Least number of non-zero coefficients to describe a degree n polynomial

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I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it.

Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be transformed into simpler expressions. For instance it is apparently well-known that the Tschirnhaus transformation allows to bring any quintic into so-called Bring-Jerrard form $x^5+ax+b$, while for degree 6 one needs at least three coefficents $x^6+ax^2+bx+c$.

Is there a name for such "generalized Bring-Jerrard form", and what is known about it? In particular there is a cryptic footnote of Arnold (page 3 of this lecture) where he says roughly that the degrees for which more coefficients are needed occur along "a rather strange infinite sequence": could someone please describe what those degrees are (I had a look at the OEIS but I believe that sequence is different from Hamilton numbers, and couldn't find a relevant one).

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2 Answers

You might have a look at Polynomial Transformations of Tschirnhaus, Bring and Jerrard. It gives more explicit detail on why you can remove the first three terms after the leading term (covering the cases of degree 5 and 6 you mention above), but it does concentrate on degree 5.

Hamilton's 1836 paper on Jerrard's original work has an elementary explanation of the technique (much of the paper concentrates on showing that certain other reductions Jerrard proposed, including a general degree 6 polynomial to a degree 5, were "illusory"). It also explains Jerrard's trick for eliminating the 2nd, 3rd and 5th terms. Finally, Jerrard has a method for eliminating the second and fourth terms, while bringing the third and fifth coefficients into any specified ratio: this only works in degree 7 or above (Jerrard had mistakenly thought this worked generally, and thus solved the general quintic by reducing it to de Moivre's solvable form -- this all predates Abel's work!)

If by "Bring-Jerrard" form you just mean a certain number of the initial terms (after the first) have been eliminated, then the Hamilton numbers you linked to are indeed exactly what you want.

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Thanks for the reply and the references. Right, I was confused, Arnold's footnote really deals with something else, namely the least number of non-zero coefficients of certain equations of degree n whose solutions are universal functions from solving any other equation of the same degree. So Arnold's sequence giving the numbers of those non-zero coefficients for each degree is definitely different from Hamilton's (and probably quite interesting too). – Thomas Sauvaget Nov 22 2009 at 19:51

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The modern notion of the essential dimension of a group gives a precise way to state your question (and generalizations), and there is some recent work extending the work mentioned in Scott's answer. To get started, see the article

J. Buhler and Z. Reichstein, On the essential dimension of a group, Compositio Math. 106 (1997), 159-179.

For instance, it is proved there that for polynomials of degree $n$, at least `$\lfloor n/2 \rfloor$` coefficients are required. (This agrees with what you mentioned for $n=5$ and $n=6$.)

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Thank you very much for this extremely interesting reference! – Thomas Sauvaget Mar 27 2010 at 8:21
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