Random Diophantine Equations

Abstract

We consider additive Diophantine equations of degree k in s variables and establish that whenever s ≥ 3. k + 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. © 2014 Elsevier Inc.