Values of Random Polynomials at Integer Points

TL;DR

This paper provides bounds on the heights of approximate solutions to quadratic equations and analyzes the distribution of values of random polynomials at integer points, using classical results on Siegel transforms.

Contribution

It introduces new bounds on solution heights and distributional properties of polynomial values, extending quantitative results in number theory and homogeneous dynamics.

Findings

Bounds on heights of approximate integral solutions

Quantitative distribution results for polynomial values at integers

Application of Rogers' bounds to random polynomial analysis

Abstract

Using classical results of Rogers bounding the $L^2$-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantiative Oppenheim theorem of Eskin-Margulis-Mozes for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.