Density of solutions for systems of forms

TL;DR

This paper provides an effective bound for the density of solutions to systems of forms over fields of characteristic zero, improving previous bounds and impacting areas like polynomial map surjectivity and the Hardy-Littlewood method.

Contribution

It establishes a practical, effective bound for the density of solutions in systems of forms, improving upon previous astronomical bounds and enabling broader applications.

Findings

Derived an explicit bound for solution density constant B

Improved upon previous bounds, making them practically applicable

Applied results to polynomial map surjectivity and Hardy-Littlewood circle method

Abstract

Let $K$ be a field of characteristic zero over which every diagonal form in sufficiently many variables admits a nontrivial solution. For example, $K$ may be a totally imaginary number field or a finite extension of a $p$-adic field. Suppose $f_1,\ldots,f_s$ are forms of degree $d$ over $K.$ Bik, Draisma and Snowden recently proved that there exists a constant $B = B(d,s,K)$ such that the rational solutions to the system of equations $f_1=\ldots=f_s = 0$ are Zariski dense, as long as the Birch rank of $f_1,\ldots,f_s$ is greater than $B.$ We establish an effective bound for this constant, improving vastly on the astronomical bound coming from their proof. Our result has applications for surjectivity of polynomial maps and for the Hardy-Littlewood circle method.