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](http://homepage.mac.com/ehgoins/ma598/lecture_27.pdf) 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](http://www.pdmi.ras.ru/~arnsem/Arnold/arnlect1.ps.gz)) 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](https://oeis.org/A000905), and couldn't find a relevant one).



Least number of non-zero coefficients to describe a degree n polynomial
You might have a look at [Polynomial Transformations of Tschirnhaus, Bring and Jerrard](http://www.apmaths.uwo.ca/~djeffrey/Offprints/Adamchik.pdf) ([Internet Archive](http://web.archive.org/web/20180816061604/http://www.apmaths.uwo.ca/~djeffrey/Offprints/Adamchik.pdf)). 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](http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Jerrard/Jerrard.pdf) ([Internet Archive](http://web.archive.org/web/20171126173729/https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Jerrard/Jerrard.pdf)) 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.


