Abstract
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 language | American English |
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Pages (from-to) | 1063-1096 |
Number of pages | 34 |
Journal | Geometric and Functional Analysis |
Volume | 30 |
Issue number | 4 |
DOIs | |
State | Published - 1 Aug 2020 |
Bibliographical note
Funding Information:T. Ziegler is supported by ERC Grant ErgComNum 682150.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.