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L’Enseignement Mathématique

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Volume 65, Issue 3/4, 2019, pp. 303–376
DOI: 10.4171/LEM/65-3/4-2

Published online: 2020-06-11

Resolvent degree, Hilbert’s 13th Problem and geometry

Benson Farb[1] and Jesse Wolfson[2]

(1) University of Chicago, USA
(2) University of California, Irvine, USA

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.

Keywords: Algebraic function, enumerative problems, resolvent degree, Hilbert’s 13th Problem

Farb Benson, Wolfson Jesse: Resolvent degree, Hilbert’s 13th Problem and geometry. Enseign. Math. 65 (2019), 303-376. doi: 10.4171/LEM/65-3/4-2