On rank in algebraic closure

TL;DR

This paper establishes polynomial bounds relating the Schmidt rank of a form over a field to its algebraic closure, with implications for counting rational and prime points on algebraic varieties.

Contribution

It provides the first polynomial bounds for the Schmidt rank over various fields in terms of the algebraic closure rank for forms of degree greater than four.

Findings

Polynomial bounds for $rk_{f k}(Q)$ in terms of $rk_{ar{f k}}(Q)$ for $d>4$

Implications for counting integer points on varieties

Implications for counting prime points on varieties

Abstract

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with $R_i, S_i \in {\mathbf k}[x_1, \ldots, x_s]$ forms of degree $<d$. When $ {\mathbf k} $ is algebraically closed, this rank is essentially equivalent to the codimension in $ {\mathbf k}^s $ of the singular locus of the variety defined by $ Q, $ known also as the Birch rank of $ Q. $ When $ {\mathbf k} $ is a number field, a finite field or a function field, we give polynomial bounds for $ rk_{\mathbf k}(Q) $ in terms of $ rk_{\bar {\mathbf k}} (Q) $ where $ \bar {\mathbf k} $ is the algebraic closure of $ {\mathbf k}. $ Prior to this work no such bound (even ineffective) was known for $d>4$. This result has immediate consequences for counting integer points (when $ {\mathbf k} $ is a number field) or prime points (when $ {\mathbf k} = \mathbb Q $) of the variety $ \{Q=0\} $ assuming $ rk_{\mathbf k} (Q) $ is large.