Tschirnhaus transformations after Hilbert

  • Jesse Wolfson

    University of California, Irvine, USA
Tschirnhaus transformations after Hilbert cover

A subscription is required to access this article.

Abstract

In this paper, we use enumerative geometry to simplify the formula for the roots of the general one-variable polynomial of degree , for all . More precisely, let denote the minimum for which there exists a formula for the roots of the general degree polynomial using only algebraic functions of 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 ). In this paper, we turn Hilbert's sketch into a general method. We show this method produces best-to-date upper bounds on for all , improving earlier results of Hamilton, Sylvester, Segre and Brauer.

Cite this article

Jesse Wolfson, Tschirnhaus transformations after Hilbert. Enseign. Math. 66 (2020), no. 3/4, pp. 489–540

DOI 10.4171/LEM/66-3/4-9