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(3) We define the non-classical rank (nc-rank) rnc(P ) to be the rank of P˜. m (4) A polynomial P is m-universal if for any polynomial Q ∈ Pd(k ) of degree d there exists an affine map φ : km → V such that Q = P ◦ φ. (5) We denote by XP ⊂ V the hypersurface defined by the equation P (v)=0 Xsing X and by P the singular locus of P . Xsing (6) s(P ) := codimXP ( 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
Theorem 1.9. Let k be an algebraically closed field, C the category of finite- dimensional affine k-vector spaces with morphisms being affine maps, let Fd be the contravariant endofunctor on C given by
Fd(V )= {Polynomials on V of degree ≤ d}, and let G⊂F be a proper subfunctor. Then there exists r such that rnc(P ) ≤ r for any finite-dimensional k-vector space V and P ∈ G(V ). 1.0.3. Extending weakly polynomial functions. Let k be a field, V a k-vector space and X a subset of V . A function f : X → k is weakly polynomial of degree ≤ a, if the restriction of f on any affine subspace L of V contained in X is a polynomial of degree ≤ a. Theorem 1.10. Let k be a field which either an algebraically closed field, or a finite field such that |k| > ad and let X ⊂ V be a hypersurface defined by a polynomial of degree d of sufficiently high nc-rank. Then any k-valued weakly polynomial function of degree ≤ a on X is a restriction of a polynomial F on V of degree ≤ a. Remark 1.11. The main difficulty in a proof of Theorem 1.10 is the non- uniqueness of F in the case when a>d.
1.0.4. Nullstellensatz over Fq. We prove the following variant of Nullstellensatz for polynomials over Fq.
Theorem 1.12. There exists r(d) such that for any finite field k = Fq, a k- vector space V and a k-polynomial P of degree d and nc-rank larger than r(d) the following holds: Any polynomial R of degree < q/d vanishing at all points x ∈ XP is divisible by P . Remark 1.13. (1) This result is a strengthening of the Proposition 9.2 from [8] in two ways: (a) We show the independence of r(d) on the degree of R, and (b) The paper [4] shows the existence of polynomials Qi of bounded de- c X grees such that R(x) = i=1 Qi(x)Pi(x), for all x ∈ (k), but does not show that R is contained in the ideal generated by {Pi}. (2) We outline ideas of proofs ofP Theorems 1.10, 1.12 at the beginning of corresponding sections. (3) The quantitative bound on the rank in all proofs depend only on the bounds in Theorem 3.4. We conjectured that the bound on r in Theorem 3.4 depends polynomially on s. This conjecture was proved in [26] (and in a slightly weaker form in [12]).
2. The formulation of results. We start with a series of definitions. 4 DAVIDKAZHDANANDTAMARZIEGLER
Definition 2.1 (Rank). Let k be field and V a k-vector space. (1) A tensor of degree d on V is a multilinear map T : V d → k. (2) Let P : V → k be a polynomial of degree d. We define the rank r(P ) as the r minimal number r such that P can be written as a sum P = j=1 QjRj where Q , R are polynomials of degrees < deg(P ) defined over k 1. We j j P define the rank of a degree 1 polynomial to be infinite, and the rank of the zero polynomial to be 0. (3) For a polynomial P : V → k of degree d we define P˜(h1,...,hd) := d ∆h1 ... ∆hd P : V → k, where ∆hP (x)= P (x + h) − P (x). (4) We define the non-classical rank (nc-rank) rnc(P ) to be the rank of r(P˜). (5) For a d tensor P : V d → k we define the partition rank (p-rank) pr(P ) as the minimal number r such that P can be written as a sum P (x1,...,xd)= c Ji⊂[1,d],i∈[r] Qj((xl)l∈Ji )Rj((xl)l∈Ji ) where Qj, Rj are degree Remark 2.2. The rank r(P¯) depends only on the linear span L(P¯) of (Pi). The following statement is immediate. ¯ c ¯ Claim 2.3. Let P =(Pi)i=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 2.4. 