text:
\section{Introduction}\label{Sec:intro}
In a series of papers \cite{FefNar93},\cite{FefNar94},\cite{FefNar96}, Fefferman and Narasimhan investigated the local behavior of a
polynomial $f$ of $N$ real or complex variables, restricted to a  given
$n$-dimensional algebraic variety $\mathcal{X}$.  Conceptually, the problem is to quantify to what extent
 the local behavior of the trace of $f$ on $\mathcal{X}$ deteriorates relative to
an  $N$-dimensional ball. Of particular interest here is to determine the
dependence of quantitative estimates on the degree of the
polynomials.
Fefferman and Narasimhan chose the classical Bernstein inequalities for polynomials of several variables
to measure the distortion of a polynomial restricted to an algebraic variety.

The authors' interest in this particular problem arose as part of their work on the  Chulaevsky-Sinai conjecture.
In their pioneering paper \cite{ChuSin89}, Chulaevsky and Sinai  analyze the  spectrum of a discrete Schr\"odinger operator on ${\mathbb Z}$ with a quasi-periodic potential
given by evaluating a generic smooth function on ${\mathbb T}^{2}$ along the orbit of an ergodic shift.
In \cite{GolSchVod16a} (building on work from \cite{GolSchVod16}) the authors found  that some ``generic
versions" of these restricted Bernstein estimates play a crucial role in addressing this conjecture.

\smallskip

There are two major differences between the current paper and \cite{FefNar96}: (i) we obtained estimates at
singular points and the estimates at regular points don't depend on the distance to the singular points
(ii) we allow analytic functions and analytic sets in place of polynomials and algebraic varieties.

As for (i),  Fefferman and Narasimhan had considered compact subsets of algebraic varieties
away from the singular points. For polynomials and algebraic varieties, Roytwarf and Yomdin~
\cite{RoyYom97} extended their Bernstein estimates to be independent of the distance to the singular points.
However, the aforementioned spectral analysis forces us to consider analytic functions and sets, rather than algebraic ones.
Our estimates for analytic functions are not as sharp as for
polynomials.

Regarding (ii), we note that Coman-Poletsky \cite{ComPol07} (for $n=1$) and Brudnyi
\cite{Bru08} (for all $n\ge1$) studied Bernstein estimates (amongst other local properties)
for the restriction of analytic functions of $n+1$ variables to the graph of an analytic function of $n$ variables.
Both these papers require a certain {\em transversality condition} of the
zeros sets of the functions in question.  We shall also impose a condition of this nature in our approach.

We proceed to discuss the main results of the paper. First we need to introduce some notation related to Cartan sets
and to Bernstein exponents. The Cartan sets will appear in our transversality condition to allow the application
of the Cartan-type estimate established in \cref{thm:C}.

\begin{defi}\label{defi:cartan0sets} $(1)$ Let $H\ge 0$, $ K\ge 1 $.
  For ${\mathcal{B}} \subset{\mathbb{C}}^2$ we say that ${\mathcal{B}} \in \mathop{\rm{Car}}\nolimits_{2,0}(H, K)$ if
  \[
    {\mathcal{B}}\subset \bigcup\limits^{j_0}_{j=1} B(\underline v_j, r)
  \]
  with $r=e^{-H}$ and $j_0 \le K$.

  $(2)$ Let $ f$ be analytic on the ball
  $B({\underline v}_0,R)\subset \mathbb{C}^{2}$, $ {\mathcal{S}}\subset \mathbb{C}^2 $, and $ \mu\in(0,1) $. Define
  \begin{gather*}
      M_f({\underline v}_0,R)=\sup_{B({\underline v}_0,R)}\log |f|,\quad
      M_f({\mathcal{S}},{\underline v}_0,R)=\sup_{B({\underline v}_0,R)\cap {\mathcal{S}}}\log |f|,\\
      B_f(\mu;{\underline v}_0,R) =M_f({\underline v}_0,R)-M_f({\underline v}_0,\mu R),\\
      B_f(\mu;{\mathcal{S}},{\underline v}_0,R) =M_f({\mathcal{S}},{\underline v}_0,R)-M_f({\mathcal{S}},{\underline v}_0,\mu R).
  \end{gather*}
  We call $B_f(\mu;{\underline v}_0,R),B_f(\mu;{\mathcal{S}},{\underline v}_0,R)$ {\em Bernstein exponents}. We make the natural
  convention that if the function $ f $ vanishes identically, its Bernstein exponents are zero.

  $ (3) $ Let $ f $ be analytic on $ B(0,1) $, $ \mu\in(0,1) $. We define
  \begin{equation*}
    B_f(\mu)=\sup_{\underline v_0\in B(0,1/4),0<R\le 1/4} B_f(\mu;\underline v_0,R).
  \end{equation*}

  $ (4) $ Given an analytic function $ f $ on a disk $ {\mathcal{D}}(z_0,R)\subset \mathbb{C} $, the quantities
  $ M_f(z_0,R) $ and $ B_f(\mu; z_0,R) $ are defined analogously to the above.
\end{defi}

\smallskip
The classical Bernstein doubling inequality for a univariate polynomial $ f $ can be expressed using the above
notation as
\begin{equation*}
  B_f(\mu;z_0,R)\le (\log \mu^{-1})\times \deg f,
\end{equation*}
where $ \mu\in (0,1) $, $ z_0\in \mathbb{C} $, $ R>0 $.

\smallskip

Throughout we will impose the following {\em transversality condition}.  Suppose
the functions $f_1,f_2$ are analytic in the ball $B(0,1)\subset \mathbb{C}^2$,  and are
normalized so that $ M_{f_i}(0,1)\le 0 $, $ i=1,2 $. We let $ F=(f_1,f_2) $ and
we define
\begin{equation*}
  {\mathcal{N}}_F(\epsilon):= \{ \underline v\in B(0,1): |F(\underline v)|<\epsilon \}.
\end{equation*}
We require that
\begin{equation}\label{eq:cartanmaincond}
  {\mathcal{N}}_F(\exp(-H_0))\cap B(0,1/2)\in \mathop{\rm{Car}}\nolimits_{2,0}(H_1, K_1),\quad \log K_1\ll H_1,
\end{equation}
for some $H_0\gg H_1\gg B_0:=\max_i B_{f_i}(1/4)$.

\begin{remark}
  A priori it might appear that $ K_1 $ can be exponentially large, i.e.,
  $ \exp(cH_0) $ for some small $ c>0 $. However, a simple argument,
  presented in \cref{lem:K1-sa-bound}, shows that we always have the polynomial bound
  $ K_1\le H_0^C $, where $ C $ is some absolute constant.
\end{remark}

Let ${\mathcal Z}=\{\underline v\in B(0,1): f_2({\underline v})=0\}$.  It is well known that there exists a discrete set of singular points
$ \sng {\mathcal Z} $ (relative to $ B(0,1) $) such that the set of regular points $ \reg {\mathcal Z}:= {\mathcal Z}\setminus \sng {\mathcal Z} $ is
a one dimensional complex manifold (see, for example, \cite{Chi89}).

\begin{thma}\label{thm:A}
  Assume the transversality condition holds and let $ {\mathcal Z} $ be as above.
  Let $ C_0=\log (K_1B_0^2H_0^2) $. Then the following statements hold.
  \begin{enumerate}[leftmargin=*]
  \item
    For any $ \underline v_0\in B(0,1/8)\cap {\mathcal Z} $ and $ 0<R\le 1/4 $,
    \begin{equation*}
      B_{f_1}(1/4;{\mathcal Z},\underline v_0,R)\lesssim \max(\log R^{-1},C_0)B_0^2H_0.
    \end{equation*}
  \item
      There exists an atlas of $  \reg {\mathcal Z} $ with charts defined on $ {\mathcal{D}}(0,1) $ such that for any chart
      $ \phi $ satisfying $ \phi({\mathcal{D}}(0,1))\cap B(0,1/8)\neq \emptyset $ and any $ {\mathcal{D}}(z_0,R)\subset {\mathcal{D}}(0,1) $,
      we have
      \begin{equation*}
        B_{f_1\circ \phi}(1/4;z_0,R)\le C(f_2)C_0B_0^2H_0.
      \end{equation*}
  \end{enumerate}
\end{thma}

\begin{remark}
  The $ \log R^{-1} $ factor from part (1) of \cref{thm:A} is needed because the estimate covers singular points.
  See \cref{ex:logR}.
\end{remark}

In \cref{thm:B} we obtain a sharper version of part (2) of the previous theorem for the polynomial case.
Such a result is also known from \cite{RoyYom97}. The work of Roytwarf and Yomdin relies on a
classical inequality for the Taylor coefficients of  $p$-valent functions due to Biernacki~\cite{Bie36}.
In turn \cite{Bie36} relies on a deeper growth bound for $p$-valent functions obtained by Cartwright~\cite{Ca35}
(see \cite{Hay94} for a more detailed account of these issues).
In \cref{thm:B} we show that in the context of algebraic curves the Bernstein estimates by
Roytwarf and Yomdin follow from more elementary arguments in the spirit of the argument principle,
without any reference to properties of $p$-valent functions.
We also require basic properties of the harmonic conjugate and of course
Bezout's theorem (which is also needed in Roytwarf and Yomdin in order to estimate the valency).
It seems that this approach can be developed for a general algebraic variety.

\begin{thmb}\label{thm:B} Assume that $f_1,f_2$ are polynomials. Let $ {\mathcal Z} $ be as above.
  Then there exists an atlas of $  \reg {\mathcal Z} $ with charts defined on $ {\mathcal{D}}(0,1) $ such that for any chart
  $ \phi $  and any $ {\mathcal{D}}(z_0,R)\subset {\mathcal{D}}(0,1) $,  we have
  \begin{equation*}
    B_{f_1\circ \phi}(1/4;z_0,R)\le C(f_2)\times \deg f_1.
  \end{equation*}
\end{thmb}

For our application in \cite{GolSchVod16a} we use the Cartan estimate for maps in
$\mathbb{C}^2$ which is \cref{thm:C} we state below. The proof of \cref{thm:A} relies on
\cref{thm:C}. The Cartan estimate for an analytic
function $f({\underline v})$, ${\underline v}\in \mathbb{C}^2$ (see \cref{lem:high_cart}), basically says that if
the set $\{|f|<\epsilon_0\}$ is ``not two-dimensional" then
$\{|f|<\epsilon\}$ is ``one-dimensional" for any
$\epsilon\ll \epsilon_0$. We prove an analogue statement for mappings. Let
$F: B(0,1)\subset \mathbb{C}^2\to \mathbb{C}^2$ be analytic.  We show  that if the set $\{|F|<\epsilon_0\}$ is
``zero-dimensional", then $\{|F|<\epsilon\}$ is ``zero-dimensional" for
any $\epsilon\ll \epsilon_0$.  Of course, the quantitative details of
the statement here are as important as the topological ones.

\begin{thmc}\label{thm:C}
  Assume the transversality condition holds. Then for any $H\gg 1$ we have
  \begin{equation*}
    {\mathcal{N}}_F(\exp(-HB_0^2H_0))\cap B(0,1/4)\in \mathop{\rm{Car}}\nolimits_{2,0}(H, K),\quad K\lesssim K_1 B_0^2H_0^2.
  \end{equation*}
\end{thmc}

The proof of Theorem~\ref{thm:C} proceeds in four steps: (a) apply the Weierstrass preparation theorem to the given analytic functions in one of the two coordinates (b) determine the resultant of the two polynomials obtained in the previous step (c) apply Cartan's theorem in one variable so as to guarantee that this resultant is not too small off of a union of small disks in $\mathbb{C}$, which in turn gives that at least one of the two analytic functions is not too small outside of thin cylinders in $\mathbb{C}^2$ (d) repeat the previous steps with respect to  the other variable. The intersection of the two families of thin cylinders gives a $\mathop{\rm{Car}}\nolimits_{2,0}$ set.

It would be interesting to extend this method   to higher dimensions, i.e., to the construction of $\mathop{\rm{Car}}\nolimits_{d,0}(H, K)$ sets with $d\ge3$ -- at least for polynomials in $d$ variables. In principle, this appears possible but it seems to require the use of multivariate resultants, which are more delicate than the univariate ones. If Theorem~\ref{thm:C} extends to $d\ge 3$, then one would obtain a Bernstein estimate as in Theorem~\ref{thm:A}.
As our applications do not require this extension, we do not pursue these matters here.

