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On the Codimension of the Singular Locus

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On the Codimension of the Singular Locus ON THE CODIMENSION OF THE SINGULAR LOCUS DAVID KAZHDAN AND TAMAR ZIEGLER Abstract. Let k be a field and V an k-vector space. For a family P¯ = X {Pi}1≤i≤c, of polynomials on V , we denote by P¯ ⊂ V the subscheme defined by the ideal generated by P¯. We show the existence of γ(c, d) such that varieties X P¯ are smooth outside of codimension m, if deg(Pi) ≤ d and rank (strength) γ(d,c) rnc(P¯) ≥ γ(d, c)(1 + m) . 1. Introduction Let k be a field and V an k-vector space. For a family P¯ = {Pi}1≤i≤c, of X V polynomials on V we denote by P¯ ⊂ the subscheme defined by a the ideal Xsing X ¯ {Pi} and by P¯ ⊂ P¯ the subscheme of singularities. We denote by κ(P ) the Xsing X X X codimension of P¯ in P¯. If c = 1 we write P instead of P¯. For a polynomial P : V → k we denote by dP : V → V∨ the differential −1 V Xsing X −1 dP (v)(h) := ∂P/∂h(v), h ∈ . Then P = P ∩ dP (0) = dP (0) where dP (v): V → A ⊕ V∨ is given by dP (v)=(P (v),dP (v)). For a polynomial P : V → k we denote by P˜ : V d → k the multilinearf symmet- ˜ d ricf form associated with P definedf by P (h1,...,hd) := ∆h1 ... ∆hd P : V → k, where ∆hP (x)= P (x + h) − P (x). The rank r(P ) is the minimal number r such r that P can be written in the form P = i=1 QiRi, where Qi, Ri are polynomials on V of degrees <d. We define the non-classical rank (nc-rank) rnc(P ) to be the P rank of P˜, and observe that if char(k) >d (or char(k) = 0) then r(P ) ≤ rnc(P ), .since P˜ ◦ ∆= d!P where ∆ : V ֒→ V d is the diagonal imbedding The following statement is proven in [1]. Claim 1.1. There exist a function C(d,c,m) such that κ(P¯) ≥ m for any family P¯ = {Pi}1≤i≤c of polynomials Pi on V , with deg(Pi) ≤ d for 1 ≤ i ≤ c, such that r(P¯) ≥ C(d,c,m) where r(P¯) is the algebraic rank (see section 2). arXiv:1907.11750v2 [math.AG] 26 May 2020 The goal of this paper is to obtain an explicit upper bound on C(d,c,m). We first consider the special case when c = 1. Theorem 1.2. Let d > 0. There exists a function γ(d) such that for any field k, and a polynomial P of degree d and of nc-rank ≥ γ(d)(1 + m)γ(d), we have κ(P ) ≥ m. The second author is supported by ERC grant ErgComNum 682150. 1 2 DAVIDKAZHDANANDTAMARZIEGLER Remark 1.3. It is sufficient to consider the case when P ∈ k[x1,...,xn] is a homogeneous polynomial. Really for such a polynomial P of degree d there exists a unique homogeneous polynomial Q ∈ k[x0, x1,...,xn] of degree d such that Q(1, x1,...,xn) = P (x1,...,xn). Since r(Q) ≥ r(P ) and κ(P ) = κ(Q) the inequality κ(Q) ≥ m implies the inequality κ(P ) ≥ m. Now we state the Theorem in the general case. Let P¯ = {Pi}1≤i≤c be a family of polynomials on V of degrees di, 1 ≤ i ≤ c. Theorem 1.4. There exists a function γ(d,c) such that for any field k and any family P¯ of degree d¯ and nc-rank γ(c,d)(1 + m)γ(c,d), we have κ(P¯) ≥ m. Remark 1.5. Our proof only shows that the dependence of C on c and d is double exponential. Here is an outline of the proof. We denote by dP¯ : V → (Λc(V ))∨ the map ¯ such that dP (v)(h1,...,hc) := det(∆P,h¯ 1,...,hc (v)) where ∆P,h¯ 1,...,hc (v) is the (c×c)- matrix with elements {∂Pi/∂hj (v)}, 1 ≤ i, j ≤ c. Let dP¯(v) := (P¯(v),dP¯(v)). The following statement is well known. −1 Claim 1.6. Xsing ¯ f P¯ = dP (0). c c ∨ For any subspace fL ⊂ Λ (V ) we denote by pL : Λ (V ) → L the natural ∨ projection and by Q¯L the composition pL ◦ dP¯ : V → L . We see that for a −1 proof of Theorem 1.4 it is sufficient to find a good upper bound on dim(dP¯ (0)). To obtain such a bound we construct a large fsubspace L ⊂ Λc(V ) such that the map Q¯L is of analytic rank > dim(L). Then (see Claim 3.3 ) all fibersf of Q¯L −1 ¯ ¯−1 are of dimension dim(V ) − dim(L). Since dP (0)) ⊂ QL (0) we see that the Xsing X codimension of P¯ in P¯ is not less than dim(L) − c. f 2. definitions Let k be a field and k¯ it algebraic closure. Definition 2.1. 