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 2.3 on Q¯. We also may assume that P¯ has no affine polynomials, since we can initially restrict to the locus of the affine polynomials. Lemma 2.5. Let P be a d-tensor. Then r(P ) ≤ pr(P ) ≤ 4dr(P ). Proof. Let P be a d-tensor. It is clear that r(P ) ≤ pr(P ). We show that pr(P ) ≤ 4dr(P ). Let k be a field with |k| ≥ d + 2 and write G = (k⋆)d. For any ¯j = ⋆ ¯ d ji {j1,...,jd}, jd ≤ d we denote by χ¯j : G → k the character t → i=1 ti where t¯ =(t1,...,td). Q 1This notion of rank is also known as Schmidt-rank in the analytic number theory literature or strength in the algebraic geometry literature. PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 5 Let Vi, 1 ≤ i ≤ d be k-vector spaces and P the space of polynomials P : d V1 ×...×Vd → k of degree ≤ d. Let J be the space of maps ¯j from [1,d] → [1,d]. For any ¯j ∈ J we denote by P¯j the subspace of polynomials P : V1 × ... × Vd → k such that d ji P (t1v1,...,tdvd)= ti P (v1,...,vd). i=1 Y Since |G| is prime to char(k) and |k| ≥ d + 2 we have a direct sum decomposition P = ⊕¯j∈J P¯j. We denote by p¯j : P → P¯j the projection. We write 1¯ = (1,..., 1) ∈ J denote by pd : P → P1¯ the projection. For any a, b, a + b = d we denote by Ja,b ⊂ J × J the set of pairs ¯ja, ¯jb such that supp(¯ja) ∩ supp(¯jb)= ∅, deg(¯ja)= a, deg(¯jb)= b. For any polynomials Q, R ∈ P of degrees a, b, a + b = d we write P (Q, R) = p (Q)p (R). We have that p¯(QR)= P (Q, R), and therefore pr(P ) ≤ (j1,j2)∈Ja,b j1 j2 1 4dr(P ). P Definition 2.6. (1) For m ≥ 1 and a k-vector space V , we denote by Affm(V) the algebraic m variety of affine maps φ : A → V and write Affm(V ) := Affm(V)(k). V Am (2) We define an algebraic morphismκ ˜P¯ : Affm( ) → Pd¯( ) byκ ˜P¯(φ) := ¯ m P ◦ φ, and denote by κP¯ the corresponding map Affm(V ) → Pd¯(k ). Definition 2.7. A map κ : M → N between finite sets is ǫ-uniform, where ǫ> 0, if for all n ∈ N we have |N||κ−1(n)|−|M| ≤ ǫ|M|. ¯ Remark 2.8. In this paper we say that a bound r(d, m, t) is effective if (1) For fixed d := maxi di, the bound is polynomial in m, t, c. (2) The dependence on d is doubly exponential. The effective lower bounds for r(d,¯ m, t) follow from the Conjecture 3.5 proven in [26]. Theorem 2.9. For any sequence d¯ and any m, t ≥ 1, there exists an effective bound r(d,¯ m, t) such that for any finite field k = Fq, a k-vector space V and a ¯ ¯ −t family P ∈ Pd¯(V ) of nc-rank ≥ r(d, m, t), the map κP is q uniform. To formulate Remark 1.3 precisely we introduce a number of additional defi- nitions. Z Definition 2.10. Fix an affine hypersurface W ⊂ V . We denote by P¯ the Y Z variety of affine m-dimensional subspaces L ⊂ XP¯ ∩ W and by P¯ ⊂ P¯ the X subvariety consisting of L ⊂ P¯ ∩ W such that there is no (m + 1)-dimensional X affine subspace M ⊂ P¯, M 6⊂ W containing L. 6 DAVIDKAZHDANANDTAMARZIEGLER Theorem 2.11. For any d¯, any m, t ≥ 1, there exists an effective bound r˜(m, t, d¯) with the following property. For any finite field k = Fq, any k-vector space V and ¯ ¯ |YP¯ | −t any family P ∈ Pd¯(V ) of nc-rank ≥ r˜(m, t, d), we have that ≤ q . |ZP¯ | Remark 2.12. The condition that r(P ) ≫ 1 does not imply that YP¯ = ∅. To see that r(P ) ≫ 1 does not imply the emptiness of the set YP¯ consider the case n n−1 d V = k , W = {w ∈ V : wn =0}, m =1,P = i=1 xi + xn and L = ke1. 2.1. Applications. In this subsection we provideP precise formulations of Theo- rems 1.4, 1.10 and 1.12. 2.1.1. The surjectivity over algebraically closed fields. We start with a formaliza- tion of Theorem 1.4. Theorem 2.13 (Effective Stillman conjecture). There exists an effective bound r(m, t, d¯) with the following property. For any sequence d¯ and m, t ≥ 1, any algebraically closed field k, any k-vector ¯ ¯ space V and any family P ∈ Pd¯(V ) of nc-rank ≥ r(m, t, d) the following holds (1) The map κP¯ is surjective. (2) All fibers of the morphism κ˜P¯ are of the same dimension. (3) The morphism κ˜P¯ is flat. (4) P¯ ⊂ k[V ∨] is a regular sequence. Remark 2.14. (1) Parts (1) and (2) follow from Theorem 