We conclude this introduction by providing some details of the aforementioned spectral theory applications.
Consider a trigonometric polynomial of two variables
\begin{equation}\label{Vnondegen53}
  \begin{split}
    V(z,w)=\sum_{|m|,|n|\le k} c_{m,n}e(mz+nw),
  \end{split}
\end{equation}
$e(\zeta):=e^{2\pi i \zeta}$.  To normalize the setting we consider
the unit sphere in the space of the coefficients
\[
  \mathcal{C}_1=\{(c_{m,n})\in\mathbb{R}^{4k+2}:\sum_{m,n}|c_{m,n}|^2=1\}.
\]
We use ${\rm mes}$ for the Lebesgue measure on the sphere.  Take arbitrary
$\omega \in {\mathbb T}^2$, $\lambda\in \mathbb{R}$. Consider the determinant
\begin{equation}\label{dirichlet1}
  \begin{split}
    \begin{aligned}
      &f_N({\underline v})=
      \begin{vmatrix}
        \lambda V({\underline v}) & -1 & 0  &\cdots &\cdots & 0\\[5pt]
        -1 & \lambda V({\underline v}+\omega)& -1 & 0 & \cdots & 0\\[5pt]
        \vdots & \vdots & \vdots & \vdots & \vdots \\
        &&&&&-1 \\[5pt]
        0 & \dotfill & 0 & -1 && \lambda V({\underline v}+(N-1)\omega)
      \end{vmatrix}
    \end{aligned}
  \end{split}
\end{equation}
For ${\underline v}\in \mathbb{R}^2$,  $f_N({\underline v})$ is the characteristic
determinant of the Schr\"{o}dinger operator with potential
$V({\underline v}+n\omega)$, $n\in \mathbb{Z}$ on the interval $[0,N-1]$ subject
to Dirichlet boundary conditions. In \cite{GolSchVod16a} we establish the
following results: {\em Given arbitrary $\epsilon>0$, there exists a set
$\mathcal{C}\subset \mathbb{R}^{4k+2}$ with
${\rm mes} (\mathcal{C}_1\setminus \mathcal{C})<\epsilon$ and
$\lambda_0=\lambda_0(\epsilon)$ depending only on $\epsilon$ such that
for any $V$ with $(c_{m,n})\in \mathcal{C}_1$ and any
$|\lambda|\ge \lambda_0$ there exists
a set $\Omega(V)\subset {\mathbb T}^2$ with ${\rm mes} (\Omega(V))<\epsilon$ such
that for any $\omega\in {\mathbb T}^2\setminus \Omega(V)$, any $N$, and any
${\underline v}_0\in {\mathbb T}^2$ the functions $f_N({\underline v}_0+r_0{\underline v})$ and
$f_N({\underline v}_n+r_0{\underline v})$, ${\underline v}_n={\underline v}_0+n\omega$, $|n|>N$,
${\underline v}\in B(0,1)$, $r_0=\exp (-(\log N)^A)$, $A\gg 1$ being an
absolute constant, obey all conditions of \cref{thm:A,thm:C} with
$B_0,H_0\le (\log N)^c$, $c\ll 1$.}

The exceptional sets in this result are not artificial.  In fact,
 the theorem fails for some
$(c_{m,n})\in\mathcal{C}$.  A similar fact is true for the exceptional
frequencies.

\section{Cartan's Estimate}

Recall the following definition from \cite{GolSch08}.

\begin{defi}\label{defi:cartansets}
  Let $H\ge 0$, $ K\ge 1 $.  For an arbitrary set ${\mathcal{B}} \subset {\mathbb{C}}$
  we say that ${\mathcal{B}} \in \mathop{\rm{Car}}\nolimits_1(H, K)$ if ${\mathcal{B}}\subset
  \bigcup\limits^{j_0}_{j=1} {\mathcal{D}}(z_j, r_j)$ with $j_0 \le K$, and
  $\sum_j\, r_j < e^{-H}$.

  If $d\ge 1$ is an integer  and ${\mathcal{B}}\subset {\mathbb{C}}^d$, then we define
  inductively that ${\mathcal{B}}\in \mathop{\rm{Car}}\nolimits_d(H, K)$ if for any $1 \le j \le d$ there
  exists ${\mathcal{B}}_j \subset {\mathbb{C}}, {\mathcal{B}}_j \in \mathop{\rm{Car}}\nolimits_1(H,  K)$, so that ${\mathcal{B}}_z^{(j)} \in \mathop{\rm{Car}}\nolimits_{d-1}(H, K)$ for any $z \in {\mathbb{C}}
  \setminus {\mathcal{B}}_j$,  here ${\mathcal{B}}_z^{(j)} = \left\{(z_1, \dots, z_d) \in {\mathcal{B}}:
    z_j = z\right\}$.
\end{defi}

The above definition of Cartan sets is motivated by the following statement, known as Cartan estimate on
the lower bound of an analytic function of several variables.

\begin{lemma}[{\cite[Lem.~2.15]{GolSch08}}]\label{lem:high_cart}
 Let $\varphi(z_1, \dots, z_d)$ be an analytic function defined
in a polydisk ${\mathcal{P}} = \prod\limits^d_{j=1} {\mathcal{D}}(z_{j,0}, 1)$, $z_{j,0} \in
{\mathbb{C}}$.  Let $M \ge \sup\limits_{{\underline{z}}\in{\mathcal{P}}} \log |\varphi({\underline{z}})|$,  $m \le \log
\bigl |\varphi({\underline{z}}_0)\bigr |$, ${\underline{z}}_0 = (z_{1,0},\dots, z_{d,0})$.  Given $H
\gg 1$ there exists a set ${\mathcal{B}} \subset {\mathcal{P}}$,  ${\mathcal{B}} \in
\mathop{\rm{Car}}\nolimits_d\left(H^{1/d}, K\right)$, $K = C_d H(M - m)$,  such that
\begin{equation}
\label{eq:cart_bd}
\log \bigl | \varphi(z)\bigr | > M-C_d H(M-m)
\end{equation}
for any $z \in \prod^d_{j=1} {\mathcal{D}}(z_{j,0},1/6)\setminus {\mathcal{B}}$. Furthermore, when $ d=1 $ we can take
$ K=C(M-m) $ and keep only the disks of $ {\mathcal{B}} $ containing a zero of $\phi$ in them.
\end{lemma}

\begin{remark}
  (1)
  The choice of the constant $ 1/6 $ in \cite[Lem.~2.15]{GolSch08} was so that one could invoke the
  one-dimensional Cartan estimate as stated in Theorem 4 of \cite[Lecture 11]{Lev96}. However,
  it is straightforward to adjust the result from \cite{Lev96}
  and the proof from \cite{GolSch08} to replace $ 1/6 $ by any $ r<1 $. Of course, the constant $ C_d $ would depend
  (explicitly) on the particular choice of $ r $.

  (2) The definition of Cartan sets gives implicit information about their measure. For example, using Fubini and
  the definition of $ \mbox{\rm Car}_d $, one gets by induction that the set exceptional set $ {\mathcal{B}} $ in the previous lemma
  satisfies $ {\rm mes}_{\mathbb{C}^d}({\mathcal{B}})\le C(d)\exp(-H) $.
\end{remark}

The following notion will be needed for our discussion of Weierstrass' preparation theorem.
\begin{defi}\label{defi:regulardir}
  Let $f$ be analytic on the ball $B({\underline v}_0,R_0)\subset \mathbb{C}^2$.
  Let $\mathfrak{e}\in \mathbb{C}^2$ be an arbitrary unit  vector.
  We say that $\mathfrak{e}$ is $m$--regular for $f$ at ${\underline v}_0$
  (or just $m$--regular if
  it is clear from the context what $ \underline v_0 $ is) if
  \[
    \sup_{z\in{\mathcal{D}}(0,R_0/4)}\log |f({\underline v}_0+z\mathfrak{e})|\ge m.
  \]
\end{defi}

We show that Cartan's estimate implies that most directions are regular.
We use $ \sigma $ to denote the standard spherical measure.

\begin{lemma}
\label{lem:6.3c}
Let $ f $ be as in \cref{defi:regulardir} and let
\begin{equation*}
  M \ge \sup_{B(\underline v_0,R_0)} \log |f|,\quad \sup_{B(\underline v_0,R_0/4)}\log |f|\ge m.
\end{equation*}
Take arbitrary $H\gg 1$ and set $\underline{m}= M-C_2H(M-m)$, with $ C_2 $ as in \cref{lem:high_cart}.
Denote by ${\mathcal{B}}$ the set of $\mathfrak{e}$ which are not $\underline{m}$--regular.
Then
 \[
\sigma ({\mathcal{B}})\lesssim\exp (-H^{1/2}).
 \]
\end{lemma}
\begin{proof} Apply the Cartan estimate
  to find a set $ \hat {\mathcal{B}} $, ${\rm mes} (\hat{\mathcal{B}}) \lesssim R_0^4\exp (-H^{1/2})$, such that
  $ \log \bigl | f({\underline v})\bigr | >\underline{m} $
for any ${\underline v} \in B({\underline v}_0,R_0/4)\setminus \hat{\mathcal{B}}$.
Using spherical coordinates  write
\[
{\rm mes} (\hat {\mathcal{B}})\ge\int_{{\mathcal{B}}}d\sigma(\mathfrak{e})\int_0^{R_0/4}r^3dr\gtrsim R_0^4\sigma ({\mathcal{B}})
\]
and the statement follows.
\end{proof}

\section{Bernstein Exponent and Number of Zeros}

In this section we provide a relation between Bernstein exponents for one variable analytic functions and the number
of their zeros.
\begin{lemma}\label{lem:Harnack-bounds}
  Let $ \phi $ be a non-vanishing analytic function on $ {\mathcal{D}}(z_0,R) $
  Then for  any $ z $, $ |z-z_0|=r<R $, we have
  \begin{equation*}
    -\frac{2r}{R-r}(M-\log|\phi(z_0)|)\le \log|\phi(z)|-\log|\phi(z_0)|
    \le \frac{2r}{R+r}(M-\log|\phi(z_0)|),
  \end{equation*}
  where $ M=M_\phi(z_0,R) $.
\end{lemma}
\begin{proof}
  The estimates follows immediately from Harnack's inequality applied to $ u(z)=M-\log|\phi(z)| $.
\end{proof}

\begin{prop}\label{prop:bernstein-analytic}
  Let $ \phi $ be an analytic function on $ {\mathcal{D}}(0,1) $ such that
  \begin{equation*}
    M_\phi(0,1)\le 0,\quad M_\phi(0,1/4)\ge m.
  \end{equation*}
  Let $ n $ be the total number of zeros of $ \phi $ in $ {\mathcal{D}}(0,3/4) $.
  Then  for any $ |z_0|<1/8 $, $ r<1/8 $, $ \mu\in (0,1) $, we have
  \begin{equation}\label{eq:B_phi}
    B_{\phi}(\mu;z_0,r)\le Cr(n-m)-n\log\mu\lesssim -(r-\log \mu) m,
  \end{equation}
\end{prop}
\begin{proof}
  Take $\zeta_0\in {\mathcal{D}}(0,1/4)$ wit $\log |f(\zeta_0)|= m$. Using Jensen's formula applied to
  \begin{equation*}
    f \left( \frac{z+\zeta_0}{1+\overline {\zeta_0}z} \right)
  \end{equation*}
  we get $ n\lesssim -m $. So, we just have to prove the first estimate in \cref{eq:B_phi}.

  Let $ a_1,\ldots,a_n $, be the zeros of $ \phi $ in $ {\mathcal{D}}(0,7/8) $,
  repeated according to their multiplicities.
  Let $ P(z)=\prod_{k=1}^n(z-a_k) $, $ h=\phi/P $, and
  $ z_1 $, $ |z_1-z_0|=\mu r $, be such that $ \log|h(z_1)|=M_h(z_0,\mu r) $.
  Note that  $ h $ is non-vanishing and analytic on $ {\mathcal{D}}(0,3/4) $.
  Using \cref{lem:Harnack-bounds}
  we have that for any $ z\in {\mathcal{D}}(z_0,\mu r) $
  \begin{multline*}
    \log|h(z)|\ge \log|h(z_1)|-\frac{2|z-z_1|}{1/2-|z-z_1|}(M_h(z_0,1/2)-\log|h(z_1)|)\\
    \ge M_h(z_0,\mu r)-C\mu r(M_h(0,3/4)-M_h(z_0,\mu r)).
  \end{multline*}
  Therefore
  \begin{equation*}
    M_\phi(z_0,\mu r)\ge M_h(z_0,\mu r)-C\mu r(M_h(0,3/4)-M_h(z_0,\mu r))+M_P(z_0,\mu r)
  \end{equation*}
  and
  \begin{multline}\label{eq:B-initial}
    B_\phi(\mu;z_0,r)\le M_h(z_0,r)-M_h(z_0,\mu r)+C\mu r(M_h(0,3/4)-M_h(z_0,\mu r))\\
    +M_P(z_0,r)-M_P(z_0,\mu r).
  \end{multline}
  Let $ z_2 $, $ |z_2-z_0|=r $, such that $ \log|h(z_2)|=M_h(z_0,r) $ and $ z_3 $, $ |z_3|=1/4 $, such that
  $ \log|h(z_3)|=M_h(0,1/4) $. Using \cref{lem:Harnack-bounds} we get
  \begin{multline*}
    M_h(z_0,r)-M_h(z_0,\mu r)=\log|h(z_2)|-\log|h(z_1)|\\
    \le \frac{2|z_2-z_1|}{1/2+|z_2-z_1|}(M_h(z_1,1/2)-\log|h(z_1)|)
    \le Cr(M_h(0,3/4)-M_h(z_0,\mu r)),
  \end{multline*}
  \begin{multline*}
	M_h(z_0,\mu r)-M_h(0,1/4)=\log|h(z_1)|-\log|h(z_3)|\\
    \ge -\frac{2|z_3-z_1|}{1/2-|z_3-z_1|}(M_h(z_3,1/2)-\log|h(z_3)|)
   \ge -C(M_h(0,3/4)-M_h(0,1/4)).
  \end{multline*}
  Plugging these estimates in \cref{eq:B-initial} we get
  \begin{equation}\label{eq:Bern-h-P}
    B_\phi(\mu;z_0,r)\le Cr(M_h(0,3/4)-M_h(0,1/4))+B_P(\mu;z_0,r).
  \end{equation}
  Recall that we know $ B_P(\mu;z_0,r)\le -n\log\mu $, so to get the conclusion we just have to estimate
  $ M_h(0,3/4)-M_h(0,1/4) $. Given $ z\in {\mathcal{D}}(0,3/4) $,  apply
  the submean value property to get
  \begin{equation*}
    \log|h(z)|\le\frac{1}{2\pi}\int_0^{2\pi} \log|\phi(z+e^{i\theta}/4)|\,d\theta-
    \frac{1}{2\pi}\int_0^{2\pi} \log|P(z+e^{i\theta}/4)|\,d\theta\le n\log 4
  \end{equation*}
  and conclude $ M_h(0,3/4)\le n\log 4 $. We used the assumption that $ M_{\phi}(0,1)\le 0 $ and
  the fact that
  \begin{equation}\label{eq:log-int}
    \frac{1}{2\pi}\int_0^{2\pi} \log|z-a_k+e^{i\theta}/4|\,d\theta
    = \begin{cases}
      \log \frac{1}{4} &, |z-a_k|\le \frac{1}{4}\\
      \log|z-a_k| &, |z-a_k|>\frac{1}{4}
    \end{cases}
    \ge \log \frac{1}{4}.
  \end{equation}
  Since clearly $ M_P(0,1/4)\le 0 $, we  have $ M_h(0,1/4)\ge M_\phi(1/4) $. So,
  \begin{equation*}
    M_h(0,3/4)-M_h(0,1/4)\le C(n-m)
  \end{equation*}
  and the conclusion follows.
\end{proof}

\begin{remark}
  (1) It is not true that conclusion of Proposition~\ref{prop:bernstein-analytic}
  can be made just in terms of the number $n$ of zeros of $\phi$. Some estimate for
  $ M_\phi(0,1/4)$ is really needed. Here is an elementary example $\phi(z)=\exp(-N+Nz)$,
  $z\in{\mathcal{D}}(0,1)$ and $N>0$ is arbitrary. Clearly,  $M_\phi(0,1)=0$, $n=0$. On the other hand
  $ M_\phi(0,1/4)\simeq -N$, $B_\phi(1/4;0,1/8)\simeq N$.

