is the extension of $K$ generated by the coordinates of $P$. The lower bound comes from the fact that torsion points generate extensions of large degree, an old result of Masser~\cite{Ma84}.
\begin{theorem} (Masser) Let $A$ be an abelian variety of dimension $g$ over a number field $K$. Then there exist $\delta = \delta(g)>0$ and $c = c(A,K)>0$ such that if $P\in A[N]$ is of exact order $N$ then
$$[K(P):K]\ge cN^{\delta}.$$
\end{theorem}
Taking $\eps = \frac{\delta}{2}$ shows that for sufficiently big $N$, the set $X\cap A[N]$ must be empty.

\section{Definable sets in o-minimal structures}

In fact, in \cite{PW06} Pila  and Wilkie obtained a much more general result than Theorem~\ref{higher}: they estimated the number of rational points of bounded degree on    arbitrary  definable sets in o-minimal structures over $\R$.

\subsection{O-minimal structures}

For our purposes the following definitions will be sufficient; the interested reader might want to consult the main reference on the subject, \cite{dries}.

\begin{definition}We define a \emph{structure} on a set $R$ to be a sequence
$$\mathcal{S}_0, \mathcal{S}_1, \mathcal{S}_2, \mathcal{S}_3, \ldots,$$
where $\mathcal{S}_m\subset \mathcal{P}(R^m)$ is a collection of subsets of $R^m$.
A set $A \subseteq R^m$ such that $A\in \mathcal{S}_m$ is called \emph{definable} in the structure.
Further we ask that

\begin{enumerate}[(S1)]\item Each $\mathcal{S}_m$ is closed under finite union and complement (and therefore  under finite intersection as well): if $A, B \in \mathcal{S}_m$, then $A\cup B \in \mathcal{S}_m$ and $\overline{A} \in \mathcal{S}_m$.
\item The structure is closed under taking Cartesian products: if $A\in \mathcal{S}_m$ and $B\in \mathcal{S}_n$, then $A\times B\in \mathcal{S}_{m+n}$.
\item Equalities are definable in the structure. Namely, the set $\{(x_1,\ldots,x_m) \mid x_i = x_j\}$ belongs to $\mathcal{S}_m$ for each $i,j\in\{1,\ldots,m\}$.
\item The structure is closed under projections: if $\pi:R^{m+1}\to R^m$ denotes the projection on the first $m$ coordinates, then for all $A\in \mathcal{S}_{m+1}$ the set
$$\pi(A) = \{\underline{x}\in\R^m \mid \exists y\in\R,\ (\underline{x},y)\in A\}$$
belongs to $\mathcal{S}_m$.
\end{enumerate}

The last axiom is called \emph{quantifier elimination}.
\end{definition}

For a map $f\colon R^m \to R^n$ we say that it is \emph{definable} if its graph is definable:

\[ \{ (\underline{x},\underline{y}) \in R^m\times R^n \mid \underline{y} = f (\underline{x}) \} \in \mathcal{S}_{m+n}. \]

Let us remark that, as was described in the previous sections, Pila's methods involve taking intersections and projections of the initial sets; this shows that definable sets in a structure are a convenient setting for possible generalizations of Pila's results, allowing us to keep track of the regularity of the sets we are working with while we perform operations on them. For our purpose we are particularly interested in the case where the underlying set is $\R$, and we would like to work with structures over $\R$ that are compatible with its properties as an ordered ring. More precisely, we are going to consider structures where
\begin{itemize}\item The operations $+,\cdot:\R^2\to\R$ are definable.
\item The singletons $\{x\}$ for all $x\in\R$ are definable.
\item The relation $<$ is definable, in the sense that
for all $m$ and for all $i,j\in\{1,\ldots,m\}$ the set $\{(x_1,\ldots,x_m)\mid x_i< x_j\}$ is definable.
\end{itemize}
From now on we will understand by ``structure over $\R$'' a structure over the set $R = \R$ satisfying these additional properties. Note that under these assumptions, all semi-algebraic sets, that is, sets given by equations and inequalities involving polynomials, are definable. It is natural to ask whether semi-algebraic sets themselves form a structure over $\R$. This is true, the non-trivial part being axiom $(S4)$, a theorem by Tarski and Seidenberg.

All semi-algebraic subsets of $\R$ are of one of the following types:
\begin{itemize}\item $\varnothing$;
\item open intervals $(a,b)$ with $a,b\in \R\cup\{\pm\infty\}$;
\item singletons $\{x\}$ for $x\in\R$;
\item finite unions of the previous sets.
\end{itemize}

\begin{definition}A structure over $\R$ is called \emph{o-minimal} if the only definable sets in $\R$ are those listed above.
\end{definition}
Though this definition seems to concern only sets in dimension $1$, it imposes strong regularity conditions on definable sets in higher dimensions; for instance, every definable set is a finite union of smooth manifolds,  see \cite{dries}.  On the other hand, the following examples show that the class of o-minimal structures is very rich and allows numerous strong extensions of Pila's results.

\paragraph{Examples:}
					\begin{enumerate}
					\item The simplest (and smallest) example of an o-minimal structure is the structure $\R_{\text{alg}}$ of semi-algebraic sets.

					\item Real semi-analytic sets do not form a structure, because a projection of an analytic set is not necessarily analytic, see \cite{BM88}. Call a set \emph{subanalytic} if it is a projection of a semi-analytic set, and \emph{globally  subanalytic} if it is a restriction to $\R^m$ of a subanalytic set inside $\mathbb{P}\R^m$. Van den Dries~\cite{Dr86} was, probably, the first one to observe that globally subanalytic sets  form an o-minimal structure, denoted by $\R_\an$.

					\item The structures $\R_{\text{alg,exp}}$ and $\R_{\text{an,exp}}$ obtained by extending $\R_{\text{alg}}$ and $\R_{\text{an}}$ to make the function $\exp$ definable are o-minimal, as shown by Wilkie, Van den Dries and others in the 1990's, see \cite{DMM94,DM96} for precise references.

					\item
					Peterzil and Starchenko~\cite{PS04} showed definability of the Weierstrass $\wp$-function. Precisely, denote by $\Delta$ the standard fundamental domain for the action of $\text{SL}_2(\Z)$ on the upper half plane $\HH$  (see Subsection~\ref{ssmodu} for more details) and by  $\wp(z,\tau)$ the Weierstrass $\wp$-function of $\Lm = \Z\tau + \Z$.  Then the function $f(x,y,\tau)=\wp(x\tau+y,\tau)$ on the set $[0,1]\times[0,1]\times\Delta$ is definable in $\R_{\text{an},\exp}$. It follows that the $j$-invariant function restricted to $\Delta$ is definable in $\R_{\text{an,exp}}$ as well.

						\end{enumerate}

\subsection{The theorem of Pila and Wilkie}
 We can now state the principal theorem of Pila and Wilkie from~\cite{PW06}.
\begin{theorem} \label{pilaw}(Pila, Wilkie, 2006) Let a subset $X\subset \R^m$ be definable in some o-minimal structure. Then for every $\eps>0$ there is a constant $c(X,\eps)$ such that
$$|\xtr(\Q,T)|\le c(X,\eps)\,T^{\eps}.$$
\end{theorem}
Applying this to the  o-minimal structure $\R_\an$,  we see that Theorem \ref{higher} we stated earlier is a special case of this one.

