Improved, sublinear projective Schwartz-Zippel and (sub)quadratic dimension growth bounds in arbitrary codimension

TL;DR

This paper introduces a sublinear version of the projective Schwartz-Zippel bound and extends quadratic dimension growth bounds to arbitrary codimension, improving theoretical limits in algebraic geometry.

Contribution

It presents a sublinear sharpened Schwartz-Zippel bound and extends quadratic dimension growth bounds to arbitrary codimension, advancing the understanding of algebraic dimension growth.

Findings

Sublinear Schwartz-Zippel bound for linear varieties

Subquadratic dimension growth bounds in low dimensions

Extension of quadratic bounds to arbitrary codimension

Abstract

We work towards a question raised by Cluckers and Glazer in [CG25], to bring the dimension growth upper bounds and lower bounds for the worst case closer together. To this end, we introduce a sublinear sharpened version of the projective Schwartz-Zippel bound. We prove several cases, including the case of configurations of linear varieties. This leads to subquadratic dimension growth bounds in some low dimensions, improving on the quadratic dependence obtained by Binyamini, Cluckers and Kato in [BCK25]. We introduce a natural projection argument with pull-backs and use this to address a second question by Cluckers and Glazer by extending the quadratic dimension growth bounds from [BCK25] to arbitrary codimension.