A uniform estimate for the density of rational points on quadrics

TL;DR

This paper provides a coefficient-independent estimate for the density of rational points on quadrics, especially in four variables, and extends to higher variables with restrictions on rational lines.

Contribution

It introduces a uniform estimate for rational point density on quadrics that is independent of coefficients for four variables and adaptable to higher variables with certain restrictions.

Findings

Coefficient-independent estimate for four-variable quadrics

Extension of estimate to higher variables with restrictions

Density bounds for rational points on quadratic varieties

Abstract

This paper is concerned with the density of rational points of bounded height lying on a variety defined by an integral quadratic form Q. In the case of four variables, we give an estimate that does not depend on the coefficients of Q. For more variables, a similar estimate still holds with the restriction that we only count points which do not lie on rational lines.