In the sequel we shall often omit for brevity the reference to o-minimality, but will assume it tacitly. By ``definable'' we shall always mean ``definable in some o-minimal structure''.

The main new tool in the proof is an o-minimal version of Gromov's and Yomdin's re-parametrization lemma.

\begin{definition}Let $X\subset\R^m$ be definable. A partial $\CC^r$-parametrization of $X$ is a definable function $f:(0,1)^m\to X$ which is injective and $\CC^r$. It is said to be \emph{bounded} if all its derivatives of order $\le r$ are bounded. A finite set $\{f_1,\ldots,f_k\}$ of (bounded) partial parametrizations of $X$ is called a (\emph{bounded}) \emph{$\CC^r$-parametrization} of $X$ if the union of the images of all the $f_i$ covers all of $X$.
\end{definition}

Existence of a $\CC^r$-parametrization (without boundedness condition) is not difficult to establish,  see~\cite{dries}, and the principal hassle is boundedness. Gromov~\cite{Gr87}, improving on a result  of Yomdin~\cite{Yo87}, showed that a semi-algebraic set admits a bounded $\CC^r$-parametrization. Pila and Wilkie \cite{PW06} extended this further to arbitrary definable sets.

\begin{lemma}(Yomdin--Gromov re-parametrization lemma) A compact definable set has a bounded $\CC^r$-parametrization.
\end{lemma}

Moreover, a similar statement holds for a definable family of definable sets.

Thanks to the bounded parametrization, one no longer has to struggle with non-uniformity, as we did in Section~\ref{shigher}: all the derivatives are bounded, and one can proceed with the inductive argument almost straightforwardly.
This is one of the best examples I know when placing the problem in the correct general context  greatly clarifies it, and the proof of a much more general statement appears to be much simpler than that of particular cases.

Later Pila in \cite{Pi09} extended Theorem \ref{pilaw} to algebraic points of bounded degree on definable sets. Denote by $\xtr(\Q,d,T)$ the set of points on $X$ of height\footnote{I intentionally do not specify what I mean here by ``height''; each reader can use her/his favorite definition, the result will always be the same.} bounded by~$T$ and degree (over~$\Q$) bounded by~$d$.

\begin{theorem}\label{pilaalg}(Pila, 2009) Let $d\ge 1$ be an integer and let $X\subset\R^m$ be a definable set. Then for all $\eps>0$ there is a constant $c(X,\eps,d)>0$ such that
$$|\xtr(\Q,d,T)|\leq c(X,\eps,d)\,T^{\eps}.$$
\end{theorem}

The idea is to reduce this to $\Q$-points in a higher dimensional set, precisely, in the symmetric product of $d$ copies of $X$. If~$P$ is a point of degree~$d$ over~$\Q$ and $P_1, \ldots, P_d$ are its conjugates, then the point $(P_1, \ldots, P_d)\in X^d$ gives rise to a $\Q$-rational point on the symmetric product.

\section{The André--Oort conjecture}
Before discussing another very important application of Pila's results, we need to recall some facts on modular curves and complex multiplication, the main reference being \cite{La73}.

\subsection{Modular curves}
\label{ssmodu}

Let $E$ be an elliptic curve over $\C$. Then there is a complex analytic isomorphism $$E(\C)\cong\C/\Lm,$$ where $\Lm$ is some lattice in $\C$. Two elliptic curves $E_1$ and $E_2$ are isomorphic if and only if the corresponding lattices $\Lm_1,\Lm_2$ are equivalent, that is, if there exists $\alpha\in\C^\times$ such that $\Lm_2 = \alpha\Lm_1$. Thus up to isomorphism we can always write $E(\C) =\C/\Lm$ where  $\Lm$ is the lattice $\langle 1,\tau\rangle$ generated by $1$ and an element~$\tau$ of the upper half-plane $\mathcal{H}$. There is a transitive action of $\Gamma(1):=\slz$ on $\mathcal{H}$ given by
$$\left(\begin{array}{cc} a & b \\
												c & d
												\end{array}\right)\cdot\tau = \frac{a\tau + b}{c\tau + d},$$
and  the modular curve $Y(1) = \Gamma(1)\backslash\HH$ of $\Gamma(1)$-orbits parametrizes isomorphism classes of elliptic curves. One can also describe it more explicitly choosing the following fundamental domain:
$$\Delta = \left\{\tau\in\HH,\ -\frac12\leq\text{Re}(\tau)<\frac12,\ |\tau|>1\right\}\cup\left\{\tau\in\HH,\ |\tau|= 1, -\frac12\le\text{Re}(\tau)\le 0\right\}.$$
Two elliptic curves are isomorphic over $\C$ if and only if they have the same $j$-invariant, and every complex number is the $j$-invariant of some elliptic curve over $\C$. Therefore, the $j$-invariant defines an analytic isomorphism $\Gamma(1)\backslash\HH\cong\C$.

Define the \emph{congruence subgroup}
$$\Gamma_0(N) = \left\{\gamma\in\slz,\ \gamma\equiv \left(\begin{array}{cc} * & * \\
												0 & *
												\end{array}\right)\pmod N\right\}.$$
												The curve $Y_0(N) = \Gamma_0(N)\backslash\HH$ parametrizes pairs $(E,f)$ where $E$ is an elliptic curve and $f$ is a cyclic $N$-isogeny, that is, an isogeny $f:E\to E'$ (where $E'$ is some other elliptic curve) of degree~$N$ with kernel cyclic of order $N$. We can define a function on $Y_0(N)$ by $j_N(\tau) = j(\tau/N)$. There is an explicit description of $Y_0(N)$ as a plane curve over $\Q$ given by an equation $\Phi_N(x,y) = 0$, where $\Phi_N$ is an irreducible polynomial in $\Z[x,y]$ such that $\Phi_N(j,j_N) = 0$.

\subsection{Complex multiplication}
Let $E = E_{\tau}$ be an elliptic curve over $\C$ with corresponding lattice $\Lm = \langle 1,\tau\rangle$. Its endomorphism ring $\mathrm{End}(E)$ contains the ring $\Z$ of rational integers, corresponding to the multiplication-by-$n$ maps for every $n\in\Z$. If $\mathrm{End}(E)$ is strictly bigger than $\Z$, the curve $E$ is said to have \emph{complex multiplication}, or to be a \emph{CM-curve}.  In this case $\tau$ is necessarily imaginary quadratic, and $\mathrm{End}(E)$ is an order $\mathcal{O}_{\tau}$ in the imaginary quadratic field $K = \Q(\tau)$. In particular, there is a positive integer $f$, called the \emph{conductor} of $\tau$, such that $\mathcal{O}_{\tau} = \Z + f\mathcal{O}_K$ where $\mathcal{O}_K$ is the ring of integers of $K$. Writing $\Q(\tau) = \Q(\sqrt{d})$ where $d<0$ is an integer such that $-d$ is square-free, $D = f^2d$ is the discriminant of the order $\mathcal{O}_{\tau}$, called the \emph{discriminant} of $\tau$. The following is a fundamental result of the theory of complex multiplication:
\begin{theorem} \label{cm}Assume $E$ has complex multiplication by the order $\mathcal{O}_{\tau}$ of conductor $f$. Then $j(\tau)$ is an algebraic integer of degree $h_{f,K}$, the class number of the order $\mathcal{O}_{\tau}$ of $K$.
\end{theorem}
Moreover, $h_{f,K}$ is related in the following way to the class number $h_K$ of $K$:
\begin{equation}\label{classnb}h_{f,K} = c\cdot h_K \, f\,\prod_{p|f}\left(1 - \left(\frac{d}{p}\right)\frac{1}{p}\right), \qquad c\in \{1,1/2,1/3\}.
\end{equation}
See \cite[Section~7]{Co89} for more details.

