Tschirnhaus transformations after Hilbert

Abstract: Let 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 $RD(9)\le 4$). In this paper, we turn Hilbert's sketch into a general method. We show this method produces best-to-date upper bounds on RD(n) for all n, improving earlier results of Hamilton, Sylvester, Segre and Brauer.