  (2) It is known from \cite{RoyYom97} that if we have control on the valency of the function $ \phi $,
  instead of just the number of zeros, then the estimate for $ M_\phi(0,1/4) $ is not needed anymore.
\end{remark}

\section{Weierstrass' Preparation Theorem and Bernstein Exponents}

We start with a statement of the classical Weierstrass' preparation theorem attuned to our purposes.
\begin{lemma}\label{lem:weier}
  Let $f(z,w)$ be analytic function
  on a polydisk
  \begin{equation*}
    {\mathcal{P}}:={\mathcal{D}}(z_0,R_0)\times {\mathcal{D}}(w_{0},R_0)\subset \mathbb{C}^2,\quad \ R_0>0.
  \end{equation*}
  Assume that $f(\cdot, w)$ has no zeros on some circle $\Gamma_{\rho_0}=\{ z:|z-z_0|=\rho_0 \}$,
  $ 0<\rho_0<R_0/2 $,
  for any $w \in  {\mathcal{D}}(w_0, r_1)$, $0<r_1<R_0$.  Then there exists a Weierstrass polynomial $P(z,w)
  = z^k +a_{k-1} (w) z^{k-1} + \cdots + a_0 (w)$ with $a_j(w)$ analytic in
  ${\mathcal{D}}(w_{0}, r_1)$ and an analytic function $g(z,w), (z,w) \in {\mathcal{P}}':={\mathcal{D}}(z_0, \rho_0) \times {\mathcal{D}}(w_{0}, r_1)$
  so that the following properties hold:
  \begin{enumerate}
  \item[(a)] $f(z,w) = P(z,w) g(z,w)$ for any $(z,w) \in {\mathcal{P}}'$.

  \item[(b)] $g(z,w) \ne 0$ for any $(z,w) \in {\mathcal{P}}'$.

  \item[(c)] For any $w \in {\mathcal{D}}(w_0,r_1)$, $P(\cdot,w)$ has no zeros in $\mathbb{C} \setminus {\mathcal{D}}(z_0,\rho_0)$.

  \item[(d)] We have
    \begin{gather}
      \label{eq:g-lb}\left( \inf_{\Gamma_{\rho_0}\times {\mathcal{D}}(w_0,r_1)}\log|f| \right)-k\log(2\rho_0)
      \le \inf_{{\mathcal{P}}'}\log|g|,\\
      \label{eq:g-ub}\sup_{{\mathcal{P}}'}\log|g|\le\left(\sup_{{\mathcal{P}}} \log |f|\right)+k\log \frac{2}{R_0}.
    \end{gather}
  \end{enumerate}
\end{lemma}
\begin{proof}
  By the usual Weierstrass argument, one notes that
  $$
  b_p(w) := \sum^k_{j=1} \zeta^p_j (w) = \frac{1}{2\pi i} \oint\limits_{\Gamma} z^p \
  \frac{\partial _z f(z, w)}{f(z, w)}\, dz
  $$
  are analytic in ${\mathcal{D}}(w_0,r_1)$.  Here $\zeta_j (w)$ are the zeros of $f(\cdot, w)$
  in ${\mathcal{D}}(z_0,\rho_0)$.  Since the coefficients $a_j(w)$ are
  linear combinations of the $b_p$,
  they are analytic in $w$.  Analyticity
  of $g$ follows by standard arguments. We just have to prove (d). Since all the roots of $ P(\cdot,w) $ are in
  $ {\mathcal{D}}(z_0,\rho_0) $, we have $ \sup_{{\mathcal{P}}'}|P|\le (2\rho_0)^k $ and \cref{eq:g-lb} follows using the
  minimum modulus principle.

  Note that actually the function $ g $ can be defined on $ {\mathcal{P}} $ as $ g=f/P $ and it is analytic there.
  Given $ (z,w)\in {\mathcal{P}}' $,  apply
  the sub-mean value property for subharmonic functions to get
  \begin{multline*}
    \log|g(z,w)|\le \frac{1}{2\pi}\int_0^{2\pi} \log|f(z+R_0e^{i\theta}/2,w)|\,d\theta-
    \frac{1}{2\pi}\int_0^{2\pi} \log|P(z+R_0e^{i\theta}/2,w)|\,d\theta\\
    \le \left(\sup_{{\mathcal{P}}} \log |f|\right)+k\log \frac{2}{R_0}.
  \end{multline*}
  The estimate on the mean value of the polynomial follows by considerations analogous to \cref{eq:log-int}.

\end{proof}

Next we describe how Bernstein exponents rule the application of
 Lemma~\ref{lem:weier}.

\begin{lemma}\label{lem:6.3}
  Let $f$ be analytic on  $B(0,1)$,
  $ M\ge \sup_{B(0,1)}\log|f| $, $ \underline m= M-B $, $ B\gg 1 $, $ \mathfrak{e}_1 $ a  $ \underline m $-regular direction for $ f $ at $ 0 $
  (recall   \cref{defi:regulardir}),
  and $ \mathfrak{e}_2 $ another non-collinear direction.
  With a slight abuse of notation we denote by $f(z,w)$ the function in the new coordinates with respect to the basis
  $\mathfrak{e}_1,\mathfrak{e}_2$.
  Then there exists a circle $\Gamma_{\rho_0} = \left\{|z| = \rho_0\right\}$,
  $1/8 < \rho_0 < 1/4$, and $r_1=\exp \left(-CB\right)$, with $ C>1 $ an absolute constant, such that
  \begin{equation}\label{eq:f-blb}
    \inf_{\Gamma_{\rho_0}\times {\mathcal{D}}(0,r_1)} \log|f|\ge \exp(M-CB).
  \end{equation}
  In particular, Lemma~\ref{lem:weier} applies for $f(z,w)$ with this choice of $\rho_0$ and~$r_1$,
  as well as with $k\lesssim B$ and $ \delta\ge M-CB $.

\end{lemma}
\begin{proof}
  Since $ \mathfrak{e}_1 $ is a $ \underline m $-regular direction, there exists $ z_1 $, $ |z_1|= 1/4 $,
  such that $ \log|f(z_1,0)|\ge \underline m $.
  Due to Cartan's estimate one has
  \begin{equation}\label{eq:fz0-Cartan}
      \log|f(z,0)| \ge M - C(M-\underline m)= M-CB
  \end{equation}
  for any $z \in {\mathcal{D}}(0, 1/4) \setminus {\mathcal{B}}$, where ${\mathcal{B}} \in \mathop{\rm{Car}}\nolimits_1\left(C', C'B\right)$, $ C'\gg 1 $.
  As a consequence of the definition of $ \mbox{\rm Car}_1 $ sets, we can choose $1/8 < \rho_0 < 1/4$ such that
  ${\mathcal{B}} \cap \Gamma_{\rho_0} =\emptyset$.
  Then
  \begin{equation}\label{eq:w0}
      |f(z, 0)| \ge \exp \left(M - CB \right)
  \end{equation}
  for any $z \in \Gamma_{\rho_0}$.  Note that due to Cauchy's estimates
  \begin{equation*}
    |f(z,w) - f(z,0)| \lesssim e^M |w|
  \end{equation*}
  for any $z \in {\mathcal{D}}(0, 1/2)$, $w \in {\mathcal{D}}(0, 1/2)$.  Taking into
  account~\eqref{eq:w0},  one obtains
  \begin{equation*}
      |f(z,w)| >\exp \left(M - CB \right)
  \end{equation*}
  for any $z \in \Gamma_{\rho_0}$, provided $w \in{\mathcal{D}}(0, r_1)$, $r_1 =\exp \left(-CB\right)$, with $ C $ large
  enough (of course, $ C $ is larger than in \cref{eq:w0}). This proves \cref{eq:f-blb} and allows us to apply
  \cref{lem:weier} as stated. For the bound on the degree of the Weierstrass polynomial note that
  by Jensen's formula applied to $f(z,0)$, $z\in{\mathcal{D}}(z_1,1/2)$,
  \begin{equation*}
    k \le \#\left\{z \in {\mathcal{D}}(0,1/4): f(z,0) = 0 \right\}
    \le \#\left\{z \in {\mathcal{D}}(z_1, 1/2): f(z,0) = 0 \right\} \lesssim B.
  \end{equation*}
\end{proof}

\begin{remark}\label{rem:Weier-simultaneous}
 $(1)$ Due to \cref{lem:6.3c}, we will always apply the previous lemma with $ B\simeq B_f(1/4;0,1) $. This is how the
  Bernstein exponent determines the size of the polydisk on which we have the Weierstrass factorization.

  $(2)$ If we are given two functions $ f_1,f_2 $ satisfying the assumptions of \cref{lem:6.3} with the same $ M $ and
  $ B $, then it is clear from the proof of the lemma that we can arrange for the conclusion to hold for both
  functions with the same choice of $ \rho_0 $ and $ r_1 $. Indeed, one only needs to choose $ \rho_0 $ such that
  $ \Gamma_{\rho_0}\cap({\mathcal{B}}_1\cup {\mathcal{B}}_2)=\emptyset $, where $ {\mathcal{B}}_i $ are the Cartan sets needed to guarantee
  \cref{eq:fz0-Cartan} for $ f_i $.
\end{remark}

\section{Resultants}

We briefly recall the definition of the resultant of two univariate polynomials and some of the basic properties
that we'll use.
Let $f(z) = a_nz^n + a_{n-1} z^{n-1} + \cdots + a_0$,
$g(z) = b_mz^m + b_{m-1} z^{m-1} + \cdots + b_0$ be polynomials,
$a_i, b_j \in {\mathbb{C}}$, $a_n\neq 0$, $b_m\neq 0$.  Let $\zeta_i$, $1 \le i \le n$ and $\eta_j$,
$1 \le j \le m$ be the zeros of $f(z)$ and $g(z)$ respectively.  The
resultant of $f$ and $g$ is defined as follows:
\begin{equation}\label{eq:resdef}
  \mathop{\rm{Res}}(f, g) = a_n^mb_m^n\prod_{i,j} (\zeta_i - \eta_j)=(-1)^{mn}b_m^n\prod_{j} f(\eta_j)
  =(-1)^{mn}a_n^m\prod_{i} g(\zeta_i).
\end{equation}
The resultant $\mathop{\rm{Res}}(f, g)$ can be expressed explicitly in terms of the
coefficients (see \cite{Lan02}):
\begin{equation}\label{eq:resdef1}
  \mathop{\rm{Res}}(f, g) = \left|\begin{array}{ll}
                       {\overbrace{\begin{array}{lll}
                                     a_n & 0 & \cdots\\
                                     a_{n-1}  & a_n & \cdots\\
                                     a_{n-2} & a_{n-1} & \cdots\\
                                     \cdots & \cdots & \cdots\\
                                     \cdots & \cdots & \cdots\\
                                     \cdots & \cdots & \cdots\\
                                     a_0 & a_1 &  \\
                                     0 & a_0 &  \end{array}}^m} &
                                                                  {\overbrace{\begin{array}{llll}
                                                                                b_m & 0 & \cdots & 0\\
                                                                                b_{m-1} & b_m & \cdots & \cdots\\
                                                                                b_{m-2} & b_{m-1} &\cdots & \cdots\\
                                                                                \cdots & \cdots & \cdots & \cdots\\
                                                                                \cdots & \cdots & \cdots & \cdots\\
                                                                                \cdots & \cdots & \cdots& \cdots\\
                                                                                &&&\\
                                                                                &&&\\
                                                                              \end{array}}^n}
                     \end{array}\right|
                 \end{equation}

\begin{lemma}\label{lem:resbasicprop1}
  Let $ f,g,\zeta_i,\eta_j $ as above. Set
  \[
   t_f=\min (|a_n|,1),\quad t_g=\min (|b_m|,1),\quad T_f=
    \max_i(\max |a_i|,1),\quad T_g=
    \max_j(\max |b_j|,1),
    \]
    \[
    R_f=t_f^{-1}T_f m,\quad R_g=t_g^{-1}T_g n.
  \]
  The following statements hold.

  $ (0) $    $ \max |\zeta_i|\le R_f,\quad \max |\eta_j|\le R_g $.

  $ (1) $
  If
  \[
    \big|\mathop{\rm{Res}}(f, g)\big|<\delta^{m}t_g^{n},\quad 0<\delta<1,
  \]
  then there exists $j$ such that
  \[
    \big|f(\eta_j)\big|< \delta.
  \]
  In particular, there exists
  $|z|\le R_g$ such that $ \max ( \big|f(z)\big|,\big|g(z)\big|)< \delta $.

