TY - JOUR
T1 - Properties of High Rank Subvarieties of Affine Spaces
AU - Kazhdan, David
AU - Ziegler, Tamar
N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85089671600&partnerID=8YFLogxK
U2 - 10.1007/s00039-020-00542-4
DO - 10.1007/s00039-020-00542-4
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AN - SCOPUS:85089671600
SN - 1016-443X
VL - 30
SP - 1063
EP - 1096
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 4
ER -