\subsection{The André--Oort conjecture}
\begin{definition} A point ${P\in\C^2}$ is said to be a \emph{CM-point} if ${P = (j(\tau_1),j(\tau_2))}$ where $\tau_1$ and $\tau_2$ are imaginary quadratic.
\end{definition}
Lines of the form $\C\times\{j(\tau)\}$ or $\{j(\tau)\}\times\C$ contain infinitely many CM-points. So does a modular curve $Y_0(N)$, as we have seen it contains all points of the form $\left(j(\tau),j\left({\tau}/{N}\right)\right).$ André's theorem states that all curves inside $\C^2$ containing infinitely many CM-points must be of one of these types:
\begin{theorem}\label{andre}(André, 1998) If $X\subset\C^2$ is an irreducible curve which is not a horizontal or vertical line nor $Y_0(N)$ for some $N$, then $X$ has finitely many CM-points.
\end{theorem}
The general André--Oort conjecture deals with Shimura varieties, an important special case being products of modular curves $Y_0(N_1)\times\ldots\times Y_0(N_k)$: in this setting it was proved by Edixhoven in 1998 assuming the GRH for imaginary quadratic fields. We are going to concentrate on the Shimura variety $\C^k$ obtained when $N_1 = \ldots = N_k = 1$.
\begin{definition} \begin{enumerate}\item A \emph{special point} in $\C^k$ is a point of the form $(j(\tau_1),\ldots, j(\tau_k))$ where $\tau_i$'s are imaginary quadratic.
\item A \emph{special subvariety} of $\C^k$ is a subvariety given by an equation of one of the following forms:
\begin{itemize}\item $x_i = j(\tau)$ for some $i$, where $\tau$ is imaginary quadratic;
\item $\Phi_N(x_i,x_j) = 0$ for some $i,j$.
\end{itemize}
\end{enumerate}
\end{definition}
Special points and special subvarieties are an analogue of torsion points and torsion subvarieties in the statement of the Manin--Mumford conjecture. Now we can state the André--Oort conjecture for $\C^k$, proved by Pila in \cite{Pi11} using his previous results on rational points in definable sets.
\begin{theorem}(André--Oort conjecture) Let $X$ be a subvariety of $\C^k$. Then it has finitely many maximal special subvarieties.
\end{theorem}
We are going to sketch Pila's proof for $k=2$, when it boils down to proving Theorem \ref{andre}. The ideas are very similar to the ones used for the proof of Manin--Mumford. Consider a map
$$\begin{array}{rcl}\HH^2&\stackrel{\pi}{\longrightarrow}&\C^2.\\
                    (\tau_1,\tau_2)&\mapsto& (j(\tau_1),j(\tau_2))
\end{array}$$
For a non-special irreducible $X\subset\C^2$, put $\tilX = \pi^{-1}(X)\cap(\Delta\times\Delta)$. Any CM-point on $X$ comes from some point of $\tilX$, so it suffices to bound the number of points from $\tilX$ mapping to CM-points in $X$. Note that for any CM-point $P$ on $X$, a point $\widetilde{P}\in\tilX$ mapping to $P$ is of the form $(\tau_1,\tau_2)$ where $\tau_1,\tau_2$ are imaginary quadratic, so $[\Q(\widetilde{P}):\Q]\leq 4.$ We therefore need a bound on the size of $\tilX(\Q,4,T)$ in terms of $T$, and so we must check that we can apply Theorem \ref{pilaalg}. The following is a consequence of the already mentioned theorem by Peterzil and Starchenko about the definability of the Weierstrass $\wp$-function in $\R_{\text{an},\exp}$:
\paragraph{Claim 1:}$\tilX$ is definable.

\bigskip

Moreover, in the same manner as in the proof of Manin--Mumford (an algebraic curve that is not special cannot be $\text{SL}_2(\Z)$-invariant) we obtain

\paragraph{Claim 2:}$\tilX^{\text{tr}} = \tilX$.

\bigskip

Assuming these claims, by Theorem \ref{pilaalg} we get
\begin{equation}\label{upper}|\tilX(\Q,4,T)|\le c(X,\eps)\,T^{\eps}.\end{equation}
Now we are going to bound the number of special points on $X$ from below. For this we can assume that $X$ is defined over some number field $L$. If $P\in X$ is special, all its conjugates over $L$ are special as well.
Bearing in mind that $P$ is a CM-point, $P = (j(\tau_1),j(\tau_2))$ where we can choose $\tau_1$ and $\tau_2$ inside the fundamental domain $\Delta$, so that $\widetilde{P} = (\tau_1,\tau_2)\in\tilX$. Therefore
\begin{equation}\label{degrees}[L(P):L]\geq \max \{[\Q(j(\tau_1)):\Q],[\Q(j(\tau_2)):\Q]\}.\end{equation}
According to Theorem \ref{cm}, for $i = 1,2$,
$$[\Q(j(\tau_i)):\Q] = h_{f_i,\Q(\tau_i)},$$
 where $f_i$ is the conductor of $\tau_i$. The following theorem tells us that the class number of an imaginary quadratic field cannot be too small:
\begin{theorem}(Siegel) Let $K$ be an imaginary quadratic field and fix $\eps'>0$. Then there is a constant ${c = c(\eps')>0}$ such that
$$h_K\geq c\,|D_K|^{\frac{1}{2}-\eps'},$$
where $D_K$ is the discriminant of the number field $K$.
\end{theorem}
Thanks to formula $(\ref{classnb})$, we then get that
\begin{equation}\label{degdisc}[\Q(j(\tau_i)):\Q]\geq c_i|D_i|^{\frac{1}{2} - \eps'}\end{equation}
 for some constant ${c_i>0}$, where $D_i = \,f_i^2\,D_{\Q(\tau_i)}$ is the discriminant of $\tau_i$.
 Moreover, for $\tau$ imaginary quadratic of discriminant~$D$ lying in the fundamental domain, a quick computation shows that the height $H(\tau)$ of $\tau$ is bounded in terms of its discriminant:
\begin{equation}\label{heightdisc}H(\tau)\leq\sqrt{|D|}.\end{equation}
Putting (\ref{degrees}), (\ref{degdisc}) and (\ref{heightdisc}) together, we get finally for $\eps'$ sufficiently small
$$ [L(P):L]\geq c\max \{H(\tau_1),H(\tau_2)\}^{1-2\eps'}  = c\,H(\widetilde{P})^{1-2\eps'}.$$
If there are infinitely many special points on $X$, the heights $H(\widetilde{P})$ can be arbitrarily large, which with the upper bound $(\ref{upper})$ yields the result.

\bigskip

For further reading on the topic we strongly recommend Scanlon's expository texts \cite{Sc11,Sc12} and Zannier's lecture notes \cite{Za12}.

As mentioned in the preface, these notes do not claim for any kind of exhaustiveness. For instance, we do not speak at all on Heath-Brown's ``determinant method'', which can be viewed as a ``~$p$-adic'' version of the Bombieri-Pila method; see \cite{HB02,BHS06} and the subsequent work of Browning, Salberger and others.

