Additive and Analytic Combinatorics

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IMA Workshops September 29-October 3, 2014 Additive and Analytic Combinatorics ORGANIZERS SPEAKERS David Conlon, University of Oxford Tim Austin, Courant Institute of Mathematical Sciences Ernie Croot, Georgia Institute of Technology Vitaly Bergelson, The Ohio State Van Vu, Yale University University Tamar Ziegler, Hebrew University Emmanuel Breuillard, Université de Paris XI (Paris-Sud) Boris Bukh, Carnegie Mellon University Mei-Chu Chang, University of California, Additive combinatorics is the theory of counting Riverside additive structures in sets. This theory has seen David Conlon, University of Oxford exciting developments and dramatic changes in Ernie Croot, Georgia Institute of direction in recent years thanks to its connections Technology with areas such as harmonic analysis, ergodic Jacob Fox, Massachusetts Institute of Technology theory, and representation theory. As it turns out, Bob Guralnick, University of Southern many combinatorial ideas that have existed in the California combinatorics community for quite some time can Akos Magyar, University of British be used to attack notorious problems in other areas Columbia of mathematics. A typical example is the Green- Hamed Hatami, McGill University Tao theorem on the existence of long arithmetic Frederick Manners, University of Oxford progressions in primes, which uses a famous Lilian Matthiesen, Institut de theorem of Szemerédi on arithmetic progressions Mathematiques de Jussieu in dense sets as a key component. The field is also Jozsef Solymosi, University of British of great interest to computer scientists; a number of Columbia the techniques and theorems have seen application Terence Tao, University of California, Los Angeles in, for example, communication complexity, Van Vu, Yale University property testing, and the design of randomness Melanie Wood, University of Wisconsin, extractors. Madison Yufei Zhao, Massachusetts Institute of Technology Tamar Ziegler, Hebrew University The IMA is a NSF-funded institute www.ima.umn.edu/2014-2015/W9.29-10.3.14.

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Properties of High Rank Subvarieties of Affine Spaces 3
PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES DAVID KAZHDAN AND TAMAR ZIEGLER 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 [5]. 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. 1. Introduction Let k be a field. For an algebraic k-variety X we write X(k) := X(k). To simplify notations we often write X instead of X(k). In particular we write V := V(k) when V is a vector space and write kN for AN (k). For a k-vector space V we denote by Pd(V) the algebraic variety of polynomials on V of degree ≤ d and by Pd(V ) the set of polynomials functions P : V → k of degree ≤ d. We always assume that d< |k|, so the restriction map Pd(V)(k) → Pd(V ) is a bijection. For a family P¯ = {Pi} of polynomials on V we denote by X V P¯ ⊂ the subscheme defined by the ideal generated by {Pi} and by XP¯ the set XP (k) ⊂ V . We will not distinguish between the set of affine k-subspaces of V and the set of affine subspaces of V since for an affine k-subspace W ⊂ V, the map W → W(k) is a bijection.
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Arxiv:1006.0205V4 [Math.NT]
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