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

Luca Dehennin

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.

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.
Paper Structure (13 sections, 29 theorems, 64 equations)

This paper contains 13 sections, 29 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Schwartz-Zippel
  3. Dimension Growth
  4. Affine Varieties
  5. Results for Higher Codimension
  6. Projective Varieties
  7. Affine Varieties
  8. Sublinear Schwartz-Zippel and Subquadratic Dimension Growth
  9. Unions of Linear Varieties
  10. Points and Curves
  11. Curves and Surfaces
  12. Surfaces and Threefolds
  13. Threefolds and Fourfolds

Key Result

Lemma 1.1

An affine variety of degree $d$ and pure dimension $k$ in ${\mathbb A}_{\mathbb Q}^n$ contains no more than $cdB^{k}$ points of height $\le B$ for some constant $c$ depending only on $k$.

Theorems & Definitions (52)

  • Lemma 1.1: Affine Schwartz-Zippel, CCDN-dgc
  • proof
  • Lemma 1.2: Projective Schwartz-Zippel
  • Conjecture 1.3: Sublinear projective Schwartz-Zippel
  • Theorem 1.4: Quadratic projective dimension growth for hypersurfaces, BinCluKat
  • Proposition 1.5: Lower bounds for the worst case in projective dimension growth, CGlaz
  • Proposition 1.6: Subquadratic projective dimension growth for curves, CGlaz
  • Theorem 1.7: Quadratic projective dimension growth in all codimensions
  • Theorem 1.8: Subquadratic projective dimension growth
  • Theorem 1.9: Subquadratic projective dimension growth
  • ...and 42 more