TITLE: Least number of non-zero coefficients to describe a degree n polynomial
QUESTION [23 upvotes]: 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).

REPLY [21 votes]: 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 $n=6$.)