Diophantine inequalities and quasi-algebraically closed fields

TL;DR

This paper proves that certain polynomial forms over fields related to finite fields can take arbitrarily small non-zero values when the number of variables exceeds the square of the degree, with implications for distribution and algebraic closure.

Contribution

It establishes new bounds for the small value property of forms over quasi-algebraically closed fields, extending classical Diophantine inequality results.

Findings

Forms of degree d in more than d^2 variables take arbitrarily small non-zero values.

Results apply to distribution modulo polynomial rings over finite fields.

Extends to quasi-algebraically closed fields generally.

Abstract

Consider a form $g(x_1,...,x_s)$ of degree $d$, having coefficients in the completion $F_q((1/t))$ of the field of fractions $F_q(t)$ associated to the finite field $F_q$. We establish that whenever $s>d^2$, then the form $g$ takes arbitrarily small values for non-zero arguments $x\in F_q[t]^s$. We provide related results for problems involving distribution modulo $F_q[t]$, and analogous conclusions for quasi-algebraically closed fields in general.