While Heath-Brown's method is beautiful and powerful, it mainly applies in the algebraic case, and has yet to show its efficiency in the transcendental case, which is the main topic of these notes. Therefore it is left out.

{\footnotesize

redpajama_set_name: RedPajamaArXiv


Sweet Dream is the Exhibition Name.
Masses of double, peachy-apricot, cupped form blooms in clusters in May-October.
Foliage plentiful with growth wide and bushy.
Heirloom Old Garden Roses states "it is the most popular selling rose in England".
"Very striking well formed blooms of soft peachy apricot borne both singly and in small clusters. A small cushion like bush with dense glossy healthy foliage. Always seems to be in flower and is ideal for the garden or container. A grand rose for low edging or borders. The blooms hold their colour and form well in bad weather. Has excellent disease resistance. Strong Sweet Scent.
AWARDS - Rose Of The Year 1988. Belfast Certificate Of Merit 1990. Royal Horticultural Society Award Of Garden Merit 1993.
Also known as "Apricot Sweet Dream" "Sweet Dreams"" from The Country Garden Plant Centre.
"Apricot, peach shading. Mild to strong fragrance. Medium, double (17-25 petals), cluster-flowered, in large clusters, cupped, globular bloom form. Prolific, blooms in flushes throughout the season.
Width of 18" to 2' (45 to 60 cm).
Juvenile Foliage through Light Green to Dark Green Mature Foliage.


text:
TEAM SOLO GET BIG TIME TV EXPOSURE ON PAN ARABIC REGIONAL & GLOBAL CHANNEL
Deputy Commissioner General and Pavilion Director Dennis Marita at the Show
Team Leader Product Marketing Garedd Porowai appeared on screen
Jenny Barile of the Ministry of Foreign Affairs and External Trade appeared on the TV show
PR Volunteer Kelepi Abariga during the TV Show
The Solomon Islands Team at the Dubai Expo 2020 were studio guests on Dubai One TV on Saturday 8 January 2021 hosted by Business Presenter Ms. Dina Barqawi.
In the show, Deputy Commissioner General and Pavilion Director Mr Dennis Marita first enlightened the viewers about our Pavilion, the history of the Solomon Islands, the intricate traditional and contemporary designs, food and body products and the carefully placed artifacts which resembled a journey in time, one which many past visitors have commented on, as a Pavilion that was easy to understand in terms of the invaluable information that was available as they entered and browsed through.
Public Relations Volunteer Kelepi Abariga was given the opportunity to elaborate further on the name Solomon, who according to Middle Eastern history was known as Sulaimān ibn Dāwūd (Arabic: سُلَيْمَان بْن دَاوُوْد,) Solomon son of David), and was revered both in the Holy Bible and the Quran.
"Our ability to establish such narratives also resonated well with the Theme for Expo 2020 Dubai of 'Connecting Minds & Creating the Future' as we were able to help the future generation understand how history had evolved and helped shape our existence and understanding of our common linkages," said Mr Abariga, a Fiji-born 4th generation Solomon Islands descendant based in Dubai.
Ms Jenny Barile from the Ministry of Foreign Affairs & External Trade gave an overview of the various aspects of investment portfolios available in the Solomon Islands adding that our presence at the Expo 2020 Dubai would help provide the platform that would enable further collaborations between the UAE and the International Community especially in the areas of Trade and Economic Development.
Team Leader Product Marketing Garedd Porowai explained that the Solomon Islands was a viable destination for UAE tourists to visit as it was an emerging destination where everything is raw, rustic and real with no kids clubs or 7 star resorts, but 7 star experiences and adventure. "As for Niche Tourism Markets, Scuba Diving, Surfing, Fishing, WWII History and Culture would attract every visitor's interest," Porowai said.
Mr Porowai further explained that Sustainable Tourism was the way forward for the Solomon Islands. With approximately 1000 more Islands to discover, it was the Cultural Diversity comprising of the 3 ethnic groups of Melanesian, Polynesian and Micronesians that reflected the Islands unique heritage and identity. He concluded that whilst Tourism's contribution to GDP was still behind, the industry will certainly become a main contributor to the country's economy in the future.
The 30 minute Prime Time Live Broadcast to more than 50 million viewers in the Arabic Region and around the world also included Live-Crossover from the Dubai One TV crew capturing the various artifacts on display at our Solomon Islands Pavilion which briefly featured Pavilion staff joyfully Joji Luita beating the Bamboo drums.
It was certainly a Window of Opportunity for our Team Solo to tell the World about our beloved Happy Isles and to also express how grateful and privileged we are to be Connecting Minds at Expo 2020 Dubai and helping to create the Future towards a better Growth and Prosperity for all.
-GCU Press

