Properties of High Rank Subvarieties of Affine Spaces
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GAFA Geometric And Functional Analysis
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 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) ⊂ k n 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 k N 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 XP¯ ⊂ V 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. In the introduction we consider only the case of hypersurfaces X ⊂ V and provide an informal outline of main results. Precise definitions appear in the next section. Definition 1.1. Let P be a polynomial of degree d on a k-vector space V . (1) We denote by P˜ : V d → k the multilinear symmetric form associated with P defined by P˜ (h1 , . . . , hd ) := Δh1 . . . Δhd P : V d → k, where Δh P (x) = P (x + h) − P (x). T. Ziegler is supported by ERC Grant ErgComNum 682150.
D. KAZHDAN AND T. ZIEGLER
GAFA
(2) The rank r(P ) is the minimal number r such that P can be written in the form P = ri=1 Qi Ri , where Qi , Ri are polynomials on V of degrees < d. (3) We define the non-classical rank (nc-rank) rnc (P ) to be the rank of P˜ . (4) A polynomial P is m-universal if for any polynomial Q ∈ Pd (k m ) of degree d there exists an affine map φ : k m → V such that Q = P ◦ φ. (5) We denote by XP ⊂ V the hypersurface defined by the equation P (v) = 0 and the singular locus of XP . by Xsing P (6) s(P ) := codimXP (Xsing P ). Remark 1.2. (1) If char(k) > d then r(P ) ≤ rnc (P ). ˜ (2) In low characteristic it can happen that P is of high rank and P is of low rank, for example in characteristic 2 the polynomial P (x) = 1 d. 1.4 Nullstellensatz over Fq . for polynomials over Fq .
We prove the following variant of Nullstellensatz
Theorem 1.12. There exists r(d) such that for any finite field k = Fq
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