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

Question

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).

Answer 1

You might have a look at Polynomial Transformations of Tschirnhaus, Bring and Jerrard (Internet Archive). 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 (Internet Archive) 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.

Answer 2

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 $n=6$.)

Answer 3

Partly more detailed lower bounds for essential dimensions of polynomials are given in Theorem 1.4. of Reichstein, Z.; Youssin, B.: Essential dimensions of algebraic groups and a resolution theorem for G-varieties. Canad. J. Math. 52 (2000) (5) 1018–1056.