  $ (2) $
  If there exists $z$ such that with $s=\max (m,n)$, $t=\min (t_f,t_g)$ holds
  \[
    \max [ \big|f(z)\big|,\big|g(z)\big|]<t\delta^{s},\quad 0<\delta<1,
  \]
  then
  \[
    \big|\mathop{\rm{Res}}(f, g)\big|< t^{2s}(2R)^{s^2}\delta,
  \]
  $R=\max (R_f,R_g)$.
\end{lemma}
\begin{proof}
  (0) follows by noting that, for example,
  \begin{equation*}
    |a_n||\zeta_i|^n\le (\max |a_i|)(|\zeta_i|^{n-1}+\dots+|\zeta_i|+1)\le (\max |a_i|) n\max(|\zeta_i|^{n-1},1).
  \end{equation*}
  (1) follows by contradiction from \cref{eq:resdef}. For (2) note that there must exist
  $ \zeta_{i_0}, \eta_{j_0} $ such that $ |z-\zeta_{i_0}|<\delta $, $ |z-\eta_{j_0}|<\delta $ and therefore,
  using (0) and \cref{eq:resdef},
  \begin{equation*}
    |\mathop{\rm{Res}}(f,g)|\le t^{2s}(2R)^{s^2-1}|\zeta_{i_0}-\eta_{j_0}|<t^{2s}(2R)^{s^2}\delta.
  \end{equation*}
\end{proof}

\section{Refinement of the Assumption \cref{eq:cartanmaincond}}\label{secCar2sets}

We give a simple argument showing that by making some small adjustments we actually have $ K_1\le H_0^C $ in
\cref{eq:cartanmaincond}.

\begin{lemma}\label{lem:K1-sa-bound}
  Using the notation and assumptions of \cref{thm:C} we have that
  \begin{equation*}
    {\mathcal{N}}(F,\epsilon_0/2)\cap B(0,1/2)\in \mbox{\rm Car}_{2,0}(H_1/2,H_0^C)
  \end{equation*}
  where $ C $ is some large absolute constant.
\end{lemma}
\begin{proof}
  Let $ f_{i,N} $ be the degree $ N $ Taylor polynomials (at the origin) associated with $ f_i $, $ i=1,2 $
  (recall that $ F=(f_1,f_2) $). Since $ M_{f_i}(0,1)\le 0 $, a standard application of the Cauchy estimates
  yields that
  \begin{equation*}
    |f_i-f_{i,N}|<\epsilon_0/100
  \end{equation*}
  for $ N=C\log \epsilon_0^{-1}=C H_0 $, $ C\gg 1 $. Let $ F_N=(f_{1,N},f_{2,N}) $. We have
  \begin{multline}\label{eq:sa-approx}
    {\mathcal{N}}(F,\epsilon_0/2)\cap B(0,1/2)\subset {\mathcal{N}}(F_N,3\epsilon_0/4)\cap B(0,1/2)\\
    \subset {\mathcal{N}}(F,\epsilon_0)\cap B(0,1/2).
  \end{multline}
  The set $ {\mathcal{N}}(F_N,3\epsilon_0/4)\cap B(0,1/2) $ is semialgebraic of degree less than $ CN $ and therefore it
  has at most $ N^C $ connected components. We refer to \cite[Ch.~9]{Bou05} for a brief review of semialgebraic
  sets and their properties. It follows from our assumptions that $ {\mathcal{N}}(F_N,3\epsilon_0/4)\cap B(0,1/2) $ is
  covered by less than $ K_1 $ balls of radius $ \exp(-H_1) $. Therefore, each connected component of
  $ {\mathcal{N}}(F_N,3\epsilon_0/4)\cap B(0,1/2) $ can be covered by just one ball of radius smaller than
  \begin{equation*}
    CK_1 \exp(-H_1)\le \exp(-H_1/2)
  \end{equation*}
  (recall that $ \log K_1\ll H_1 $) and so
  \begin{equation*}
    {\mathcal{N}}(F_N,3\epsilon_0/4)\cap B(0,1/2)\in \mbox{\rm Car}_{2,0}(H_1/2,N^C).
  \end{equation*}
  The conclusion now follows from \cref{eq:sa-approx}.
\end{proof}

\section{Proofs of Theorems A,B,C}\label{sec:thmABC}

We start with the proof of \cref{thm:C}.

\begin{proof}[Proof of \cref{thm:C}]
  Take $ \underline v_0=(z_0,w_0)\in B(0,1/4) $. By our assumptions
  \begin{equation*}
    B_{f_i}(1/4;\underline v_0,1/4),\quad M_{f_i}(\underline v_0,1/4)\le 0.
  \end{equation*}
  Due to \cref{lem:6.3c}, we can find unit vectors $ \mathfrak{e}_1,\mathfrak{e}_2$, $ |\langle \mathfrak{e}_1,\mathfrak{e}_2 \rangle|\ll 1 $,
  that are $ \underline m $-regular at $ \underline v_0 $ for both $ f_1,f_2 $ restricted to $ B(\underline v_0,1/4) $, with $ \underline m=-CB_0 $,
  $ C\gg 1 $. Then Lemma~\ref{lem:6.3} applies to both $f_1,f_2$ and to both directions $ \mathfrak{e}_1,\mathfrak{e}_2 $. As in
  Lemma~\ref{lem:6.3}, with a slight abuse of notation we denote by $ f_i(z,w) $ the functions in the coordinates
  with respect to the basis $ \mathfrak{e}_1,\mathfrak{e}_2 $ centered at $ \underline v_0 $ and with the obvious rescaling needed to apply
  the lemma.
  Applying Lemma~\ref{lem:6.3} (see also \cref{rem:Weier-simultaneous})
  in the direction of $ \mathfrak{e}_1 $ (and $ \mathfrak{e}_2 $ as the choice of non-collinear direction)
  we can write
  \begin{align*}
    f_i(z,w) &= P_i(z,w) g_i(z,w),\\
    P_i(z,w)
             & = z^{k_i} +a_{i,k_i-1} (w) z^{k_i-1} + \cdots + a_0 (w)
  \end{align*}
  on $ {\mathcal{P}}:={\mathcal{D}}(0,\rho_{0})\times {\mathcal{D}}(0,r_1) $, $ 1/8<\rho_{0}<1/4 $, $ r_1=\exp(-CB_0) $, where
  the coefficients $ a_{i,j}(w) $ are analytic
  on $ {\mathcal{D}}(0,r_1) $, $ g_i $ are analytic and non-vanishing on $ {\mathcal{P}} $, the polynomials $ P_i(\cdot,w) $,
  $ w\in {\mathcal{D}}(0,r_1) $, have no  zeroes in $ \mathbb{C}\setminus {\mathcal{D}}(0,\rho_0) $, and $ k_i\lesssim B_0$. Furthermore,
  using part (d) of \cref{lem:weier},
  \begin{equation}\label{eq:g-lbub}
    -B_0\lesssim    \inf_{{\mathcal{P}}}  \log \bigl | g_i \bigr|
    \le \sup_{{\mathcal{P}}}\log \bigl | g_i\bigr|\lesssim B_0.
  \end{equation}
  Let
  \[
    R(w)=\mathop{\rm{Res}} \left(P_1(\cdot,w),P_2(\cdot,w)\right).
  \]
  Note that by \cref{eq:resdef1}, $ R $ is analytic on $ {\mathcal{D}}(0,r_1) $.
  Since we chose $ \underline v_0\in B(0,1/4) $, the polydisk $ {\mathcal{P}} $ is a subset of
  $ B(0,1/2) $, as a set in the standard coordinates. This allows to use the hypothesis to guarantee that
  there exist points $ \underline v_j=(z_j,w_j) $ (expressed in the $ \mathfrak{e}_1,\mathfrak{e}_2 $ coordinates), $ 1\le j \le J\le K_1 $
  such that for
  \begin{equation*}
    (z,w)\in {\mathcal{P}}\setminus \left( \bigcup_{j=1}^J B(\underline v_j,C\exp(-H_1))\right)
  \end{equation*}
  we have
  \begin{equation*}
    \max(|f_1(z,w)|, |f_2(z,w)|)\ge \exp(-H_0)/\sqrt{2}
  \end{equation*}
  and by \cref{eq:g-lbub}
  \begin{equation}\label{eq:P-lb}
    \max(|P_1(z,w)|,|P_2(z,w)|)\gtrsim \exp(-H_0-CB_0).
  \end{equation}
  Note that we used the radius $ C\exp(-H_1) $ instead of $ \exp(-H_1) $ to account for the distortion under the
  change of coordinates. Since we are assuming that $ H_1\gg B_0 $ and $ \log K_1\ll H_1 $, we can find
  $$ w\in {\mathcal{D}}(0,r_1/4)\setminus \bigcup_{j=1}^J {\mathcal{D}}(w_j,C\exp(-H_1)).$$ For any such $ w $ \cref{eq:P-lb} holds for
  any $ z\in {\mathcal{D}}(0,\rho_0) $ and by part (1) of \cref{lem:resbasicprop1}
  \begin{equation*}
    \log|R(w)|\gtrsim -B_0H_0-B_0^2\gtrsim -B_0H_0.
  \end{equation*}
  Note that by the definition of the resultant \cref{eq:resdef}, we have $ \sup_{{\mathcal{D}}(0,r_1)}|R(w)|\le 1 $.
  Take $ H\gg 1 $.
  Applying
  Cartan's estimate, we get
  \begin{equation*}
    \log|R(w)|\gtrsim -HB_0H_0
  \end{equation*}
  for any $$ w\in {\mathcal{D}}(0,r_1/4)\setminus {\mathcal{B}},\quad  {\mathcal{B}}=\bigcup_{1\le k\le K} {\mathcal{D}}(w_k',r_1\exp(-H)),\quad
  K\lesssim B_0 H_0. $$ By part (2) of \cref{lem:resbasicprop1},
  \begin{equation*}
    \max(|P_1(z,w)|,|P_2(z,w)|)\ge \exp(-CHB_0^2H_0)
  \end{equation*}
  for any $ w\in {\mathcal{D}}(0,r_1/4)\setminus {\mathcal{B}} $ and $ z\in \mathbb{C} $. Using \cref{eq:g-lbub}, we get
  \begin{equation}\label{eq:F-lb}
    |F(z,w)|\gtrsim \exp(-CHB_0^2H_0-CB_0)\ge \exp(-HB_0^2H_0)
  \end{equation}
  for any $ w\in {\mathcal{D}}(0,r_1/4)\setminus {\mathcal{B}} $ and $ z\in {\mathcal{D}}(0,\rho_0) $.
  Applying Lemma~\ref{lem:6.3} again
  in the direction of $ \mathfrak{e}_2 $ (and with $ \mathfrak{e}_1 $ as the choice of non-collinear direction) and repeating the
  above argument we get that there exist  $ 1/8<\tilde \rho_0<1/4 $, $ \tilde r_1=\exp(-CB_0) $, such that
  \cref{eq:F-lb} also holds for any $ z\in {\mathcal{D}}(0,\tilde r_1/4)\setminus \tilde {\mathcal{B}} $
  and $$ w\in {\mathcal{D}}(0,\tilde \rho_0),\quad  \tilde {\mathcal{B}}=\bigcup_{1\le \ell \le L} {\mathcal{D}}(z_\ell',\tilde r_1\exp(-H)),\quad
  L\lesssim B_0H_0, $$ In particular, \cref{eq:F-lb} holds for any
  \begin{equation*}
    (z,w)\in {\mathcal{D}}(0,\tilde r_1/4)\times {\mathcal{D}}(0, r_1/4)
    \setminus \left( \bigcup_{k,\ell} {\mathcal{D}}(z_\ell',\tilde r_1\exp(-H))\times{\mathcal{D}}(w_k',r_1\exp(-H))\right).
  \end{equation*}
  Going back to standard coordinates we obtained that there exist less than $ CB_0^2H_0^2 $
  points $ \underline v_j' $ such that
  \cref{eq:F-lb} holds for any
  \begin{equation*}
    (z,w)\in B(\underline v_0,\exp(-CB_0))\setminus \left( \bigcup_{j} B(\underline v_j',\exp(-H)) \right).
  \end{equation*}
  Since \cref{eq:F-lb} holds outside the initial $ \mbox{\rm Car}_{2,0} $ set, we only need to apply the above argument
  on $ K_1 $ balls covering the initial set to get the conclusion.
\end{proof}

We will need the following lemma for the proof of \cref{thm:A}.

\begin{lemma}\label{lem:complexzerosetextends}
  Let $f$ be analytic on the ball
  $B(0,1)$, ${\mathcal Z}=\{\underline v \in B(0,1): f(\underline v)=0\}$. Let ${\mathcal{B}}\in \mathop{\rm{Car}}\nolimits_{2,0}(H_1,K_1)$, $H\gg 1$, $ \log K\ll H $.
  If $ 0\in {\mathcal Z} $, then
  \[
    B(0,1/4)\cap {\mathcal Z} \setminus {\mathcal{B}}\neq \emptyset.
  \]
\end{lemma}
\begin{proof}
  We argue by contradiction. Assume $ B(0,1/4)\cap {\mathcal Z} \subset {\mathcal{B}} $. By the assumptions on $ {\mathcal{B}} $ we can find
  $ 1/8<r<1/4 $ such that $ {\mathcal{B}}\cap B(0,r) $ is compactly contained in $ {\mathcal{B}} $. Therefore the zero set of $ f $
  restricted to $ B(0,r) $ is compactly contained in $ B(0,r) $ and $ {\mathcal Z}\cap B(0,r) $ is a compact analytic
  variety in $ \mathbb{C}^2 $. This cannot be, because compact analytic varieties in $ \mathbb{C}^2 $ are necessarily
  finite sets (see for example \cite{Chi89
  ) and analytic functions of several variables cannot have
  isolated zeros (recall that $ 0\in {\mathcal Z} $).
\end{proof}

\begin{proof}[Proof of \cref{thm:A}]
  (1)
  Take $ \underline v_0\in B(0,1/8)\cap {\mathcal Z} $, $ {\mathcal Z}=\{ f_2=0 \} $, $ 0<R\le 1/4 $. Let $ H=C\max(\log R^{-1},C_0) $
  with $ C $ large enough (recall that $ C_0=\log (K_1 B_0^2 H_0^2) $). By \cref{thm:C} we have
  \begin{equation*}
    |F(\underline v)|\ge \exp(-HB_0^2H_0)
  \end{equation*}
  for all $ \underline v\in B(0,1/4)\setminus {\mathcal{B}} $, $ {\mathcal{B}}\in \mbox{\rm Car}_{2,0}(H,K) $, $ K\lesssim K_1B_0^2H_0^2 $.
  Note that $ B(\underline v_0,R)\subset B(0,1/4) $ and our choice of $ H $ is such that we can apply
  \cref{lem:complexzerosetextends} to $ f_2 $ restricted to $ B(\underline v_0,R) $ and the above $ {\mathcal{B}} $ (after an obvious
  rescaling). So,
  there exists $ \underline v_1\in {\mathcal{B}}(\underline v_0,R/4)\cap {\mathcal Z}\setminus {\mathcal{B}} $. Note that we have
  \begin{equation*}
    |f_1(\underline v_1)|=|F(\underline v_1)|\ge \exp(-HB_0^2H_0)
  \end{equation*}
  and therefore
  \begin{equation*}
    M_{f_1}({\mathcal Z},\underline v_0,R/4)\ge -HB_0^2 H_0.
  \end{equation*}
  The first statement now follows by recalling that
  \begin{equation*}
    M_{f_1}({\mathcal Z},\underline v_0,R)\le M_{f_1}(0,1)\le 0.
  \end{equation*}

