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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
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 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. Claim 3.1 ([2, 5]). ForP any d > 0 and a polynomial P : V → k of degree d we have ar(P ) ≤ pr(P˜), i.e. 2d −pr(P˜) E d ˜ E h∈V ψ(P (h1,...,hd)) = k x∈V ψ(P (x))kUd > q . The following result is proven in [6] (and independently, with a slightly weaker bound that depends on the field k, by [4]) Theorem 3.2. There exists a triply exponential function C(d) and double expo- nential function D(d) such that pr(P˜) ≤ C(d)(ar(P )D(d) +1) for all polynomials of degree d. We rewrite Theorem 3.2 in the following way: For any d > 0 there exists Ad, Bd > 0 such that for any polynomial P : V → k of degree d, we have that if E ˜ −s x∈V d ψ(P (x)) > q , Bd then pr(P˜) < Ad(s + 1). The dependence of the constants on d is double exponential for Bd snd triply exponential for Ad. Alternatively, for any polynomial P : V → k of degree d there exists αd, βd > 0 such that it pr(P˜) >r we have that β E E −(αdr−1) d | x∈V ψ(P (x))|≤k x∈V ψ(P (x))kUd < q . Claim 3.3. If ar(L) > dim(L) then |f(λ)/f(µ) − 1| ≤ 1/2 for all λ,µ ∈ L∨(W ). ONTHECODIMENSIONOFTHESINGULARLOCUS 5 −1 ˆ ˆ Proof. Since f(λ)= |L| l∈L ψ(hλ,li)f(l) and f(0) = 1 it is sufficient to show that |fˆ(l)| ≤ 1/qdim(L)+1 for l ∈ L −{0}. But this inequality follows immediately P from the definition of ar(L) and the inequality ar(L) > dim(L). 4. The case of one a hypersurface In this section we present a proof of Theorem 1.4 in c = 1. Theorem 4.1. There exists a function γ(d) such that κ(P ) ≥ m for any polyno- mial P degree d and of nc-rank γ(d)(1 + m)γ(d). Proof. To simplify notations we define 1/Bd T = T (r, m, d) := [(r/Ad) − m]/2Cd, where Ad, Bd are from Theorem 3.2, and Cd is from Remark 2.2. We start with the following statement. Proposition 4.2. There exists a function r(d) such that the following holds. For any finite field k of characteristic >d, for any m ≥ 1 and a polynomial P : V → k of degree d and nc-rank rnc(P ) > r(d) there exist h1,...,hm ∈ V such that the ¯ family Q := {P,Ph1 ,...,Phm } is of nc-rank > T . Proof. Fix d, m. Since P is of degree d and Pw is of degree Suppose for anyw ¯ we have that {Pw1 ,...,Pwm } is of nc-rank ≤ T . Then for m anyw ¯ there exists b ∈ k \ 0 such that biPwi is of nc-rank ≤ T . By the m pigeonhole principle there exists b ∈ k \ 0 such that the nc-rank of biPwi ≤ T m m P for a set F ofw ¯ of size ≥|V | /q . Fix this b. W.l.o.g. b1 =6 0. This implies that P there exists (w2,...,wm) such that (w1,w2,...,wm) ∈ F for a set F1 if w1 of size m −m 2 ′ ≥|V |/q . Now subtract to get for q |V | many pairs w1,w1 we have ′ b1(Pw1 − Pw1 ) is of nc-rank < 2T . So that P ′ is of nc-rank < 2T for q−m|V |2 many w ,w′ . w1−w1 1 1 But this implies by Remark 2.2 that the p-rank of