Properties of High Rank Subvarieties of Affine Spaces

We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional 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...

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Published inGeometric and functional analysis Vol. 30; no. 4; pp. 1063 - 1096
Main Authors Kazhdan, David, Ziegler, Tamar
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.08.2020
Springer Nature B.V
Subjects
Analysis
Combinatorial analysis
Fields (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Properties (attributes)
Tensors
Vector spaces
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Summary:We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional 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 Bik et al. (Polynomials and tensors of bounded strength, arXiv:1805.01816 ). We also show that for k -varieties X ⊂ A n of high rank any weakly polynomial function on a set X ( k ) ⊂ k n extends to a polynomial.
Bibliography:ObjectType-Article-1
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content type line 14
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-020-00542-4