Hilbert’s 13th problem in prime characteristic

Abstract

The resolvent degree ${\mathrm{rd}}_{\mathbb{C}}(n)$ is the smallest integer $d$ such that a root of the general polynomial

can be expressed as a composition of algebraic functions in at most $d$ variables with complex coefficients. It is known that ${\mathrm{rd}}_{\mathbb{C}}(n)=1$ when $n⩽5$. Hilbert was particularly interested in the next three cases: he asked if ${\mathrm{rd}}_{\mathbb{C}}(6)=2$ (Hilbert’s Sextic conjecture), ${\mathrm{rd}}_{\mathbb{C}}(7)=3$ (Hilbert’s 13th problem) and ${\mathrm{rd}}_{\mathbb{C}}(8)=4$ (Hilbert’s Octic conjecture). These problems remain open. It is known that ${\mathrm{rd}}_{\mathbb{C}}(6)⩽2$, ${\mathrm{rd}}_{\mathbb{C}}(7)⩽3$ and ${\mathrm{rd}}_{\mathbb{C}}(8)⩽4$. It is not known whether or not ${\mathrm{rd}}_{\mathbb{C}}(n)$ can be $>1$ for any $n⩾6$.
In this paper, we show that all three of Hilbert’s conjectures can fail if we replace $\mathbb{C}$ with a base field of positive characteristic.