P˜ ′ is < 2TC , so that w1−w1 d d d−1 2 E 2 −2T Cd−m kψ(P )kUd = wkψ(Pw)kUd−1 > q , so that P is of nc-rank Bd < Ad(2TCd + m) . To get a contradiction we need Bd Ad(2TCd + m) < r. Namely 1/Bd T < [(r/Ad) − m]/2Cd. 6 DAVIDKAZHDANANDTAMARZIEGLER β Corollary 4.3. |fˆ(l)| ≤ q−(αdT −1) d for l ∈ L \{0}. Now we prove Theorem 4.1. We write r = ma, a> 0. Then Proposition 4.2 guaranties the existence ¯ of h1,...,hm ∈ V such that the family Q := {P,Ph1 ,...,Phm } is of nc-rank 1/Bd −1 > [(r/Ad) − m]/2Cd. Assuming that a(Bd) ≥ 2 we see that for m > m0(d) −1 a(Bd) we have that rnc(Q¯) ≥ c(d)m . As follows from Proposition 3.2 we have −1 ¯ ′ aBd βd ′ −1 ar(Q) ≥ c (d)(m − 1) for some c (d) > 0. Choose a such that aBd βd ≥ 2. Then for m > m1(d) we have ar(Q¯) ≥ m + 1 It is easy to see the existence of γ(d) > 0 such that T (m) > m for m = rγ(d). X s(dim(V )−rγ(d)) It follows now from Claim 3.3 that | Q¯ (ks)| ≪ q ,s ≥ 1 where X ks/k is the extension of degree s. Therefore (see Section 3.5 of [3]) dim( Q¯ ) = dim(V ) − rγ(d). The arguments of Section 3.6 of [3] show the the same equality Xsing X is true in the case of an arbitrary field. Since (by Claim 1.6) P ⊂ Q¯ we see that κ(P ) ≥ m. 5. higher codimension Fix numbers s and n and denote by m = ⌊n/s⌋, and by l := n − sm Let s n L = k ,V = L ⊗ k = L1 × ...Ln, Li = L, Wj = Vsj ×···× Vs(j+1)−1 for l 1 ≤ j ≤ m. So V = W1 ×···× Wm × k . j We denote by yi for 1 ≤ i ≤ n, 1 ≤ j ≤ s the natural coordinates on V and n define a polynomial of degree s, Fs : V → k by n 1 s 1 s Fs (y ,...,y ) := det(yi1 ,...,yis ) 1≤i <... 0 such that b(Fs ) ≤ q . Proof. Consider first the case s = 2. Then P (a1,...,an; b1,...,bn)= (aibj − ajbi). 1≤i Qi := ∂P/∂ai = bj − bj, 1 ≤ i ≤ n − 1. j>i j Qi+1(a1,...,an; b1,...,bn) − Qi(a1,...,an; b1,...,bn)= bi+1 − bi, n 1−n we see the Y is defined by the system bi+1 −bi = 0 of equations. So b(F2 )= q . ONTHECODIMENSIONOFTHESINGULARLOCUS 7 t t t 1 Consider now the general case. We write y =(a1,...,an). Let Qi := ∂P/∂yi n(d−1) for 1 ≤ i ≤ n. We define Ys ⊂ W := k as the set of solutions of the system 1 {Qi (w)=0}1≤i≤n−1. n −sn As before ar(Fs )= q |Ys|. (s−1)n Let Zs ⊂ k be defined by the system {Rj =0}1≤j≤n/s of equations where 1 1 Rj := Qs(j+1) − Qsj, 1 ≤ j ≤ m. Since Ys ⊂ Zs it is sufficient to show that (ns−m) |Zs| ≤ q . j By the construction Rj are non-zero polynomials on Wj,1 ≤ j ≤ m. Let Zs ⊂ Wj be defined by the equation Rj(w) = 0. Since Rj is irreducible it follows from j 2 −1/2 s2−1 1 m e the Weil’s bounds that |Zs | ≤ (1+ s q )q . Since Zs = Zs ×···× Zs × k (ns−m) we see that |Zs| ≤ q ′ ′ e Denote di = di − 1, and let e = di, and. denote fi = ′ . Denote S(d; e) di subsets of [e]= {1,...,e} of size d. Denote by Xi the set of subsets of [1, e] of size ′ P di. We write Vi for the spaces of k[Xi] of k-valued functions on Xi. These spaces n have bases parametrized by elements of Xi and define Wi := Vi ⊗ R, R := k i,xi and W := ⊕iWi. So a point w ∈ W has coordinates wm , 1 ≤ m ≤ n, 1 ≤ i ≤ s, xi ∈ Xi. For any sequence λ = (i1 < ··· < is) andx ¯ = (x1,...,xs), xj ∈ Xj we denote by hλ,x¯ the polynomial on W which is the determinant of an j,xj s × s matrix with elements aij = wli . Consider the function n 1,x1 s,xs 1,x1 s,xs Gs ((y )x1∈X1 ,... (y )xs∈Xs )= det(yi1 ,...,yis ) s x ∈X , x =[e] 1≤i1<... n −c0n/s Claim 5.2. b(Gs ) ≤ q . n snm Consider the collection of polynomials Fi,s : k → k defined by n 1 1 s s n 1 s Fi,s(y1,...,ym,...,y1,...,ym) := Fs (yi ,...,yi ); 1 ≤ i ≤ m. Then the bias of any non trivial linear combination of this collection is ≤ q−c0n/s. n m Similarly {Gi,s}i=1. Theorem 5.3. For any m,d,s > 0, 1 < di ≤ d, 1 ≤ i ≤ s there exists C = C(m,d,s) > 0 such for any polynomial family {P 1,...,P s} on a k vector space i C V , P of degrees di for 1 ≤ i ≤ s, and nc-rank > Cr the following holds: there exist t = t(r, s, m) and w1,...,wtm ∈ V , such that the collection Q¯ of functions 1 s t 1 1 s s {P ,...,P , Fi,s(Pw1 ,...,Pwtm ,...,Pw1 ,...,Pwtm ); i ∈ [m]} is of nc-rank >r. I will assume that di > 1 this is ok ? 8 DAVIDKAZHDANANDTAMARZIEGLER −1 a2(s,d) Proof. To simplify notations we write T = T (m,s,d) := m a1(s,d)r − m, where the constants a1(s,d), a2(s,d) will be chosen later and will depend on the constants Ad, Bd,αd, βd from Proposition 3.2. 1 s Proposition 5.4. For any polynomials P ,...,P : V → k of degrees di, 1 < i ≤ s, and nc-rank r>r(d,s) there exist t > 0 and w1,...,wtm ∈ V , such that the family ¯ 1 s t 1 1 s s Q := {P ,...,P , Fi,s(Pw1 ,...,Pwtm ,...,Pw1 ,...,Pwtm ); i ∈ [m]} is of nc-rank > T . Proof. Fix d, m, s> 1. We need to findw ¯ =(w1,...,wtm) such that 1 s t 1 1 s s {P ,...,P , Fi,s(Pw1 ,...,Pwtm ,...,Pw1 ,...,Pwtm ), i ∈ [m]} is of nc-rank > T . Suppose for anyw ¯ =(w1,...,wtm), the system 1 s t 1 1 s s (∗) {P ,...,P , Fi,s(Pw1 ,...,Pwtm ,...,Pw1 ,...,Pwtm ), i ∈ [m]} of polynomials is of nc-rank < T , but the system {P 1,...,P s} is of nc-rank > T . Then for anyw ¯ there exists b ∈ ks,c ∈ km \ 0 such that s m j t 1 1 s s bjP + ciFi,s(Pw1 ,...,Pwtm ,...,Pw1 ,...,Pwtm ) j=1 i=1 X X is of nc-rank < T . By the pigeonhole principle there exists b ∈ ks,c ∈ km \ 0 such s j m t tm m+s that j=1 bjP + i=1 ciFi,s is of nc-rank < T forasetofw ¯ of size ≥|V | /q . m t s ′ Fix these b, c. Since the degree of i=1 ciFi,s is e = i=1 di > max di, we get thatP the p-rank ofP P P m t 1 1 s s ∆ue ... ∆u1 ciFi,s(Pw1 (x),...,Pwtm (x),...,Pw1 (x),...,Pwtm (x)) By Claim 5.2 the function m 1 1 s s t 1,x1 1,xtm s,x1 s,xtm g(y)= ciGi,s((y1 ),... (ytm ),..., (y1 ),,...,, (ytm )), i=1 i X i i,xj i,xj e i ′ where (yj ) denote the tuple (yj )x ∈Xi , is of p-rank ≥ c0t/2sc. Denote fi = , j di and let f = s f i=1 i Consider the function ψ ◦ g : ktmf → C. By Fourier analysis, P ψ ◦ g(y)= aχχ(y) χ∈Xkˆtmf Since g is multilinear, and by Claim 5.2 