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 4variable 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 besttodate upper bounds on RD(n) for all n, improving earlier results of Hamilton, Sylvester, Segre and Brauer.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.06515
 Bibcode:
 2020arXiv200106515W
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14G25 (Primary) 11C08;
 12E12 (Secondary)
 EPrint:
 46 pages. Comments welcome!