  (2) Take $ \underline v_0\in B(0,1/8) $. By our assumptions
  \begin{equation*}
    B_{f_2}(1/4;\underline v_0,1/4)\le B_0,\quad M_{f_2}(\underline v_0,1/4)\le 0.
  \end{equation*}
  Due to \cref{lem:6.3c}, we can find a unit vector $ \mathfrak{e}_1$,
  that is $ \underline m $-regular at $ \underline v_0 $ for $ f_2 $ restricted to $ B(\underline v_0,1/4) $, with $ \underline m=-CB_0 $,
  $ C\gg 1 $. Let $ \mathfrak{e}_2 $ be another unit vector orthogonal to $ \mathfrak{e}_1 $.
  As in
  Lemma~\ref{lem:6.3}, with a slight abuse of notation we denote by $ f_i(z,w) $ the functions in the coordinates
  with respect to the basis $ \mathfrak{e}_1,\mathfrak{e}_2 $ centered at $ \underline v_0 $ and with the obvious rescaling needed to apply
  the lemma.   Applying Lemma~\ref{lem:6.3}
  in the direction of $ \mathfrak{e}_1 $ (with $ \mathfrak{e}_2 $ as the choice of non-collinear direction)
  we can write
  \begin{align*}
    f_2(z,w) &= P(z,w) g(z,w),\\
    P(z,w)
             & = z^{k} +a_{k-1} (w) z^{k-1} + \cdots + a_0 (w)
  \end{align*}
  on $ {\mathcal{P}}:={\mathcal{D}}(0,\rho_{0})\times {\mathcal{D}}(0,r_1) $, $ 1/8<\rho_{0}<1/4 $, $ r_1=\exp(-CB_0) $, where
  the coefficients $ a_{j}(w) $ are analytic
  on $ {\mathcal{D}}(0,r_1) $ and $ k\lesssim B_0 $. Since we also have that  $ g $ is analytic and non-vanishing on $ {\mathcal{P}} $,
  \begin{equation*}
    {\mathcal Z}\cap {\mathcal{P}}={\mathcal Z}_P\cap {\mathcal{P}},\quad {\mathcal Z}_P:= \{ (z,w)\in \mathbb{C}\times {\mathcal{D}}(0,r_1) : P(z,w)=0 \}.
  \end{equation*}
  It is well known (see \cite{Chi89}) that for any point $ (z,w) $ of the variety
  $ {\mathcal Z}_P $, there exist  $ \epsilon>0 $, $ \delta>0 $ such that the following statements hold.
  \begin{enumerate}[leftmargin=1.5em,label=(\roman*)]
  \item
    If $ (z,w) $ is a regular point, then there exists an analytic function
    $ \zeta: {\mathcal{D}}(0,\epsilon)\to {\mathcal{D}}(0,\delta) $
    such that
    \begin{equation*}
      {\mathcal Z}_P\cap \left( {\mathcal{D}}(z,\delta)\times {\mathcal{D}}(w,\epsilon) \right)= \{ (z+\zeta(w'-w),w') : w'\in {\mathcal{D}}(w,\epsilon) \}.
    \end{equation*}
  \item
    If $ (z,w) $ is a singular point, then there exist integers $ p_i\ge 1 $ and analytic functions
    $ \zeta_i: {\mathcal{D}}(0,\epsilon)\to {\mathcal{D}}(0,\delta) $, $ 1\le i\le i_0(z,w)\le k $, such that $ \sum_i p_i\le k $ and
    \begin{equation*}
      {\mathcal Z}_P\cap \left( {\mathcal{D}}(z,\delta)\times {\mathcal{D}}(w,\epsilon) \right)
      = \bigcup_i\{ (z+\zeta_i((w'-w)^{\frac{1}{p_i}}),w') : w'\in {\mathcal{D}}(w,\epsilon) \}.
    \end{equation*}
  \end{enumerate}

  By compactness we can cover $ B(0,1/8)\cap {\mathcal Z} $ by finitely many polydisks  $ \frac{1}{2}{\mathcal{P}} $ (more precisely,
  by their preimages under the change of variables we assumed above) and in turn
  $ {\mathcal Z}\cap \frac{1}{2}{\mathcal{P}} $ can be covered by finitely many polydisks
  $ {\mathcal{D}}(z_j,\delta_j)\times {\mathcal{D}}(w_j,\epsilon_j/8)$ with $ (z_j,w_j)\in {\mathcal Z}_P $ and $ \epsilon_j,\delta_j $ as above.
  We will also use $ \zeta_j $ and  $ \zeta_{i,j} $ the functions associated with $ (z_j,w_j) $.
  Let $ r_0>0 $ be the minimum over all the $ \epsilon_j $ needed to cover $ B(0,1/8)\cap {\mathcal Z} $. Near each
  $ (z_j,w_j) $ we will define
  local charts and show we can control the Bernstein exponent of $ f_1 $ in the local charts. The control over the
  Bernstein exponent will follow from \cref{thm:C} and \cref{prop:bernstein-analytic}. To this end we take
  $ H=C(\log r_0^{-1}) C_0 $, with $ C\gg 1 $ large enough and we note that, with this choice of $ H $,
  \cref{thm:C} guarantees that
  \begin{equation}\label{eq:F-lb-2}
    |F(z,w)|\ge \exp(-HB_0^2H_0),\ \forall (z,w)\in {\mathcal{P}}\setminus (\mathbb{C}\times {\mathcal{B}})
  \end{equation}
  where $ {\mathcal{B}} $ is a union of disks with the sum of the radii much smaller than $ r_0^k $ (recall that
  $ k\lesssim B_0\ll H_0 $).
  To define the charts we distinguish two cases.

  \noindent (i) $ (z_j,w_j) $ is regular. Let
  \begin{equation*}
    \psi_j(w)=(z_j+\zeta_j(w),w_j+w),\  w\in {\mathcal{D}}(0,\epsilon_j).
  \end{equation*}
  It follows from \cref{eq:F-lb-2} that
  \begin{equation*}
    M_{f_1\circ \psi_j}(0,1/4)\ge -HB_0^2H_0.
  \end{equation*}
  By \cref{prop:bernstein-analytic} (recall that $ M_{f_1}(0,1)\le 0 $) it is clear that
  \begin{equation*}
    B_{f_1\circ \psi_j}(1/4;z,r)\lesssim H\le C(f_2)C_0 B_0^2H_0
  \end{equation*}
  when $ {\mathcal{D}}(z,r)\subset {\mathcal{D}}(0,\epsilon_j/8) $. This shows the conclusion of part (2) holds if we define the local
  chart by rescaling $ \psi_j\vert_{{\mathcal{D}}(0,\epsilon_j/8)} $.

  \noindent (ii) $ (z_j,w_j) $ is singular. Let
  \begin{equation*}
    \psi_{i,j}(w)=(z_j+\zeta_{i,j}(w),w_j+w^{p_i}),\ w\in {\mathcal{D}}(0,\epsilon_j).
  \end{equation*}
  It follows from \cref{eq:F-lb-2} that
  \begin{equation*}
    M_{f_1\circ \psi_{i,j}}(0,1/4)\ge -HB_0^2H_0
  \end{equation*}
  (recall that $ p_i\le k $) and therefore
  \cref{prop:bernstein-analytic} guarantees that
  \begin{equation*}
    B_{f_1\circ \psi_{i,j}}(1/4;z,r)\lesssim HB_0^2H_0\le C(f_2)C_0 B_0^2H_0
  \end{equation*}
  when $ {\mathcal{D}}(z,r)\subset {\mathcal{D}}(0,\epsilon_j/8) $. This shows that the conclusion holds if corresponding to each
  $ w\in {\mathcal{D}}(0,\epsilon_j/8)\setminus \{ 0 \} $ we define a local chart  by rescaling
  $ \psi_{i,j}\vert_{{\mathcal{D}}(w,r)} $, where $ {\mathcal{D}}(w,r) $ is the largest disk about $ w $ in $ {\mathcal{D}}(0,\epsilon_j/8) $ on
  which $ w^{p_i} $ is one-to-one.

  Clearly the above charts cover $ \reg {\mathcal Z} \cap B(0,1/8) $ and we can complete an atlas of $ \reg {\mathcal Z} $ by adding
  charts whose ranges don't intersect $ B(0,1/8) $. This concludes the proof.
\end{proof}

Next we give an example showing that the $ \log R^{-1} $ is actually necessary in part (1) of \cref{thm:A}.

\begin{example}\label{ex:logR}
  Let
  \begin{equation*}
    f_1(z,w)=z^2+w,\quad f_2(z,w)=zw,\quad {\mathcal Z}= \{ f_2=0 \}.
  \end{equation*}
  Let $ R\ll 1 $, $ \underline v_0=(R/4,0) $. Then straightforward computations show that
  \begin{equation*}
    \sup_{{\mathcal{B}}(\underline v_0,R/4)\cap {\mathcal Z}}\log|f_1(z,w)|=\sup_{|z-R/4|<R/4}\log|z^2|=\log \left( \frac{R}{2} \right)^2,
  \end{equation*}
  and
  \begin{multline*}
    \sup_{{\mathcal{B}}(\underline v_0,R)\cap {\mathcal Z}}\log|f_1(z,w)|
    =\max \left( \sup_{|z-R/4|<R}\log|z^2| ,\sup_{|w|^2+(R/4)^2<R^2}\log|w| \right)\\
    =\max \left( \log \left( \frac{5R}{4} \right)^2,\log \frac{\sqrt{15}R}{4} \right)
    = \log \frac{\sqrt{15}R}{4},
  \end{multline*}
  provided $ R $ is small enough ($ R<1/2 $ is enough). Therefore,
  \begin{equation*}
    B_{f_1}(1/4;{\mathcal Z},\underline v_0,R)= C+\log R^{-1}.
  \end{equation*}
\end{example}

Finally, we will prove \cref{thm:B}, but we first establish an auxilliary result.
To this end we will need the following extension of the classical  B\'ezout theorem.
Suppose we have a system of $ n $ complex polynomial equations $ f_i(z_1,...,z_n)=0 $, $i=1,..,n$.
Let $ {\mathcal Z}_1,\ldots,{\mathcal Z}_s $ be the irreducible components of the variety defined by the system.
Then
\begin{equation}\label{eq:Bezout}
  \deg({\mathcal Z}_1)+\dots+\deg({\mathcal Z}_s)\le \deg f_1\times \dots \times \deg f_n.
\end{equation}
The authors are grateful to J\'anos Koll\'ar  and Mihnea Popa for pointing out this version of the B\'ezout
bound (for a more general result see \cite[Thm.~12.3]{Ful98}).

