Random diophantine equations of additive type

TL;DR

This paper demonstrates that for most additive Diophantine equations of degree k with sufficiently many variables, the local-global principle applies and small solutions exist, using advanced number-theoretic techniques.

Contribution

It introduces a novel combination of the circle method and lattice point counting to establish the local-global principle for a broad class of Diophantine equations.

Findings

Local-Global principle holds for almost all such equations

Existence of very small non-trivial solutions for these equations

Bound on solutions is close to optimal

Abstract

Using the circle method in combination with lattice point counting arguments, we show that for almost all homogeneous diophantine equations of additive type and degree $k$ in more than $4k$ variables, the Local-Global principle holds true. Moreover, our approach shows that almost all such equations having a non-trivial integer solution have a very small such solution, the bound being close to the best possible one.