redpajama_set_name: RedPajamaCommonCrawl


text:
Open access peer-reviewed chapter - ONLINE FIRST
Management of Abiotic Stress in Forage Crops
By Amanpreet Singh and Harmandeep Singh Chahal
Submitted: June 6th 2020Reviewed: September 2nd 2020Published: November 30th 2020
Home > Books > Abiotic Stress in Plants [Working Title]
Downloaded: 55
Forage plays a key role in rearing ruminants and protecting the environment. Apart from serving as the primary source of food for domestic and wild animals, forages also contribute to human civilization in different ways like protecting soil through crop over and fertility by addition of organic matter. It also provides habitat for wild animals. A survival strategy plays a more important role than a growth strategy to improve the sustainability of forage production, especially in extreme environmental conditions . Climate change is likely to affect the forage production and nutritional food security for domestic animals. Long-term rainfall data in India indicate that rainfed areas experience 3 to 4 years of drought in every 10 years. Of these, one or two of it occur in severe form. Forage crop production is largely affected by abiotic factors related stress such as drought, salinity, etc. There is need to adopt various conventional and genetic approaches to improve stress tolerance of forage crops.
breeding and micronutrients
Amanpreet Singh*
Department of Agriculture, Khalsa College Amritsar, 143001, Punjab, India
Harmandeep Singh Chahal
*Address all correspondence to: nandgharia@gmail.com
Abiotic Stress in PlantsEdited by Shah Fahad
Abiotic Stress in Plants [Working Title]
Dr. Shah Fahad, Dr. Shah Saud, Prof. Yajun Chen, Dr. Chao Wu and Dr. Depeng Wang
In the agricultural context, stress has been defined as the conditions in which plants are prevented from fully expressing their genetic potential for growth, development, reproduction, and, ultimately, crop productivity [1]. Abiotic stress negatively affects the livelihoods of farmers and their families, the sustainability of livestock, as well as national economies and food security. Forages are generally described as plants and its parts consumed by domestic livestock. Forage plays a key role in rearing ruminants and protecting the environment. Apart from serving as the primary source of food for domestic and wild animals, forages also contribute to human civilization in different ways like protecting soil through crop over and fertility by addition of organic matter. It also provides habitat for wild animals. In the biological soil–plant–animal system, forage is highly demanded by livestock. Escalation in the human population in the coming decades will put the higher burden on land for food crops and fiber production. As a result, we may face forced forage cultivation in those areas having poorer soils regarding fertility and management [2]. The water use for irrigation is incredibly high and this trend could increase considerably in the future leading to shortage of water availability [3]. For perennial forage and natural vegetation, the ability to survive during adverse environmental periods is a life saving feature. A survival strategy plays a more important role than a growth strategy to improve the sustainability of forage production, especially in extreme environmental conditions [4]. Forage crop production is largely affected by abiotic factors related stress such as drought, salinity, etc. There is need to adopt various conventional and genetic approaches to improve stress tolerance of forage crops.
2. Forage status
Currently, India faces a critical imbalance in its natural resource base: around 18 percent of humans and 15 percent of the world's animal population are only served by 2.4 percent of the geographical area, 1.5 percent of forests and pastures, and 4.2 percent of water resources [5]. The three main sources of forage supply in India are crop residues, cultivated forage, and forage from common property resources such as forests, permanent pastures, and pastures. Due to the multiplicity of forage crops produced in different seasons and regions, the surplus and deficit in different regions, the non-commercial nature of crops and forage production with minimal inputs from degraded and marginal land, there has been a large gap in the availability and need for forage. Currently, the country faces a net deficit of 35.6 percent of green forage, 10.95 percent of residues from dry crops, and 44 percent of concentrated ingredients for animal feed [6]. Supply and demand for the forage scenario are presented in Figure 1 . Furthermore, in the case of forage, regional and seasonal deficiencies are more important than national deficiencies, since it is not economical to transport forage over long distances. Furthermore, the available forages are of low quality and deficient in available energy, protein, and minerals. Farmers maintain large herds of animals to compensate for low productivity, adding pressure on forage and other natural resources [7]. Almost two-thirds of the total cost of animal production is due to food and fodder. Consequently, any attempt to improve the availability of food and fodder and save the cost of food would result in better remuneration for farmers. The area under cultivated forage is only 8.4 million hectares and has been static for the past two decades. The potential for further increases seems very small due to demographic pressure for food crops. Recent crop diversification, where cash crops replace traditional cereal crops, especially coarse grains, is likely to have an impact on the availability of crop residues for animal production [8]. Likewise, the productivity of certain important cultivated forages is highly variable. Among Kharif forages, sorghum, corn, cowpea, Napier-bajra hybrid, and guinea they have a wide range. However, during rabi, the choice is limited to oats, alfalfa, and berseem. Emphasis should be placed on new area-specific crops that can break down yield barriers and meet the challenges of the food deficit.
Deficient trend of fodder crop concerning future demand. *IGFRI vision, 2050.
3. Different types of abiotic stress faced by crops
3.1 Temperature restriction
The tropical climate is cursed by higher temperatures and radiation that limit the growth and development of plants. High temperatures cause burns, sunburn, and discoloration of the leaves, reducing plant growth [9]. Limiting growth, metabolism, and performance potential due to exposure to a temperature below or above the thermal threshold for optimal biochemical, physiological, and morphological development is called thermal stress [10]. Plants are classified into psychophilic, mesophilic, and thermophilic according to their tolerance to low, medium, and high temperatures [1]. The adversity of heat stress varies with the duration, stage, and intensity of stress [11]. Increased heat stress adversely affects the spikelets number, the number of florets per plant in rice crop, and the seeds in forage crop like sorghum [12]. It also reduces quality due to reduced production of oil, starch, and protein [13]. Stress at low temperatures causes wilting, bleaching, darkening, necrosis, and death of plants [1]. Approximately 15percent of arable land is said to be affected by frost stress [14].
3.2 Moisture stress
About 28 percent of the world's land is too dry for agricultural support [15]. The estimated annual yield loss due to extraction in the tropics is almost 17 percent [16]. Increasing the draft with the changing climate scenario leads to a decrease in plant physiology, growth, and reproduction [17]. The moisture deficit causes greater transpiration and reduces the availability of water from the roots of the plants [18], which tends to balance the water on the negative side that affects growth, the relationship between nutrients and water, photosynthesis and assimilation of sharing and, ultimately, performance [19]. The stress response plan in plants varies according to the species according to its stages and other growth factors [20]. High-temperature stress affects enzyme activity, cell division in plants [21] and also changes the growth period and distribution of crops [22].
3.3 Heavy metal stress
Heavy metals are those metals that have a specific weight greater than 5 g cm−3 or an atomic mass greater than 20 and are generally toxic even at low concentrations [23]; some of heavy elements or metals are cadmium (Cd), lead (Pb), arsenic (As), silver (Ag), etc. Heavy metal contamination in the soil is mainly due to human activities such as mining, smelting, intensive agricultural practices, fuel production, electroplating, etc. [24] and may also be due to natural processes such as soil erosion, excessive weathering of rocks and minerals, and volcanic eruption. Among heavy metals, some have known physiological functions in the plant system called non-essential heavy metals, namely arsenic (As), lead (Pb), cadmium (Cd), mercury (Hg), and selenium. (Sc) and some are involved in different physiological functions of plants as a cofactor of enzymatic reactions [25] or role in redox reactions [26] called essential heavy metals, namely cobalt (Co), copper (Cu), manganese (Mn), zinc (Zn), iron (Fe), molybdenum (Mo) and nickel (Ni).
3.4 Salt stress
Crops are said to be subject to salt stress when they cannot express their full genetic potential in terms of growth, development, and reproduction, since the salinity of the soil exceeds the critical level [27] and dissolved salts in the soil and irrigation water vary from place to place [28]. The detrimental effect of soils affected by salt may be due to a high concentration of salt in the soil solution, i.e. osmotic effects or a high concentration of specific ions such as sodium or chloride that can damage sensitive crops, i.e. a specific ionic effect. The harmful effect of saline soil is due to the concentration of soluble salt, while the harmful effects of sodium soil are due to deterioration of the physical state of the soil [29]. The harmful effect of salt stress may be due to a specific ionic effect, that is, Na+ and Cl− [30] or to interact with other dynamics of mineral nutrients [31].
3.5 Nutritional stress
Several mineral elements contribute to the growth and development of a plant, 17 of which are called essential nutrients according to the essentiality criteria defined by Arnon and Stout. Since mineral nutrition is discipline independent of plant physiology [32, 33] divide essential minerals into four groups according to their biological structures and metabolic functions. There is some nutritional stress (deficiency or excess) reported by various scientists in different plants. Nitrate plays pivotal role cytokinin biosynthesis and transport, and a higher level of nitrate (NO3−) inhibits root growth and the root: shoot ratio [34]. Phosphorus deficiency limits the lengthening of the primary roots and improves the formation of lateral roots, decreases the proportion of the dry weight of the roots of the shoots [35], reduces the leaves [36] and affects the reproductive organs formation [37], plants with potassium deficiency (K+) are sensitive to lodging and airflow [38]. A sulfur deficiency decreased the net photosynthesis and the hydraulic conductivity of the roots [39], the reduction in the dry weight ratio of the roots of the shoots [40], an alteration in the metabolism of carbohydrates followed by an induced accumulation of starch [41].
4. Impact of abiotic stress on physiology of forage
4.1 Photosynthesis
Moderate stress in water deficit plants reduces photosynthesis which is accompanied by closing of stoma [42]. Measurement of the photosynthetic response and the activity of the ribosose bisphosphate carboxylase vifro (RUBISCO) in alfalfa (Medicago sativa L.) exposed to an increasing water deficit and found evidence of adverse osmotic effects [43].
4.2 Forage quality
The digestibility of legumes and their fiber have been largely affected by water availability [44, 45]. Drought affects the forage composition and quality by altering plant maturity and ratio of leaf mass to stem mass [46].
4.3 Establishment of seedlings in forages
Water availability highly affects the forage seedlings growth and maturity [47]. Seminal roots support seedlings for a short time. Seminal root system absorbs by the hydraulic conductivity of the suboptoptic internode. Redmann and Qi (1992) found that the diameter of the xylem vessels in warm-season grass seedlings that emerged from different planting depths and length of suboptoptic internode plays an important role in transport of water from the root to the shoot and reducing hydraulic conductivity.
5. Impact of climatic anomalies on forages in terms of stress
Climate change has become a serious threat to life on earth. There is also a global trend of increased storms on most lands. Glaciers are continuously melting, while daily high temperature with heat waves became more common [48]. Coping with climate variability is becoming a major challenge for human civilization. Higher seasonal variability regarding the distribution of precipitation, extreme events of temperature, and precipitation cause damage to crops and raise serious concerns about agricultural production. Among adverse weather events, drought is the major factor to directly affect the population. A warmer climate with increasing climatic variability will increase the risk of climatic extremes. Meteorological data analyzed over 5 decades from CRIDA's Gunegal research farm, a typical rain region, showed low precipitation. Climate change is likely to affect the forage production and nutritional food security for domestic animals. Long-term rainfall data in India indicate that rainfed areas experience 3 to 4 years of drought in every 10 years. Of these, one or two of it occur in severe form [49].
6. Abiotic stress management in major fodder crops
6.1 Sorghum
6.1.1 Water stress and its management
Sorghum with its persistent green character, well developed root system, higher water-use efficiency and epicuticular wax represents a good system for studying physiological features related to drought tolerance. Depending on stress development at any growth stage, sorghum shows a stress response before flowering and after flowering, respectively. All these different responses are affected by various genetic processes [50]. Pre-flowering stress affects plant biomass, panicle size, kernel quantity, and grain yield [51], whereas posttesthetic dryness leads to premature senescence of leaves and stems, lodging and the reduction of seed size [52]. Post-synthesis drought also increases plant sensitivity to biotic stress, such as charcoal rot (Macrophomina phaseolina) and fusarium stem rot (Fusarium moniliforme) [52]. For drought tolerance before flowering, six distinct genomic regions were Recombinant inbred sorghum lines (RIL) derived from the cross between the genotypes Tx7078 (tolerant before flowering, sensitive to post-flowering) and B35 (sensitive to pre-flowering, tolerant after flowering) [53]. The response to dryness after flowering is associated with the persistent green character of sorghum. Staying green is essentially the retention of the surface of mature green leaves (GLAM). Maintaining the remaining green character during the grain filling phase under stress conditions of soil water deficit constitutes an important element of drought tolerance [54].
6.1.2 Epicuticular wax
Epicuticular wax (EW) forms a glaucous upper coating that is visible on many cultivated plants called waxy bloom. Species, organ, stage of development, and environmental conditions are all those things that affect buildup of wax. Composition and structure of epicuticular wax is very diverse which is considered a potential useful trait and has been related to resistance against different adverse environmental conditions [55]. Sorghum differs from other field crops in its ability to produce sufficient amount of EW that is placed on the leaf blade as well as leaf sheath generally during pre-flowering and stages of maturity. Sorghum leaf sheath bloom is composed of large amount of free fatty acids with a 16 to 33 carbon chain length [56].
6.1.3 Osmotic adjustment
Two traits named osmotic adjustment and antioxidant capacity have been related with drought tolerance mechanisms. Osmotic adjustment has been associated with sustained performance under water limiting conditions in many crops and is an inherited characteristic. Two major independent genes namely OA1 and OA2 in sorghum have been reported to control Osmotic adjustment inheritance.
6.1.4 Cold tolerance
