# Resolvent degree, Hilbert’s 13th Problem and geometry

### Benson Farb

University of Chicago, USA### Jesse Wolfson

University of California, Irvine, USA

## Abstract

We develop the theory of resolvent degree, introduced by Brauer [Brau2] in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert’s 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a nite group. As one application of this point of view, we prove that Hilbert’s 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of nding lines on a smooth cubic surface or bitangents on a smooth planar quartic.

## Cite this article

Benson Farb, Jesse Wolfson, Resolvent degree, Hilbert’s 13th Problem and geometry. Enseign. Math. 65 (2019), no. 3/4, pp. 303–376

DOI 10.4171/LEM/65-3/4-2