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 ˜ d P defined by P (h1,...,hd) := ∆h1 ... ∆hd P : V → k, where ∆hP (x) = P (x + h) − P (x). ¯ V X V (2) For a family P = {Pi}1≤i≤c, of polynomials on we denote by V¯ ⊂ the subscheme defined by the system {v : Pi(v)=0} of equations. (3) For a polynomial P ∈ k[x1,...,xN ] of degree d > 1 we define the rank r(P ) as the minimal number r such that P can be written as a sum r P = j=1 QiRi where Qj, Rj ∈ k[x1,...,xN ] are polynomials of degrees P ONTHECODIMENSIONOFTHESINGULARLOCUS 3 <d 1. For d = 1 we define the rank to be infinite and for d = 0 we define it as 0. (4) We define the non-classical rank (nc-rank) rnc(P ) to be the rank of P˜. (5) For a d tensor T : V d → k we define the partition rank (p-rank) pr(T ) as the minimal number r such that T can be written as a sum T (x1,...,xd)= c Ji⊂[1,d],i∈[r] Qj((xl)l∈Ji )Rj((xl)l∈Ji ) where Qj, Rj are degree <d tensors. (6) If k = F and P : V → k, we define b(P )= ψ(T (v))/qdim(V ) where P q v∈V ψ : k → C⋆ is a non-trivial additive character and define the analytic rank ˜ P ar(P ) as − logq(b(P )). (7) For a linear subspace L ⊂ k[x1,...,xN ] we define r(L) := minP ∈L−{0} r(P ), and rnc(L), ar(L) similarly. (8) For a linearly independent family P¯ = {Pi}1≤i≤c, of polynomials we define r(P¯) := r(L(P¯)), where L(P¯) is the linear span of P¯, and rnc(P¯), ar(P¯) similarly. (9) For a subscheme X ⊂ V we define r(X),rnc(X), ar(X), as the maximal ¯ ¯ ¯ ¯ X X rank of r(P ),rnc(P ), ar(P ) for families P = {Pi}1≤i≤c, such that = P¯. (10) For a family P¯ we denote by κ(P¯) the codimension of the singular locus Xsing X P¯ ⊂ P¯. Remark 2.2. (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<i<j<k<l≤n xixjxkxl is of rank ∼ n, but of nc-rank 3, see d P (3) rnc(P ) ≤ pr(P˜) ≤ Cdrnc(P ), Cd ≤ 4 . (see [3]). In char >d, Cd = 1. Remark 2.3. (1) For any tensor T we have that b(T ) ≥ 0 and does not depend on a choice of a character ψ. (2) For any tensor T , we have that pr(T ) ≤ − logq b(T ) (see [2], [5]). (3) |b(P )|2d ≤ b(P˜). ¯ c ¯ Claim 2.4. Let P = {Pi}l=1 be a family of polynomials with r(P ) =6 0. Then (1) dim(L(P¯)) = c. j ¯ (2) There exists a basis Qi in L(P ) such that j (a) deg(Qi )= j. ¯j j ¯ (b) For any j the family Q = {Qi } is of rank ≥ r(P ). X ¯ Remark 2.5. Since the subscheme P¯ depends only on the space L(P ), we can (and will) assume that all the families P¯ we consider satisfy the conditions of Claim 2.4 on Q¯. Definition 2.6. Let L ⊂ k[x1,...,xN ] be a finite dimensional subspace. 1 Also known as h-invariant, or strength. For tensors it is known as the partition rank. 4 DAVIDKAZHDANANDTAMARZIEGLER ∨ (1) We denote by φL : V → L the map given by φL(v)(P ) := P (v) for P ∈ L and v ∈ V . ∨ −1 −1 (2) For λ ∈ L we define f(λ) := |V | |φL (λ)|. (3) We denote by fˆ : L → C the Fourier transform of f given by fˆ(l) := ψ(hλ,li)f(λ). ∨ λX∈L (4) For a polynomial P on V, h ∈ V we write Ph(v)= P (v + h) − P (v). (5) For a subspace W ⊂ V we denote by L(W ) the subspace of polynomials on V spanned by P and Pw, w ∈ W . 3. Results from the additive combinatorics From now on we fix a non-trivial additive character ψ : k → C⋆. We denote E −1 s∈S = |S| s∈S. Denote by Ud the Gowers Ud-uniformity norm for functions f : V → C defined by: P d 2d E ω kfkUd = x,v1,....vd∈V f (x + ωivi) d i=1 ω∈{Y0,1k X ω ¯ d ω where f = f if |ω| = i=1 ωi is odd and otherwise f = f.
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