we have that the coefficient of the trivial character is bounded : −c0t/s |Eyψ ◦ g(y)| < q We get 1 1 s s (⋆) E a χ((P˜1,x1 ),..., (P˜1,xtm ),..., (P˜s,x1),..., (P˜s,xtm )) u χ w1 wtm w1 wtm ˆtmf χ6=1X,χ∈k is greater equal than 1,x1 1,x1 s,xs s,xs −c t/s E a χ((P˜ 1 ),..., (P˜ tm ),..., (P˜ 1 ),..., (P˜ tm)) − q 0 u χ w1 wtm w1 wtm ˆtmf χ∈Xk so that (⋆) is ≥ q−T Ce−m−s − q−c0t/s. Choose t so that q−c0t/s < q−2(T Ce+m+s), i.e. t = ⌈((s/c0)2(TCe + m + s))⌉. Thus for anyw ¯ we can find χ =16 , χ ∈ kˆtmf such that 1 1 s s E ˜1,x1 ˜1,xtm ˜s,x1 ˜s,xtm −2(T Ce+m+s) uaχχ((Pw1 ),..., (Pwtm ),..., (Pw1 ),..., (Pwtm )) ≥ q . Set D = 2(TCe + m + s). By the pigeonhole principle there is a set A ofw ¯ of size ≥ q−tmf |V |tm, and χ =16 , χ ∈ kˆtmf such that for allw ¯ ∈ A we have 1 1 s s E ˜1,x1 ˜1,xtm ˜s,x1 ˜s,xtm −D uaχχ((Pw1 ),..., (Pwtm ),..., (Pw1 ),..., (Pwtm )) ≥ q This implies that there is a non trivial linear combination of 1 1 s s ˜1,x1 ˜1,xtm ˜s,x1 ˜s,xtm (Pw1 ),..., (Pwtm ),..., (Pw1 ),..., (Pwtm ) Bd for allw ¯ ∈ A, which is of p-rank ≤ Ad(D + 1). Without loss of generality the coefficient of the first term is not zero. This ′ −tmf 2 implies that for a set of w1,w1 of size ≥ q |V | , we have that 1 1 1,x 1,x1 P˜ 1 (u) − P˜ ′ (u) w1 w1 10 DAVIDKAZHDANANDTAMARZIEGLER Bd is of p-rank ≤ 2Ad(D + 1) 1 −tmf 1,x1 But this implies that for a set of size ≥ q |V | of w we have P˜w is of p-rank Bd ≤ 2Ad(D + 1) So that Bd ′ 1 −2Ad(D +1)−tmf E E d ψ(P˜ (u)) ≥ q . w∈V u∈V 1 w But this now implies that d B kψ(P 1)k2 1 ≥ q−2Ad(D d +1)−tmf , Ud1 and thus P is of p-rank Bd Bd ≤ Ad((2Ad(D +1)+ tmf) + 1) To get a contradiction we need the above to be Bd Bd Ad((2Ad((2(TCe + m + s)) +1)+ tmf) + 1) −1 a2(s,d) TCe < m a1(s,d)r − m. with a1(s,d), a2(s,d) depending polynomially on s and on Ad, Bd, from Proposi- tion 3.2. We can consider the collection Q¯ of 5.3 as a map Q¯ : AN → Af . Since (by Xsing ¯−1 Claim 1.6) P¯ ⊂ Q (0) we derive Theorem 1.4 from Proposition 5.4 using the same arguments as in the derivation of Theorem 1.2 from Proposition 4.2. References [1] Ananyan, T., Hochster, M. Small Subalgebras of Polynomial Rings and Stillman’s conjec- ture arXiv:1610.09268. [2] Kazhdan, D., Ziegler, T. Approximate cohomology. Sel. Math. New Ser. (2018) 24: 499. [3] Kazhdan, D., Ziegler, T. Properties of high rank subvarieties of affine spaces. arXiv:1902.00767. [4] Janzer, O. Low analytic rank implies low partition rank for tensors. arXiv:1809.10931. [5] Lovett, S. The Analytic rank of tensors and its applications. Discrete Analysis 2019:7. [6] Mili´cevi´c, L. Polynomial bound for partition rank in terms of analytic rank arXiv:1902.09830. Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel E-mail address: [email protected] Einstein Institute of Mathematics, Edmond J. Safra Campus, Givaat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel E-mail address: [email protected]