Hilbert's 13th problem in prime characteristic

TL;DR

This paper explores how Hilbert's classical conjectures about resolvent degrees of polynomials change when the base field is of positive characteristic, revealing potential failures of these conjectures in such settings.

Contribution

It demonstrates that Hilbert's conjectures on resolvent degrees can fail over fields of positive characteristic, contrasting with the known results over complex numbers.

Findings

Hilbert's conjectures do not necessarily hold in positive characteristic.

Potential failure of the conjectures for degrees 6, 7, and 8.

Shows the difference in polynomial solvability between characteristic zero and positive 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.