Random diophantine equations, I

TL;DR

This paper proves that for additive Diophantine equations of degree k with sufficiently many variables, almost all satisfy the Hasse principle, have solutions of positive density, and nearly all soluble equations have small solutions.

Contribution

It establishes that most such equations satisfy the Hasse principle and provides bounds on the size of the smallest solutions, nearly optimal.

Findings

Almost all equations satisfy the Hasse principle.

Soluble equations form a set of positive density.

Most soluble equations have small solutions.

Abstract

We consider additive diophantine equations of degree $k$ in $s$ variables and establish that whenever $s\ge 3k+2$ then almost all such equations satisfy the Hasse principle. The equations that are soluble form a set of positive density, and among the soluble ones almost all equations admit a small solution. Our bound for the smallest solution is nearly best possible.