Resolvent degree, Hilbert's 13th Problem and geometry

TL;DR

This paper develops the theory of resolvent degree to analyze the complexity of algebraic solutions, extending it to enumerative geometry and linking Hilbert's 13th Problem to geometric problems involving lines and bitangents.

Contribution

It introduces an extended framework for resolvent degree as an invariant in algebraic geometry, connecting classical problems to modern enumerative geometry.

Findings

Hilbert's 13th Problem is equivalent to geometric enumeration problems.

Resolvent degree serves as an invariant linking algebraic complexity and geometry.

The theory applies to problems like lines on cubic surfaces and bitangents on quartics.

Abstract

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} 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 finite 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 finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.