Tschirnhaus transformations after Hilbert

Abstract

In this paper, we use enumerative geometry to simplify the formula for the roots of the general one-variable polynomial of degree $n$, for all $n$. More precisely, let $\mathrm{RD}(n)$ denote the minimum $d$ for which there exists a formula for the roots of the general degree $n$ polynomial using only algebraic functions of $d$ or fewer variables. In 1927, Hilbert sketched how the 27 lines on a cubic surface could be used to construct a 4-variable formula for the general degree 9 polynomial (implying $\mathrm{RD}(9)\le 4$). In this paper, we turn Hilbert's sketch into a general method. We show this method produces best-to-date upper bounds on $\mathrm{RD}(n)$ for all $n$, improving earlier results of Hamilton, Sylvester, Segre and Brauer.