\begin{lemma}\label{lem:bezoutpuiseoux1}
  Let $f(z,w)$, $g(z,w)$ be non-constant polynomials with no common factors.
  Let $\zeta(w)$ be an analytic function on $ {\mathcal{D}}(0,r_0)$
  such that
  \begin{equation}\label{eq:regularity-hyp}
    \{ (\zeta (w),w) : w\in {\mathcal{D}}(0,r_0) \}\subset \reg \{ f(z,w)=0 \}.
  \end{equation}
  Then there exists at most one straight line ${\mathcal{L}}\subset \mathbb{C}$ through
  the origin such that
  \begin{equation}\label{eq:bezoutpuiseux2}
    \#\{\xi\in (-r_0,r_0):g(\zeta(\xi ),\xi )\in {\mathcal{L}}\}> (\deg f)^2\deg g.
  \end{equation}
\end{lemma}
\begin{proof}
  Let $ {\mathcal{L}} $ be a line through the origin. We first argue that if \cref{eq:bezoutpuiseux2} holds, then we must
  have $ \{ g(\zeta (\xi),\xi): \xi\in (-r_0,r_0) \}\subset {\mathcal{L}} $.
  Write
  \begin{equation*}
    f(z,\xi):=P(x+iy,\xi)+iQ(x+iy,\xi)=\hat P(x,y,\xi)+i\hat Q(x,y,\xi)
  \end{equation*}
  where $P,Q$ are the real and imaginary parts of $f$, and
  $\hat P,\hat Q$ are real polynomials of three real variables $x,y,\xi$.
  Clearly $ \deg \hat P=\deg \hat Q= \deg f $.
  Similarly write
  \begin{equation*}
    g(z,\xi):=U(x+iy,\xi)+iV(x+iy,\xi)=\hat U(x,y,\xi)+i\hat V(x,y,\xi).
  \end{equation*}
  Without loss of generality we may assume that the line
  $ {\mathcal{L}} $ is horizontal. If \cref{eq:bezoutpuiseux2} holds,
  then the system
  \begin{equation}\label{eq:system}
    \hat P=0,\quad \hat Q=0,\quad \hat V=0
  \end{equation}
  has more than $\deg \hat P\times\deg \hat Q \times \deg \hat V$ solutions $\underline v_j=(x_j,y_j,\xi_j)$ with
  \begin{equation*}
    \xi_j\in (-r_0,r_0),\quad x_j+iy_j=\zeta (\xi_j),\quad \xi_{j_1}\neq \xi_{j_2}.
  \end{equation*}
  Complexify the variables $x,y,\xi$, and let $ {\mathcal Z}_1,\ldots,{\mathcal Z}_s $ be the irreducible components of the complex
  variety defined by the system \cref{eq:system}. By the B\'ezout bound \cref{eq:Bezout}, there exists a component
  $ {\mathcal Z}_k $ that contains at least two of the solutions $ \underline v_j $ and therefore has dimension at least one.
  Let $ \underline v_0 $ be one of the solutions contained in $ {\mathcal Z}_k $.
  We will argue that there exists an analytic mapping
  \begin{equation}\label{eq:mapping}
    t\to \underline v(t)=(x(t),y(t),\xi(t))\in {\mathcal Z}_k,\  t\in {\mathcal{D}}(0,\delta)
  \end{equation}
  such that $ \underline v(0)=\underline v_0 $ and $ \xi(t) $ is non-constant. By \cite{Shi70} we know that
  there exists a neighborhood $ N $ of $ \underline v_0 $ in $ {\mathcal Z}_k $, such that for any
  $ \underline v\in N\setminus \{ \underline v_0 \} $, there exists a one dimensional irreducible variety $ \mathcal{V} $  through both
  $ \underline v $ and $ \underline v_0 $. Since $ \mathcal{V} $ can be parametrized by a Riemann surface (see \cite[Prop.~6.2]{Chi89})
  we get the existence of a mapping of the form \cref{eq:mapping}. If $ \xi(t) $ is constant, then by the
  uniqueness theorem (see \cite[Cor.~5.3.2]{Chi89}), we must have $ \mathcal{V}\subset \{ \xi=\xi_0 \} $.
  If this happens for all such mappings obtained by choosing different $ \underline v\in N\setminus \{ \underline v_0\} $, then
  $ N\subset \{ \xi=\xi_0 \} $, and by the uniqueness theorem, $ {\mathcal Z}_k\subset \{ \xi=\xi_0 \} $.
  This would contradict
  the fact that $ {\mathcal Z}_k $ contains two of the solutions $ \underline v_j $ (recall that $ \xi_{j_1}\neq \xi_{j_2} $). So we
  proved the existence of the mapping \cref{eq:mapping} with the desired properties. We have
  \begin{equation*}
    f(x(t)+iy(t),\xi(t))=0,\quad V(x(t)+iy(t),\xi(t))=0,\ t\in {\mathcal{D}}(0,\delta).
  \end{equation*}
  By the assumption \cref{eq:regularity-hyp}, we get
  \begin{equation*}
    x(t)+iy(t)=\zeta (\xi(t)),
  \end{equation*}
  provided we choose $ \delta $ small enough. Therefore
  \begin{equation*}
    V(\zeta(\xi(t)),\xi(t))=0,\ t\in {\mathcal{D}}(0,\delta)
  \end{equation*}
  and since $ \xi(t) $ is non-constant, there exists $ \epsilon>0 $ so that
  \begin{equation*}
    V(\zeta(\xi),\xi)=0,\ \xi\in (\xi_0-\epsilon,\xi_0+\epsilon).
  \end{equation*}
  So $ V(\zeta (\xi),\xi)=0 $ for all $ \xi\in(-r_0,r_0) $, that is
  $ \{ g(\zeta(\xi),\xi): \xi\in (-r_0,r_0) \}\subset {\mathcal{L}} $.

  Now we can finish the proof by arguing by contradiction.
  If the conclusion doesn't hold, it follows that we have
  $ \{ g(\zeta(\xi),\xi): \xi\in (-r_0,r_0) \}\subset {\mathcal{L}}_1\cap {\mathcal{L}}_2= \{ 0 \} $ and
  therefore the system $ f=g=0 $ has infinitely many solutions. By the classical
  B\'ezout theorem, this would contradict the assumption that $ f $ and $ g $
  don't have common factors.
\end{proof}

\begin{proof}[Proof of \cref{thm:B}]
  Let $ {\mathcal Z}_1,\ldots,{\mathcal Z}_s $ be the irreducible components of $ {\mathcal Z} $. Each of them is the zero set of
  an irreducible factor of $ f_2 $. Let $ f_{2,1},\ldots,f_{2,s} $ be such irreducible factors.
  Fix $ k\in \{ 1,\ldots,s \} $ and $ (z_0,w_0)\in \reg {\mathcal Z}\cap {\mathcal Z}_k $.
  We can make a change of variables (as in the proof of part (2) of \cref{thm:A}) such that
  $ (z_0,w_0) $ is mapped to the origin and
   we can find an analytic function
  $ \zeta: {\mathcal{D}}(0,\epsilon_0)\to {\mathcal{D}}(0,\delta_0) $ so that
  \begin{equation*}
    \phi(w)=(\zeta(w),w),  w\in {\mathcal{D}}(0,\epsilon_0)
  \end{equation*}
  is a chart for $ \reg {\mathcal Z} \cap {\mathcal Z}_k $ around the origin.

  If $ f_{2,k} $ divides $ f_1 $, then $ f_1 $ vanishes identically on $ {\mathcal Z}_k $, and its Bernstein exponent
  is $ 0 $ by convention (in any chart). So, we just need to treat the case when
  $ f_{2,k} $ and $ f_1 $ have no common factors. Let $ \psi(w)=f_1(\phi(w)) $. We
  claim that
  \begin{equation}\label{eq:Bern-claim}
    B_\psi(1/4;w_0,R)\le C(f_2) \deg f_1
  \end{equation}
  provided $ {\mathcal{D}}(w_0,R)\subset {\mathcal{D}}(0,\epsilon_0/8) $. We will check this claim by using the previous lemma and
  \cref{prop:bernstein-analytic}. Let $ a_1,\ldots,a_n $ be the zeros of $ \psi $.
  Since $ f_1 $ and $ f_{2,k} $
  are co-prime, using the classical B\'ezout theorem, we have
  \begin{equation*}
    n\le \deg f_{2,k}\times \deg f_1.
  \end{equation*}
  Factorize
  \begin{equation*}
    \psi(w)=h(w)P(w),\quad P(w)=\prod_{k=1}^n(w-a_k).
  \end{equation*}
  From the proof of
  \cref{prop:bernstein-analytic} (see \cref{eq:Bern-h-P}) we have
  \begin{equation*}
    B_\psi(1/4;w_0,R)\le CR(M_h(0,3\epsilon_0/4)-M_h(0,\epsilon_0/4))+B_P(1/4;w_0,R).
  \end{equation*}
  Recall that $ B_P(1/4;w_0,R)\le n\log 4\le C(f_2)\deg f_1 $. So, to check the claim \cref{eq:Bern-claim}
  we just need to estimate $ M_h(0,3\epsilon_0/4)-M_h(0,\epsilon_0/4) $. Without loss of generality we can
  assume $ h(0)=1 $ and therefore $ M_h(0,\epsilon_0/4)\ge 0 $.
  Since $h $ does not vanish we have
  \[
    h(w)=e^{u(w)+iv(w)},
  \]
  where $u+iv$ is analytic and $ u,v $ are real-valued. Then
  \begin{equation*}
    M:=M_h(0,3\epsilon_0/4)=\sup_{w\in {\mathcal{D}}(0,3\epsilon_0/4)}|u(w)|.
  \end{equation*}
  Due to the Borel-Carath\'eodory estimate (see \cite[Thm.~11.1.1]{Lev96})
  \begin{equation*}
    N:=\sup_{w\in {\mathcal{D}}(0,7\epsilon_0/8)} |v(w)|\gtrsim M.
  \end{equation*}
  Choose $|\hat w|= 7\epsilon_0/8$ such that $|v(\hat w)|\ge N/2$ and at the same time no root
  $a_k$ falls on the straight line  through $\hat w$ and the origin. This allows us to define the
  continuous functions
  $\theta_k(\xi):=\arg (\xi \hat w-a_k)\in [0,2\pi]$, $\xi\in (-\infty,+\infty)$. Set
  \[
    \theta(\xi)=\sum_{1\le k\le n} \theta_k(\xi).
  \]
  Take $ \theta\in (0,2\pi) $ arbitrary. We have
  \begin{equation*}
    \Im e^{-i\theta}\psi(\xi \hat w )=e^{u(\xi \hat w)}|P(w)|\sin (v(\xi \hat w)+\theta(\xi)-\theta).
  \end{equation*}
  It is clear form this formula that if $ N\gg n $, then for any $ \theta $,
  \begin{equation*}
    \# \{ \xi\in (-7\epsilon_0/8,7\epsilon_0/8) : f_1(\zeta(\xi \hat w),\xi \hat w)\in {\mathcal{L}}_{\theta} \}
    \ge N/4,
  \end{equation*}
  where $ {\mathcal{L}}_{\theta} $ is the line of angle $ \theta $ through the origin.
  This and Lemma~\ref{lem:bezoutpuiseoux1} imply that we must have
  \[
    N\lesssim (\deg f_{2,k})^2 \deg f_1.
  \]
  Putting the above together we have
  \begin{equation*}
    M_h(0,3\epsilon_0/4)-M_h(0,\epsilon_0/4)\lesssim C(f_2) \deg f_1,
  \end{equation*}
  which completes the proof of claim \cref{eq:Bern-claim}.

  Finally, it is clear that the conclusion holds by choosing the charts to be rescaled versions of
  $ \phi\vert_{{\mathcal{D}}(0,\epsilon_0/8)} $, for each $ (z_0,w_0)\in \reg {\mathcal Z} $.
\end{proof}

\def$'${$'$}

redpajama_set_name: RedPajamaArXiv


The Vintage Racer Group (VRG) club and Summit Point Motorsports Park will hold the "Jefferson 500" at Summit Point on May 15 - May 18, 2014. This multi-day racing event, sanctioned by the VRG, has been one of the Northeast's largest vintage races.
During the Jefferson 500 event, on Saturday, May 17, the Country Roads Car Club will be holding a car show on the grounds of Summit Point. The $15.00 car show registration fee provides both entry into Summit Point and the car show. Also car show participants will be eligible to sign-up for one of two short three (3) lap parade sessions (12:15pm or 5:15pm, not both) on the track. The car show times are listed for 10:00am – 4:00pm. But the car show organizers recommend that entrants arrive between 8:30am-9:00am to get parked on the show field and signed-up for the parade laps.
For those POG members interested in going to the Jefferson 500 and entering the car show as a group, we will leave in a caravan from the Reston Town Center to Summit Point. We will gather at the Reston Town Center surface parking lot closest to La Madeleine Bakery & Café at 7:30am. (La Madeleine opens at 6:30am.) We will be wheels up no later than 7:45am and driving west via the Toll Road and the Greenway on our way to VA Route 7. This should make our arrival at Summit Point Raceway around 8:45am.


