Properties of High Rank Subvarieties of Affine Spaces

David Kazhdan, Tamar Ziegler*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional Fq-vector spaces. In the first, analytic part of the paper we prove a number properties of high rank systems of polynomials. In the second, we use these properties to deduce results in Algebraic Geometry , such as an effective Stillman conjecture over algebraically closed fields, an analogue of Nullstellensatz for varieties over finite fields, and a strengthening of a recent result of Bik et al. (Polynomials and tensors of bounded strength, arXiv:1805.01816). We also show that for k-varieties X⊂ An of high rank any weakly polynomial function on a set X(k) ⊂ kn extends to a polynomial.

Original languageAmerican English
Pages (from-to)1063-1096
Number of pages34
JournalGeometric and Functional Analysis
Issue number4
StatePublished - 1 Aug 2020

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