Sorghum from the tropical and subtropical regions of Africa [57] is well adapted to warm growing conditions. Cool temperatures at the beginning of the growing season are therefore an important limitation for the growth of temperate sorghum areas [58]. Cross developed from local Chinese races, ShanQui Red (SQR, cold-tolerant), and SRN39 (cold-sensitive) was used for QTL analysis of early-season cold yields on sorghum [59].
6.2 Bajra
Bajra [Pennisetum glaucum] is a C4 plant with very high photosynthetic efficiency. Bajra also have high dry matter production capacity. It is generally cultivated under the most adverse agroclimatic conditions, where other crops such as sorghum and corn do not stand well.
6.2.1 Selecting genotypes is a good approach to managing abiotic stress
Pearl millet germplasm screening helped in the development of highly advanced breeding techniques, an improvement in the population, including OPVs, genetic pools and compounds, possible parental hybrids, and accessions of the high-throughput genetic material of cereals and forages, presumably with a high degree of salt tolerance ( Table 1 ).
Genotypes
Drought CZP 9802; 863B and PRLT 2/89-33ICMP 83,720 [60]
Heat H77/833–2, H77/29–2 and CVJ 2–5–3-1-3, 77/371XBSECT CP1 [61]
Salinity 33, 10,876 and 10,878 (Sudan), 18,406 and 18,570 (Namibia), and ICMV93753 and ICMV 94474 (India); 863-B, CZI 98–11, CZI 9621, HTP 94/54 [62]
Available genotypes for abiotic stress tolerance in pearl millet.
6.2.2 Low soil fertility
Soils in the areas where pearl millet is grown are often poor infertility because they contain a small amount of organic matter (0.05–0.40percent) due to low ground cover, coarse soil texture, and prevailing high temperatures [63]. Soils also contain low to moderate levels of available phosphorus (10–25 kg ha−1). This problem was mainly solved through nutrient management. The possibilities of genetic improvement for the efficient use of nutrients are increasingly explored in some cultures [64]. Only recently has strategic research been launched at ICRISAT in the West and Central Africa region to identify QTL to increase the efficiency of phosphorus and examine the stability of its expression across genetic environments.
6.3 Forage corn
Corn forage (Zea mays L.) has become an important component of ruminant rations in recent years. It is the only crop among non-leguminous forages that combines better nutritional quality. With a large amount of biomass [65]. Although the crop has great adaptability [66], it is the least tolerant of abiotic stress among cereals. Drought, salinity, and high temperatures are among the major abiotic stresses that negatively impact corn production in most regions of global corn production [67]. Soils with saline stress are present on all continents and in almost all climatic conditions. However, its distribution is relatively more extensive in arid and semi-arid regions than in humid regions [68]. Mohammed and Mohammed 2019 stated that the appropriate genotype based on stress selection is the inexpensive and manageable stress method based on salt, water, and heat or combined form and also concluded that the reduction in stress performance would be reduced to 20–40 parents.
6.4 Cowpea
The cowpea (Vigna unguiculata) is one of the most important legumes cultivated by subsistence farmers for human and animal consumption, mainly in the semi-arid regions of Africa and Brazil. In Africa, it is used for the livelihood of millions of people in the semi-arid regions of the West and Center [69] and is considered the most important grain legume crop in the sub-Saharan region.
6.4.1 Reproductive improvements
Cowpea is relatively drought tolerant. Despite this feature, however, drought can cause a considerable loss of performance. Efforts have been made to select the cowpea genetic material to identify lines with better drought tolerance than currently available varieties. According to Watanabe et al. [70], certain lines of genetic material, in particular, TVu 11,979 and TVu 14,914, were consistently very drought tolerant under real field conditions. Drought can occur at the beginning of the season, mid-season, or the crop development stage. Studies have shown that cowpea plants can show drought tolerance in the vegetative stage [71] and the reproductive stage [32]. Some cowpea lines exhibit a green persistence feature, also called delayed leaf senescence (DLS), which can help plants tolerate terminal and mid-season drought [32].
6.4.2 Gene selection
In cowpea plants, overexpression of the CPRD 8, CPRD12, CPRD14, CPRD22 and CPRD46 genes that confer tolerance to water stress [72], as well as the production of VusAPX genes connected to VucAPX, VupAPX and VutAPX of antioxidant enzymes [73], it is reported, in addition to the expression of the high level of the PvP5CS gene associated with the production of proline, an amino acid that fulfills the function of osmotic adjustment between species during drought.
6.5 Abiotic stress tolerance mechanism
Climate and soil determine many plant adaptations and the ecogeographic distribution of species and ecotypes show differences in physiology and development patterns that provide good evidence of adaptation mechanisms. Plants respond to environmental change as individuals through phenotypic plasticity and in populations through the selection and associated evolutionary processes. Determining the genetics underlying adaptation processes is not always easy because environmental factors can be complex or poorly defined. However, extreme environmental pressures, such as heavy metal contamination from the soil or harsh winter conditions [74] can produce detectable genetic changes. Multiple genes may be responsible for a response to a certain factor, or the same gene may be involved in different adaptive responses specific genetic interactions can be in a state of change or become fixed, limiting the possibilities for future evolution. Phenotypic plasticity acts as a buffer to prevent excessive gene flow in response to short-term changes.
7. Improving forages for abiotic stress response based on breeding techniques
7.1 Greater tolerance to stress through genetic transformation
Genetic improvement of forages through the selection of conventional plants is slow because most forage species are self-incompatible, limiting inbreeding to concentrate the desired genes to be used in the rapid development of new cultivars. Genetic transformation allows the direct introduction of desirable genes, thus offering new opportunities for forage molecular selection. Like many other crops, drought tolerance is an important goal in improving alfalfa. Since cuticle waxes play a central role in limiting the breathable loss of water from the plant surface, the genetic engineering of plant waxes is expected to eventually increase tolerance to environmental stress in crops such as agronomic importance [75].
7.2 Improvement of stress tolerance through intergeneric hybridization
Extensive hybridization with relative species followed by introgression of chromosomes and/or chromosome fragments has been considered an effective means of transferring salt and other stress tolerance genes to target species to extend the gene pool. Intergeneric hybrids between species of Lolium (Ryegrass) and Festuca (Fescue) have attracted much attention from forage breeders. Rye grasses are considered ideal grasses due to their fast establishment, their ability to resist intense grazing, their good palatability, and their high nutritional value [76].
Alfalfa (Medicago sativa L.) is widely cultivated in temperate and tropical regions for green forage, hay, silage, and grass. As a perennial forage plant, alfalfa is a fairly hardy species and has a relatively high level of drought tolerance compared too many other legume forage plants [77]. Alfalfa's increased drought tolerance is due in part to deeper roots and the ability to extract more available water from the root zone [78]. Detection of salt-sensitive proteins in two contrasting alfalfa cultivars using a comparative proteome approach revealed two new proteins, NAD synthetase, and biotin carboxylase-3, as being salt sensitive. These results provide new information on alfalfa salt stress tolerance [79]. The effects of rhizobia strains on the amino acid composition of alfalfa under salt stress indicate that proline, glutamine, arginine, GABA, and histidine accumulate significantly in salt-stressed nodules, suggesting increased production of amino acids associated with osmoregulation, nitrogen storage, or energy metabolism to counteract salt stress [80] is a widely allogeneous forage legume species distributed worldwide due to its wide range of climate adaptation [81]. But it is less drought tolerant than other temperate perennial forage legumes due to its shallow root system and its inability to effectively control transpiration [82]. Biochemical studies have indicated that when white clover was stressed by a water deficit, De novo synthesis of amino acids, including proline, has increased in both leaves and roots [83]. This phenomenon may serve as an adaptive response during the first days of drought since the transient increase in amino acid concentration has been followed by a decrease in protein synthesis that slows plant growth.
Cowpea (Vigna unguiculata L.) growing in a variety of environments from tropical to arid/semi-arid regions, increased tolerance to drought and heat would be desirable. The cowpeas (Vigna marina) that grow on sandy beaches in the tropical and subtropical regions closest to the sea have the potential to be a source of genes for breeding salt-tolerant cultivars. Chankaew et al., [84] first reported QTL mapping for salt tolerance in the Vigna marina, and multiple internal mapping consistently identified an important QTL that can explain 50 percent of the phenotypic variation. The flanking marker can facilitate the transfer of salt tolerance of this subspecies in related Vigna cultures.
8. Micronutrient stress management
Sorghum (Sorghum bicolor) is one of the important forage crops for high agricultural production and good nutritional value for animals. Nutrient requirements for growing sorghum are high; they are grown for forage, in part from organic sources, and are supplemented primarily with inorganic fertilizers. The growth, development, and biological yield of crops affected by balanced fertilization have shown positive effects. Micronutrients increase crop productivity and also maintain soil health. A very small amount is required. Soil application of micronutrients is preferable for what is desired. Choudhary et al. [85] observed that the combined application of micronutrients, that is, a considerably higher yield of cereals, stems, and organic, is obtained through a soil + leaf application. The results showed a significant increase in grain yield (14.15 and 12.13 percent), biological yields (11.37 and 9.31 percent), and in stem yield (10.75 and 8.60 percent) and through the combined spraying of soil and foliar on the soil and foliar application, respectively.
8.2 Pearl millet
Pearl millet (Pennisetum glaucum L.) is one of the main millet crops in arid and semi-arid areas. Weather. Due to the drought-tolerant nature, it grows well in poor sandy soils. Sustainable production of pearl millet can be achieved through the balanced use of nutrients in crops with the fusion of organic and inorganic sources. Intensive farming is followed in the current system, most farmers use high-yielding whole crop varieties, ultimately a significant removal of nutrients from the soil in recent years, and the consumption of fertilizers has remained well less than the elimination one. So that the qualitative and quantitative improvement of the crop yield goes through mineral fertilization and that its quality can be improved through adequate practices of nutrient management and soil cultivation [86].
8.3 Maize
The third most important cereal crop is maize (Zea mays L.) worldwide and India. It is cultivated in temperate and tropical regions of the world. It is the most important cereal for animal feed. In India, 45 percent of maize production is used in various forms of staple foods [87]. Corn, rice, and wheat are estimated to provide at least 30percent of food calories to more than 4.5 billion people in 94 developing countries. The demands for animal feed and biofuels can be met by increasing maize production [88]. The application of micronutrients can be carried out in several ways, such as seed treatment, soil and foliar application [89], which depends on the characteristics of the soil and the climate of the region. Corn productivity can be improved by applying Zn and B to the soil.
Cowpea (Vigna unguiculata) is a legume and is used as a forage crop that is grown during the Kharif season, requiring only an initial dose of nitrogen (15–25 kg N ha−1). Most nitrogen requirements are met by symbiotic nitrogen fixation. The strong application of NPK fertilizers has led to micronutrient deficiencies in many parts of the country. To achieve high yields and maintain them over the years, it becomes highly relevant to predict emerging nutrient deficiencies and to develop appropriate breeding technologies. Balanced fertilization is inevitable to increase the productivity of the crop. Among the micronutrients, Zn, Fe, B, Mn, and Mo significantly improved yield, and micronutrient foliar spray is economical on legumes.
2.Forage status
3.Different types of abiotic stress faced by crops
4.Impact of abiotic stress on physiology of forage
5.Impact of climatic anomalies on forages in terms of stress
6.Abiotic stress management in major fodder crops
7.Improving forages for abiotic stress response based on breeding techniques
8.Micronutrient stress management
Amanpreet Singh and Harmandeep Singh Chahal (November 30th 2020). Management of Abiotic Stress in Forage Crops [Online First], IntechOpen, DOI: 10.5772/intechopen.93852. Available from:
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Reboot Workshop - Hire G.I.
Each year over 200,000 service members transition from the military with over 50% of them going on 22 weeks of unemployment insurance. Survey results show that 81 percent of transitioning military personnel do not "feel fully prepared for the process of entering the job market." And unfortunately, the military's Transition Assistance Program is not designed to address the cultural needs of members leaving the service.
This lack of adequate support services presents daunting challenges related to finding meaningful employment and relating to the social environment of the civilian world, causing unintended consequences when service members are reintegrated back into society. This has resulted in high unemployment, homelessness, suicide, domestic violence and drug abuse and highlighted the need for a transition program that addresses reintegration issues at the core.
To answer the call National Veterans Transition Services, Inc. developed the REBOOT Workshop™, a three-week, cohort-based program that builds the emotional, psychological, social, and professional skills necessary for successful reintegration into civilian life. Designed for transitioning military and veterans, the workshop's insightful and personalized approach to reintegration utilizes cognitive restructuring to help veterans redefine their life – from the inside out for a successful transition home.
In 2010 the National Veterans Transition Service Inc. (NVTSI) created REBOOT Workshops™. REBOOT Workshops™ are designed to meet an acute need for robust military to civilian transition program and close the gap. By addressing transition issues at their root cause, NVTSI and its network of partners helps transitioning service members, veterans and spouses successfully transitioning from the military-to-civilian world through a three phased, 15-day intensive workshop that empowers them with resiliency and self-sufficiency.
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text:
It has been said that art is a tryst, for in the joy of it maker and beholder meet. ~Kojiro Tomita
Warigami: Combined Arms Origami
Origami enthusiasts with a particular interest in weaponry will appreciate this unique book, which presents instructions for folding fourteen war machines: six jets, three missiles, and five ground vehicles. Illustrated in full color, the step-by-step directions show how to assemble the models. Origami aircraft include a spy plane, strike fighter, and bomber, plus impaler, javelin, and harpoon missiles that can be mounted on some of the jets. Models of ground vehicles include the predator battle tank and guardian battle walker.