text:
Robert Cicchetti/Getty Images
The GOP controls half the state governments, most of them with supermajorities in the legislature. However, absent a plan for surgery by the conservative base, these red states will continue to operate as lobbyist havens for the woke corporations with just a pro-gun tinge to an otherwise left-of-center state government. We should have at least 20 states that are our equivalents of California, but with the absence of meaningful Democrat numbers in deep red states, many Republicans operate as proxy Democrats, thereby giving a governing majority to the left. Are we on the cusp of this finally changing?
With a 29-2 GOP majority in the Wyoming Senate and a 57-5 majority in the House, there aren't even enough Democrats to populate the committees. It's a reflection of the state voting for the Republican presidential candidate typically by 40+-point margins since 1996. Wyoming should easily be our version of California. Yet thanks to the fact that half the Republicans are really Democrats, it's often more like their version of California, with the exception of legacy issues like guns and abortion.
The core of what it means to be a conservative is to conserve the most basic individual liberties. It should be clear that forcing an individual to cover his mouth or inject a substance that alters his immunological system is a non-starter in the Equality State, which supposedly exudes a sense of rugged individualism. Yet to this day, children are forced to get any shot pharma deems necessary in order to attend school. And to this day, there hasn't even been a bill meaningfully limiting lockdowns and emergency powers, much less one banning COVID shots and masks, despite two years of data repudiating them.
Enter HB66 and the House Labor, Health, and Social Services Committee. It's a very simple bill that should be the baseline law in every state. It prohibits any public or private institution from forcing a jab or a mask on any human being, including in schools and hospitals, where every state continues, to some degree, to push these immoral and inhumane policies.
Let's put aside the safety issues with masks and the COVID shots specifically. What any of us should have learned from the past few years is that there is never a legal or moral justification to force another person to utilize even an effective medical procedure. If it's so safe and effective to justify such a mandate, then the person who wants it has the prerogative to go and utilize it himself, regardless of the other person's choice. If we are at the point where we can force another human being to take an action against his own body in order to function in society, there is quite literally no limit to government's power and no contours of human liberty preserved. This is why even more accepted and established vaccines should never be forced on someone. If they work so well, that is the loss of the refuser.
Yet with an 8-1 GOP majority on the House Labor Committee, the one Democrat, Mike Yin, was able to work with the committee chair and other Republicans last week to gut the bill. The prohibition on mandates was limited to just COVID shots, and then the enforcement mechanism – criminal penalties with up to a year in prison – were removed from the bill. Even with the slimmed-down bill, it only passed 5-4, and one of the Republicans, Rep. Kevin O'Hearn, indicated he'd vote against it on the floor.
Rep. Ken Clouston, one of the GOP no votes, indicated that he supported the effort but was concerned about severe COVID variants arising in the future.
"We've lost the trust in the CDC; a lot of our personal rights were violated," he said. "We haven't controlled it, these vaccines haven't worked. I just don't want to handcuff us in the future that if something worse with COVID comes on that we're not able to take precautions that we may need."
What in the world does that have to do with anything? The shots and masks do not work and cannot work for any variant. Moreover, he is missing the entire point. Even if something does work, why should that be forced upon another person?
What is the core definition of a fundamental human right? During the floor debate over the 14th Amendment, Senator Jacob Howard described a fundamental right affecting an aspect of a citizen "lying at the basis of all society and without which a people cannot exist except as slaves, subject to a despotism."
Very few things truly meet that definition. Most of what we consider as "muh rights" today don't necessarily fit that description. But when it comes to biologically altering medical procedures on or in our bodies, we quite literally cannot live freely if any government has the right to force that upon us. The harm to liberty is irreparable, whereas any benefit to society from a potentially "safe and effective" mandate would only be lost by the individual who refuses to participate.
For example, a meta-analysis of 40 pooled studies found that vitamin D supplementation correlated with a 65% reduction of risk for ICU admission. Thus far, there have been 104 treatment and 144 sufficiency studies showing a positive association with COVID outcomes, many of them showing that a high vitamin D level reduces the chance of even contracting COVID, something even the most ardent supporters of the vaccine acknowledge it doesn't do. So, should we mandate that every hospital worker and school child supplement with vitamin D and take a test to ensure his serum level is above 50 before entering?
Obviously, that should be a personal choice. People who believe in it so strongly that they are inclined to coerce others to join them have the ability to unilaterally seek that protection for themselves without drafting others into their medical choices. The same should apply to all vaccines and certainly the COVID therapies. The notion that your bodily autonomy should be subservient to "the greater good" of a genetic purity medical nationalism is a road that dead-ends in a dark place.
So here we have a shot that should have been taken off the market a long time ago, yet we are struggling to get a basic anti-mandate bill passed in the reddest state in America. This is because conservatives have failed to focus on legislative affairs in their own back yards for decades. This has allowed the lobbyists and special interests like the medical cartel to rule the roost.
There are signs that things are changing. The sponsor of HB66, Jeanette Ward, represents a new crop of legislators getting elected to be responsive to the people's interests, not the corrupt and dangerous medical cartel. Wyoming, like ten other states, now has a state freedom caucus seeking to push back against special interests and expose the fraud of sleeper Democrats running as Republicans.
The question is whether the people who voted against Biden for president by a 43-point margin rise up and demand that the political representation reflect their values and not what they so resoundingly repudiated.
Horowitz: The failure of the mRNA shots is on display for all with open eyes
Horowitz: NH Senate passes bill that could empower hospitals to arrest family members who argue with doctors
FACT CHECK: Did A TV Station Post This Tweet Linking Gas Stoves And Cardiovascular Events?
checkyourfact.com
Your Taxes Paid Thousands Of Government Staff Covid Unemployment While They Worked From Home
Supreme Court's cowardice allows Colorado to keep persecuting Christians
When Study Shows Transgender Medicine Hurts Kids, Activists And Media Allies Slant The Science
Records Show Interior Dept. Officials Knew Climate Protesters Might Turn Violent But Didn't Stop Them
Classified Documents Are A New Potential Trap For Any Politician Who Crosses The Deep State

redpajama_set_name: RedPajamaCommonCrawl


On this page you can download the audio track for the movie WALL-E russian lang. Russian dubbing audio track will allow you to fully enjoy movies WALL-E 2008 year. Download audio track WALL-E for movie for free and see the detailed characteristics of track can be below.


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Q: How to create a Minecraft chunk in opengl? I'm learning OpenGL and I have tried to make a voxel game like Minecraft. In the beginning, everything was good. I created a CubeRenderer class to render a cube with its position. The below picture is what I have done.
https://imgur.com/yqn783x
And then I got a serious problem when I try to create a large terrain, I hit a slowing performance. It was very slow and fps just around 15fps, I thought.
Next, I figured out Minecraft chunk and culling face algorithm can solve slowing performance by dividing the world map into small pieces like chunk and just rendering visible faces of a cube. So how to create a chunk in the right way and how the culling face algorithm is applied in Chunk?
So far, that is what I have tried
I read about Chunk in Minecraft at https://minecraft.gamepedia.com/Chunk
I created a demo Chunk by this below code (it is not the completed code because I removed it out)
I created a CubeData that contains cube position and cube type.
And I call the GenerateTerrain function to make a simple chunk data (16x16x16) like below (CHUNK_SIZE is 16)
for (int x = 0; x < CHUNK_SIZE; x++) {
        for (int y = 0; y < CHUNK_SIZE; y++) {
            for (int z = 0; z < CHUNK_SIZE; z++) {
                CubeType cubeType = { GRASS_BLOCK };
                Location cubeLocation = { x, y, z };
                CubeData cubeData = { cubeLocation, cubeType };
                this->Cubes[x][y][z] = cubeData;
            }
        }
    }

After that, I had a boolean array which is called "mask" contains two values are 0 (not visible) or 1 (visible) and matches with their cube data. And then I call Render function of Chunk class to render a chunk. This code below like what I have done (but it is not complete code because I removed that code and replaced with new code)
for (int x = 0; x < CHUNK_SIZE; x++) {
        for (int y = 0; y < CHUNK_SIZE; y++) {
            for (int z = 0; z < CHUNK_SIZE; z++) {
                for(int side = 0; side < 6;side++){
if(this->mask[x][y][z][side] == true) cubeRenderer.Render(cubeData[x][y][z]);
}
            }
        }
    }

But the result I got that everything still slow (but it is better than the first fps, from 15fps up to 25-30fps, maybe)
I guess it is not gpu problem, it is a cpu problem because there is too many loops in render call.
So I have kept research because I think my approach was wrong. There may have some right way to create a chunk, right?
So I found the solution that puts every visible verticle to one VBO. So I just have to call and bind VBO definitely one time.
So this below code show what I have tried
cout << "Generating Terrain..." << endl;

    for (int side = 0; side < 6; side++) {
        for (int x = 0; x < CHUNK_SIZE; x++) {
            for (int y = 0; y < CHUNK_SIZE; y++) {
                for (int z = 0; z < CHUNK_SIZE; z++) {
                    if (this->isVisibleSide(x, y, z, side) == true) {
                        this->cubeRenderer.AddVerticleToVBO(this->getCubeSide(side), glm::vec3(x, y, z), this->getTexCoord(this->Cubes[x][y][z].cubeType, side));
                    }
                }
            }
        }
    }

    this->cubeRenderer.GenerateVBO();

And call render one time at all.
void CubeChunk::Update()
{
    this->cubeRenderer.Render(); // with VBO data have already init above
}

And I got this:
https://imgur.com/YqsrtPP
I think my way was wrong.
So what should I do to create a chunk? Any suggestion?

redpajama_set_name: RedPajamaStackExchange


It must have been quite a scene! A man riding on a donkey is accompanied by some followers who are praising God in loud voices and spreading their cloaks on the road in front of him! [Other versions refer to the palm branches ( cf Mark 11:8), but this year is Luke's year and even if he doesn't mention them, we will still receive a piece of palm!] No doubt the Pharisees are worried by the content of the praises as well as casting a wary eye at any Roman soldiers who might intervene! Riots were common during Passover in Jerusalem! Such a tumultuous welcome is quite a contrast to our orderly Palm Sunday processions. (I have heard of congregations that manage to find a donkey for the pastor or someone to ride!) In the end, we need to gather our thoughts and our imagination to allow ourselves to enter into this grand scene that begins the sacred "drama" of Holy Week. In a few minutes, while holding that "palm of praise," we will be "yelling" "CRUCIFY HIM! CRUCIFY HIM!" Those words ought to give us pause, for sure!!!! It is our loud cries that convince a skeptical Pilate to sentence Jesus to death!


Evil Academy - Migrants attack trucks in Calais.
Evil Academy > Discourse > Freethinker's Academy > Migrants attack trucks in Calais.
Full Version: Migrants attack trucks in Calais.
This seems to be getting way out of control. What is wrong with Europe letting this happen.
Migrants or so-called "refugees" in Calais show their true nature. They attack, harass and board trucks / trailers on their way to the U.K.
It's kind of cool that they are there giving medical treatment. It's not like they are the ones bringing them there.
Not sure what this has to do with what's happening in Calais?
You don't think the wars for the Greater Israel Project aren't displacing people?
What about the ISIS fighters Israel is treating in their hospitals?
What about Zionists like Barbara Spectre who admitted that this was the plan?
I was just looking at what the migrants were doing in Calais. Any thoughts about that or is Israel all that you can think of?
(06-25-2017 02:30 PM)pilgrim Wrote: [ -> ] .
What is wrong with Europe letting this happen.
I'm answering your question. Zionists want Europe to be multicultural.
Maybe that's what they wanted to happen with America until Trump stopped that plan. They were flooding America with South American, Central Americans and Mexicans under Obama.
Looking at the Calais refuge camp, I don't understand what the end goal is there. Shouldn't they be trying to integrate those people into society? It seems like leaving them in large camps like that only leads to the stuff we see in that video. What is Europe's plans for them? Do they plan on leaving them clustered like then and then shipping them back to their own country eventually? It just seem crazy what is going on there.
(06-25-2017 02:20 PM)jho Wrote: [ -> ] You don't think the wars for the Greater Israel Project aren't displacing people?
I would be interested in learning more about this Greater Israel Project? Any chance you could start a new thread explaining this?
Your video that you posted doesn't seem to work.
Quote: Spectre is a Zionist and head of Paideia - The European Institute for Jewish Studies in Sweden, an organisation funded by the Swedish government and the Wallenberg Foundation to promote Jewish ideology. Through an organisation called One Sweden, which she heads, she promotes multiculturalism.
Although born in the United States, Spectre moved to Palestine ("Israel") in 1967 where she worked at the Shalom Hartman Institute in Jerusalem, though she decided to move to Sweden in 1999. Spectre is married to rabbi Philip Spectre, who was the official rabbi at the Great Synagogue in Stockholm and the Chief Rabbi in Sweden during the early 2000s.


text:
Doubleday, Abner
Life Summary
Abner Doubleday
Ballston Spa, NY
Burial Place
Arlington National Cemetery, Arlington, VA
Birth Date Certainty
Death Date Certainty
Sectional choice
West Point (US Military Academy)
Relation to Slavery
White non-slaveholder
US military (Pre-Civil War)
Union Army
US military (Post-Civil War)
Abner Doubleday (American National Biography)
Although during his lifetime Doubleday was noted for his military accomplishments, the historical significance attributed to him has been based primarily on his supposed invention of the game of baseball. In 1905 a commission was established to determine whether baseball had uniquely American origins or was descended from rounders, a game of English origin….The de facto leader of the commission was Abraham G. Mills, a former president of the National League, who in this capacity spent two years collecting written anecdotal recollections from interested persons nationwide on the matter, but he acquired little substantive evidence to support either claim. One such anecdote that piqued Mills's interest was provided by Cooperstown resident Abner Graves, who testified that he and Doubleday were schoolmates and that in 1839 Doubleday redesigned a game played by local residents known as "town ball." Doubleday, he said, instituted a smaller number of participants and a new set of rules and renamed the game "base ball." Doubleday may have played a game similar to that of baseball as a child or teenager in Cooperstown, but his status in 1839 as a second-year cadet at West Point makes a prolonged appearance in Cooperstown at that time unlikely. Moreover, no record has been found, even in Doubleday's many writings, that he ever played baseball.
Brooks D. Simpson and Matthew E. Van Atta, "Doubleday, Abner," American National Biography Online, February 2000, http://www.anb.org/articles/05/05-00198.html.
04/20/1861 Remarkable gathering of 100,000 New Yorkers in Union Square in support of suppressing the rebellion
04/27/1863 - 04/30/1863 The Army of the Potomac concentrates on Chancellorsville in preparation for an attack on Lee
05/01/1863 Union and Confederate armies collide near Chancellorsville in Spotsylvania County, Virginia
05/02/1863 "Stonewall" Jackson's flanking movement seizes the initiative in the Battle of Chancellorsville
05/03/1863 Lee's Army of Northern Virginia forces back entrenched Union forces at the Battle of Chancellorsville
05/05/1863 - 05/06/1863 The beaten Union Army retreats across the Rappahannock, ending the Battle of Chancellorsville
06/30/1863 The leading First Corps of the Army of the Potomac camps four miles west of Gettysburg
07/01/1863 - 07/03/1863 Battle of Gettysburg
04/14/1865 In Charleston Harbor, General Robert Anderson re-hoists the United States flag over Fort Sumter
Gettysburg Campaign
04/14/1865 Reverend Henry Ward Beecher's remarks at the ceremony restoring the flag to Fort Sumter, South Carolina, April 14, 1865
Abner Doubleday, detail
How to Cite This Page: "Doubleday, Abner," House Divided: The Civil War Research Engine at Dickinson College, https://hd.housedivided.dickinson.edu/index.php/node/8969.


