Hilbert's 13th problem in prime characteristic

Abstract

The resolvent degree ${\text{rd}}_{\mathbb{C}}(n)$ is the smallest integer $d$ such that a root of the general polynomial $$f(x)=x^n+a_1{x}^{n-1}+\dots +a_n$$ can be expressed as a composition of algebraic functions in at most $d$ variables with complex coefficients. It is known that ${\text{rd}}_{\mathbb{C}}(n)=1$ when $n⩽5$. Hilbert was particularly interested in the next three cases: he asked if ${\text{rd}}_{\mathbb{C}}(6)=2$ (Hilbert's Sextic Conjecture), ${\text{rd}}_{\mathbb{C}}(7)=3$ (Hilbert's 13th Problem) and ${\text{rd}}_{\mathbb{C}}(8)=4$ (Hilbert's Octic Conjecture). These problems remain open. It is known that ${\text{rd}}_{\mathbb{C}}(6)⩽2$, ${\text{rd}}_{\mathbb{C}}(7)⩽3$ and ${\text{rd}}_{\mathbb{C}}(8)⩽4$. It is not known whether or not ${\text{rd}}_{\mathbb{C}}(n)$ 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.

12 pages, 2 figures