Schmidt rank of quartics over perfect fields

TL;DR

This paper establishes an upper bound on the Schmidt rank of quartic polynomials over perfect fields, linking it to the rank over algebraic closures, thus advancing understanding of polynomial complexity in algebraic geometry.

Contribution

It provides a bound on the Schmidt rank of quartic polynomials over perfect fields based on their rank over algebraic closures, a novel connection in algebraic geometry.

Findings

Schmidt rank of quartics over perfect fields is bounded by their rank over algebraic closures

The result applies to fields with characteristic not equal to 2

Advances understanding of polynomial complexity over different fields

Abstract

Let $k$ be a perfect field of characteristic $\neq 2$. We prove that the Schmidt rank (also known as strength) of a quartic polynomial $f$ over $k$ is bounded above in terms of only the Schmidt rank of $f$ over $\overline{k}$, an algebraic closure of $k$.