Sure Start Day Nursery based in Wooler Children's Centre has received an "outstanding" grade from industry watchdog Ofsted.
Inspected at the start of August, the setting has been rated outstanding in all three main areas.
The setting is currently amidst some controversial changes, which has seen them amalgamate with another setting. This means there is currently no provision to take on further 0-3 year olds, with the streamlined provision also causing some to lose their jobs.


text:
The DELTA GREEN: 1939-1945 is a companion site to the DGWW2 Yahoogroups mailing list, which provides discussion, research, and resources for setting CALL OF CTHULHU and DELTA GREEN campaigns during the Second World War. The contents of this site include articles, histories, biographies, chronologies, new rules, agency templates, equipment lists, scenarios, links, bibliographies and more.
Subscribe to the dgww2 mailing list
Powered by groups.yahoo.com
Our Darkest Hour
Why there isn't as much here as we both might wish
Histories, biographies, and chronologies
New rules for character creation, combat, vehicles, and new skills
Weapons, uniforms, and vehicles
Ideas for one-shots and campaign play
On-line links and bibliographies
02/20/08 Game stats for Karotechia security personnel has been added to the Campaign section.
03/24/04 The Book of Phrases has been updated to include the other Russian phrases.
12/20/03 I now have a blog to track my design work on Our Darkest Hour.
11/28/03 The Book of Phrases has been updated to include some Russian phrases.
11/13/03 The Book of Phrases has been updated to include Italian phrases.
07/23/02 The Office of Strategic Services has been added to the Order of Battle.
06/06/02 Mark Romaniw has written a history of the Sluzhba Bezpeky, the Security Service of the Organization of Ukrainian Nationalists, and its involvement with the Mythos. The writeup on the Office of the Coordinator of Information has been revised.
06/06/02 A history of Kokuryukai, also known as the Black Dragon Society, has been added, as well as a description of the secret cabal behind Kokuryukai, the Order of the Green Dragon.
05/23/02 The Office of the Coordinator of Information has been added to the Order of Battle.
04/28/02 The Naval Intelligence Division has been added to the Order of Battle, and German phrases have been added to the Phrasebook.
01/27/02 A revised version of the biography of Colonel David Cornwall has been added to the Personnel Records.
12/27/01 The general history of the Karotechia has been put up in the Campaign section.
12/02/01 The Special Air Service and the Combined Operations Pilotage Parties have been added to the Order of Battle, and the new skills Navigation (Desert) and Sabotage have been added to the Rules section.
09/04/01 30 Commando and the Ahnenerbe have been added to the Order of Battle.
08/18/01 Rules for Chases on foot and Moving While Firing and a number of new skills have been put up.
08/17/01 The Small Scale Raiding Force has been added to the Order of Battle section. A slight modification to the Optional Wounds system has been made.
08/16/01 A biography for the head of PISCES, Colonel David Cornwall is up on Personnel Records. The Optional Wounds system has been revised. The Land Vehicle Rules has been revised as well, and a more complete listing of Land Vehicles and their stats has been provided in EQUIPMENT section.
06/21/01 An Optional Wounds system has been put up in the RULES section.
06/18/01 A history of PISCES from its early roots in the Delphi Project to MI-13 to its creation in 1940 is available in the Order of Battle section in CAMPAIGN.
06/16/01 A table of the Comparison of Military Ranks among the major combatants of the war is up on the Order of Battle.
06/14/01 A biography has been put up for Karl Maria Wiligut. Templates have been added for GRU SV-8 and the SS. The Chronology has been expanded, now from 1933 to 1945. An OSS Table of Equipment, circa 1943 is available in EQUIPMENT, along with some new and edited entries in the Small Arms section. The beginnings of a Book of Phrases relevant to WWII DELTA GREEN is online, waiting for many entries to be added. A few titles have been added to the bibliography on the RESOURCES page.
05/03/01 Templates have been added to the Order of Battle for Merrill's Marauders, 624 Squadron of the RAF, the Royal Naval Commandos, and The Brandenburgers.
04/14/01 The Armor section of vehicles in EQUIPMENT is now complete, with all the common AFVs used during the war now up with stats and descriptions. In addition, some of the kinks have been worked out of the Ground Vehicle Movement and Ground Vehicle Combat rules. SCENARIOS now has more filled-out descriptions of Operations LIFEGUARD, LUNACY, and SUMMER BREEZE. Ian Fleming and J. Edgar Hoover have been added to Personnel Records, and templates for the British intelligence agencies MI-5 and SIS are up as well.
03/30/01 The Armor section of vehicles in EQUIPMENT is now up, though there is only one entry: the ubiquitous PzKw III Medium Tank.
03/29/01 Added Ground Vehicle Movement, Ground Vehicle Combat, and Called Shot to the RULES section.
03/15/01 Template for the Federal Bureau of Investigation added to Order of Battle under CAMPAIGN. Basic data on Operation LIFEGUARD added to SCENARIOS. More listings have been added under RESOURCES and more weapons added under the Small Arms section. Reinhard Galt added to the Personnel Records section.
01/21/01 Several new sections added. New links have been added under RESOURCES. An incomplete template has been made for the Office of Strategic Services in Order of Battle under CAMPAIGN. Rules for characters on death's door and for special attacks while sentry stalking have been put up in Combat under RULES.
01/15/01 The Chronology and Small Arms sections are up.
Published by arrangement with the Delta Green Partnership. The intellectual property known as Delta Green is ™ and © the Delta Green Partnership, who has licensed its use here. The contents of this document are � Gil Trevizo excepting those elements that are components of the Delta Green intellectual property.
Original content for pages within the www.mindspring.com/~furrylogic/deltagreen domain other than that covered by the above, or as specified on that individual page, is copyright 2003 Gil Trevizo and may be freely copied, posted on other websites, or used in other media in whole or in part with the following mandatory conditions imposed on usage: (1) any usage must respect and protect copyrights on all material, and specifically must obey restrictions placed on use by Pagan Publishing on its copyrighted material, and (2) regardless of alterations or additions, Gil Trevizo must be credited as author of parts � Gil Trevizo.


What kind of training is available on Beam?
But this list is not exhaustive - the Beam model is highly personalised and each member has an opportunity to pursue a journey that is as unique as them.
Beam also works with employers to place individuals in jobs before they have started their training so far as possible.


In his grace and mercy, God condescends to hear our prayers wherever we are, however far we have strayed from Him.
In his mercy, he will lead us into the belly of the beast if it is there that we will finally call upon his name.
In more remarkable humility, God answers us even when we have not answered Him.
He is ready to answer us even before we know that we shall call upon Him: God had prepared a fish. He was waiting, longing to minister to Jonah's need.
He is the Lord of all and can answer us from wherever we might call upon Him.
Jonah recognizes that God has been sovereign over his journey and suffering. He recognizes that God's mercies can be terrible…that He may wound us in order to heal.
As Nate points out, Jonah seems not to attribute these terrors to himself. This is not simply a matter of giving God the glory, but of leaving out his own rebellion from the prayer.
Again, is this Jonah recognizing the sovereignty of God, or something like "the woman who You gave me"?
Curious. It is the Lord who has watched over Jonah throughout. Does he mean that he cannot see God as in the next verse? Or does this refer to God's displeasure?
He has faith in further deliverance, and this may even prophecy Jonah's eventual return to Jerusalem.
Jonah is convinced of God's unwavering love, even for him.
What will Jonah's sacrifice be? Will it please the Lord? Is it to carry out God's command, and if so, does he perform it in a manner pleasing to the Lord? Is Jonah's fulfillment of his vows any more pleasing to God than that of the men who did so aboard the ship (Jonah 1:16)?


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Home » Music » Choral » C4: Choral Composer/Conductor Collective: Devotion
C4: Choral Composer/Conductor Collective: Devotion
Superb contemporary choral music presented by an established New York City ensemble with an equally fine youth chorus.
Posted on April 7, 2018 by archive in Choral, Music
Kristopher Burke conducting the Brooklyn Youth Chorus Men's Ensemble in Martha Sullivan's "Devotion"
Jean Ballard Terepka, Music Critic
The Choral Composer/Conductor Collective, better known as C4, presented a late-winter concert that featured both its usual strengths – collaboratively presented, compelling performances of contemporary choral music and daring commitment to new compositions – and an exciting new partnership. For three of its six world premiere works, C4 was joined by the Brooklyn Youth Chorus Men's Ensemble, a group whose mission and goals are closely consonant with C4's.
The evening's works were loosely organized around the theme of devotion, a big conceptual basket into which works about reverence, affection, awe and loves of all sorts could deftly and effectively be fit. Although the concert's texts ranged from ancient to contemporary, from sacred to profane, from unknown to known authorship, the considered objects of affection included peace, nature, mystery, creativity, divinity – all love's siblings – and love itself.
The first half of the concert consisted of five works, including two world premieres by current C4 members. Heavy, Like Water, by Joseph Prestamo, was set to poems "When It Rains" and "Elegy" by West Coast poet Brandon Jordan Brown. Themes of interconnected grief and joy were reflected in musical chords of only ambivalent resolution. As might be expected from a composer who is himself a tenor, the writing for tenor voice was especially beautiful: the line "Yes, it's beginning to be the only thing I can see" was marvelous. The 11 minute work might have benefitted from some tightening in the second half of the first movement, but its quiet conclusion was strong.
The second world premiere in the first half of the program was Hayes Biggs' who are you?, a richly complex piece set to a linguistically dense and rewarding poem by frequent C4 artistic collaborator Zsuzsanna Ardó. Although the work features cerebral speculations on an elusive "identity vortex of time and space," this work for women's voices follows musical ideas divergently, giving them each wide individual travel space before bringing them back together in an intellectually satisfying conclusion. The chamber choir for this piece consisted of 11 women; the soloists, Karen Siegel and Rebecca Ehren, sopranos, and Maya Ben-Meir and Jamie Klenetsky Fay, altos, were particularly fine.
Martha Sullivan conducting Peter Louis van Dijk's "Chariots" for C4: Choral Composer/Conductor Collective
The third world premiere in the first half of the concert was current C4 singer Carlos Pérez Tabares' Simple Motets, a short setting of two familiar ancient Christian texts, Ave, Regina Caelorum and Cantate Domino. Though the music is more complex than might be singable by choirs of all levels – a goal Tabares indicated he was aiming for – the work was pleasingly accessible, alternating between intimate directness and inviting lightness. The concluding flourish emphasized the joy of the two prayer texts.
The other pieces of the concert's first half included works by Peter Louis van Dijk and Gilad Cohen, both set to Old Testament texts, and sung with the particular affinity that C4 has for musical literature inspired by ancient Hebrew poetry.
The first half of this concert was very good. The second was thoroughly marvelous: the concluding piece, Vier geistliche Gesänge – beautifully performed by C4 and written by well-known contemporary German composer Wolfram Buchenberg provided a satisfying shape to the evening as a whole by offering a quasi-liturgical presentation of Christian love. But the real triumph of the evening belonged to the Brooklyn Youth Chorus Men's Ensemble and the three works – world premieres – composed for them by three long-standing current C4 members.
Founded in 1992 and currently incorporating afterschool programs for more than 600 students enrolled in multi-level musical training divisions and advanced choral programs (http://www.brooklynyouthchorus.org), the Brooklyn Youth Chorus has long successfully realized its goals of artistic integrity and effective collaboration with a wide variety of organizations. Kristopher Burke, a member of the Brooklyn Youth Chorus administrative staff since 2011, is now conductor of the Men's Ensemble and Men's Training Division. Burke is also a former member of C4 and, as such, was in a particularly strong position to create the close collaborative coherence of the Brooklyn Youth Chorus Men's Ensemble participation in this C4 concert.
The second half of the concert opened with the fifteen boys and men of the Brooklyn group; ranging in age from twelve to eighteen, they were conducted in all three pieces they sang – the first two on their own, and the third with the men of C4 – by Burke. The first piece was Karen Siegel's Infinite Hour. This five-minute work, based on a poem by Siegel herself, examines questions of purpose and being "in the force of gravity and dark energy." The act of questioning itself becomes, for Siegel, a form of prayer, and her sophisticated musical writing, matching harmonic expressivity to the themes of her texts, made this a work right in the center of the concert's focus on devotion.
In Jamie Klenetsky Fay's I remain, based on one of her own poems, a chantlike element provided a context for the slow marcato emphasis on the text – "I remain for the world" – and made for musically gorgeous textures.
For Martha Sullivan's Devotion, based on an excerpt of Wallace Stevens' luminous 1923 Sunday Morning, the Brooklyn boys and young men sang with the C4 tenors and basses. Devotion both presented and embodied a lush, Romantic sound as a form of worship: words and men raised up a "chant of paradise … returning to the sky." Though only four minutes long, this profound piece is one of Sullivan's best to date.
All three of these C4 composers demonstrated a keenly attuned ear for the singers for whom they were writing: they captured and illuminated that particular moment of young manhood's new wisdom when the just-recently-changed voices of young men still retain the childhood innocence that foreshadows wisdom. Siegel, Fay and Sullivan have all written music that highlights both the vulnerabilities and strengths of these voices, using these characteristics to enrich the meanings of the texts they're setting to music. The three pieces together formed a fabulous musical triptych.
By itself, C4 contributes in a major way to the wealth and diversity of the New York City cultural landscape. With collaborations as successful as this winter's one with the Brooklyn Youth Chorus Men's Ensemble, the C4 contribution to New York City was additionally and felicitously expanded.
C4: Choral Composer/Conductor Collective: Devotion (March 10, 2018)
296 Ninth Avenue at 29th Street, in Manhattan
For more information: visit http://www.c4ensemble.org
Running time: one hour and 40 minutes including one intermission.
Brandon Jordan Brown
Brooklyn Youth Chorus Men's Ensemble
Carlos Pérez Tabares
Gilad Cohen
Hayes Biggs
Jamie Klenetsky Fay
Joseph Prestamo
Karen Siegel
Kristopher Burke
Martha Sullivan
Maya Ben-Meir
Peter Louis van Dijk
Rebecca Ehren
Wolfram Buchenberg
Zsuzsanna Ardo
C4: Unusual – Music of the strange, the absurd, and the surreal
Committed to new choral music of all sorts, from glorious to goofy, a brilliant group of musicians revels in daring and integrity.
C4: The Choral Composer/Conductor Collective Presents "Water Music"
An extremely fine concert of contemporary choral music featuring three world premieres on the theme of water.
C4: The Choral Composer/Conductor Collective: Vocal – C4 honors its foundations: music for voices alone
Stunning lucidity of mission and execution marks the eleventh season opening of this daring, disciplined and artistically eloquent group of musicians.
C4, The Choral Composer/Conductor Collective: Electric – Choral Works with electrified ensemble
An exciting concert by a new well-established, highly gifted group of singers, composers and conductors.
C4: Choral Composer/Conductor Collective: Organic – New Works for Choir and Organ
A daring and elegant collective of contemporary choral composers and conductors, no longer "new," now well established, presents an evening of gorgeous music.