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