Landscaping and vegetation management include a wide variety of activities, including vegetation installation and removal, lawn mowing, fertilizer and pesticide application, watering and other landscaping practices. Many of these activities can result in the contamination of stormwater runoff with sediment, nutrients, organic materials, toxic chemicals, heavy metals, oils and trash.
By implementing the proper practices and procedures below during your landscaping and vegetation management work, your business and project site can help to prevent stormwater pollution and keep our waters healthy for everyone.
Schedule any soil disturbance activities such as grading and excavation projects during dry weather.
Use mulch or other appropriate erosion control measures when soils are exposed.
Place landscaping materials, including soil and mulch, away from stormwater drains, drainage ditches and surface waters.
Berm and cover stockpiles with secured waterproof tarps or plastic sheeting to prevent exposure to rainfall and stormwater.
Cover all stormwater drains before hydro-seeding. Avoid seeding lawns on steep slopes or areas that do not drain well.
Sod these areas to protect from erosion.
Do not refuel or clean lawn mowers and other landscaping equipment near a stormwater drain, drainage ditch or surface water. Inspect all landscaping equipment for leaks on a daily basis and repair as needed.
Always follow the manufacturers' recommendations and directions for the proper amounts of fertilizer and application instructions. Make sure to calibrate granular spreaders and liquid applicators to avoid excessive application.
Avoid applying fertilizers when it is raining or when heavy rain is forecast.
Fertilizers should be worked into the soil rather that broadcast and left on the surface.
Close fertilizer spreaders when going over pavement, bare soil and other non-vegetated surfaces. Sweep up dry fertilizer granules that fall on pavement or other hard surfaces. Do not hose or blow off.
Properly identify weeds, diseases and insects before applying a pesticide.
Use non-toxic and more environmentally friendly pest control methods, whenever possible.
Read and follow the label directions and apply all pesticides as directed. Follow all federal, state, and local laws and regulations governing the use, storage, and disposal of fertilizers and pesticides and training of applicators and pest control advisors.
Do not apply pesticides when it is raining or when rain is forecast. Check the irrigation schedule to ensure that pesticides applied will not be applied prior to watering. Apply pesticides only when wind speeds are low.
Do not mix, prepare or apply pesticides near a stormwater drain, drainage ditch or surface water.
Spot treat only affected areas instead of using widespread pesticide application.
Sweep up any dry pesticide that falls onto pavement or other impervious surfaces. Do not hose or blow off. Follow manufacturers' instructions for any spills and leaks.
Sweep and bag waste materials rather than blowing or washing off paved surfaces.
Never blow or dump grass clippings, leaves and other waste into the street, stormwater drain, drainage ditch or surface water.
If disposing of yard waste off-site, bag as solid waste for pickup or delivery to a composting facility or permitted landfill.
Always cover waste materials when hauling off-site.


