Power-free values of polynomials on symmetric varieties

TL;DR

This paper investigates conditions under which polynomials take r-free values on symmetric varieties, using homogeneous dynamics, and provides explicit bounds in the case of quadrics, along with an asymptotic count of such points.

Contribution

It introduces a new approach combining homogeneous dynamics with algebraic geometry to analyze r-free values of polynomials on symmetric varieties, including explicit bounds for quadrics.

Findings

Established conditions for r-freeness of polynomial values on symmetric varieties.

Derived an asymptotic formula for the density of such points.

Provided explicit bounds for the case of quadric hypersurfaces.

Abstract

Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics.