Properties of high rank subvarieties of affine spaces. (English) Zbl 1460.14138
Summary: We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional \(\mathbb{F}_q\)-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 A. Bik et al. [Commun. Contemp. Math. 21, No. 7, Article ID 1850062, 24 p. (2019; Zbl 1425.15021)]. We also show that for \(k\)-varieties \(\mathbb{X} \subset \mathbb{A}^n\) of high rank any weakly polynomial function on a set \(\mathbb{X}(k) \subset k^n\) extends to a polynomial.
MSC:
| 14R05 | Classification of affine varieties |
| 05E14 | Combinatorial aspects of algebraic geometry |
| 14G15 | Finite ground fields in algebraic geometry |
| 15A69 | Multilinear algebra, tensor calculus |
Citations:
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