The Reading Room offers an ideal setting for intimate meetings or private dining. Natural daylight.
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Smart gadgets, drones, 3D printers, Droids, self-balancing unicycle, humanoids, and more.
A team of professionals who can create high-quality prototypes.
Creation of electronic devices and modeling of electronic circuits.
Creation of 3D models, development of 3D printers, printing of 3D cases and spare parts.
Programming of microcontrollers AVR, STM32 on C / C ++ and Python.
Designing, robots and robotic systems development.
Increase the effectiveness of integration solutions, modernization of obsolete systems and mechanisms.
Designing prototypes of varying complexity using software from solidworks.
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Updated on 2018-11-26T22:03:05+00:00, by bogdan.


This tutoring role is NOT paid.
This tutoring role is NOT paid (if tutor travels to tutees home there will be a £10 contribution).
This tutoring role is NOT paid but a £10 contribution will be made towards the tutoring/travel per session.


text:
require 'puma/runner'
require 'puma/detect'
require 'puma/plugin'

module Puma
  # This class is instantiated by the `Puma::Launcher` and used
  # to boot and serve a Ruby application when no puma "workers" are needed
  # i.e. only using "threaded" mode. For example `$ puma -t 1:5`
  #
  # At the core of this class is running an instance of `Puma::Server` which
  # gets created via the `start_server` method from the `Puma::Runner` class
  # that this inherits from.
  class Single < Runner
    def stats
      {
        started_at: @started_at.utc.iso8601
      }.merge(@server.stats)
    end

    def restart
      @server.begin_restart
    end

    def stop
      @server.stop(false) if @server
    end

    def halt
      @server.halt
    end

    def stop_blocked
      log "- Gracefully stopping, waiting for requests to finish"
      @control.stop(true) if @control
      @server.stop(true) if @server
    end

    def run
      output_header "single"

      load_and_bind

      Plugins.fire_background

      @launcher.write_state

      start_control

      @server = server = start_server

      log "Use Ctrl-C to stop"
      redirect_io

      @launcher.events.fire_on_booted!

      begin
        server.run.join
      rescue Interrupt
        # Swallow it
      end
    end
  end
end

redpajama_set_name: RedPajamaGithub