Hilbert's 13th problem in prime characteristic

Abstract

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