2.9. (2) The part (3) follows from the parts (1),(2) and Theorem 23.1 in [22]. The part (4) is an immediate consequence of the part (3). (3) As shown in [1] the part (4) of the theorem implies the validity of an effective Stillman conjecture. In the formulation of the next result we use notation introduced in Definition 2.10. Theorem 2.15. For any d¯, m, t ≥ 1, there exists effective bound r˜(m, t, d¯) with the following property. For any algebraically closed field k,a k-vector space V and ¯ ¯ Z Y a family P ∈ Pd¯(V ) of nc-rank ≥ r˜(m, t, d), we have that dim( P¯ )−dim( P¯) ≥ t. 2.1.2. Extending weakly polynomial functions. Definition 2.16. (1) Let V be a k-vector space and X ⊂ V . We say that a function f : X → k is weakly polynomial of degree ≤ a if the restriction f|L to any affine subspace L ⊂ X is a polynomial of degree ≤ a. (2) X satisfies ⋆a if any weakly polynomial function of degree ≤ a on X is a restriction of a polynomial function of degree ≤ a on V . Below is a formalization of Theorem 1.10 from the introduction. PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 7 Theorem 2.17. For any d¯ and a ≥ 1 there exists an effective bound r(d,¯ a) such for any field k which is either finite with |k| > ad or algebraically closed, a k- ¯ ¯ vector space V and a family P ∈ Pd¯(V ) of nc-rank ≥ r(d, a) the subset XP¯ ⊂ V has the property ⋆a. 2.1.3. Nullstellensatz over Fq. Below is a formalization of Theorem 1.12 from the introduction. Theorem 2.18. For any sequence d¯, there exists an effective bound r(d¯) > 0 such that the following holds. Let k = Fq be a finite field and let V be a k-vector space and P¯ = {Pi} be a family of k-polynomials of degrees ≤ di on V of nc-rank ¯ ˜ ˜ c greater than r(d). Then any polynomial R of degree < q/d, d := i=1 di, such that R(x)=0 for each x ∈ X ¯(k), belongs to the ideal J(P¯) := (P1,...,Pc). P Q 2.2. Acknowledgement. We thank U. Hrushovski for his help with simplifying the proof of Theorem 2.18. 3. Analysis In the main part of this section we prove Theorems 2.9 and 2.11 using the results on equidistribution of high rank families of polynomials. These are based on the technique of the additive combinatorics. At the end of the section we apply these results to prove Theorems 2.13, 2.15 and 1.9. 3.1. Equidistribution of high rank families of polynomials. The most ba- sic result is the following proposition on equidistribution of high rank families of polynomials: ¯ Proposition 3.1. For any sequence d =(d1,...,dc) and any s ≥ 1 there exists an effective bound r(d,¯ s) with the following property. For any finite field k = Fq,a k- ¯ ¯ ¯ c vector space V and a family P ∈ Pd¯(V ) of nc-rank ≥ r(d, s) the map P : V → k is q−s-uniform. The main ingredient of this proof comes from the relation between the bias of exponential sums and algebraic rank. Let k be a finite field, char(k) = p, |k| = q. Let V be a vector space over k. 2πiψ(x)/p We denote eq(x) = e , where ψ : k → Fp is the trace function. Then eq is a non trivial additive character. Let P : V → k be a polynomial of degree ˜ d d. Recall that P (h1,...,hd) == ∆hd ... ∆h1 P (x) is the multilinear form on V associated with P , and we can write d d |ω| P˜(h1,...,hd)= (−1) P (x + ω · h¯); |ω| = ωi, ω · h¯ = ωihi. d i=1 i=1 ω∈{X0,1} X X E −1 We denote by x∈Sf(x) the average |S| x∈S f(x). P 8 DAVIDKAZHDANANDTAMARZIEGLER Definition 3.2 (Gowers norms [6]). For a function g : V → C we define the norm kgkUd by 2d E ω kgkUd = x,v1,...vd∈V g (x + ω · v¯), d ω∈{Y0,1} where gω = g if |ω| is even and gω =g ¯ otherwise. Definition 3.3 (Analytic rank). The analytic rank of a polynomial P : V → k of degree d is defined by arank(P )= − logq keq(P )kUd . The key analytic tool in this paper is the following theorem relating bias and rank. Theorem 3.4 (Bias-rank). (1) Let s,d > 0. There exists r = r(s,d) such that for any finite field k of size q, any vector space V over k, any polynomial P : V → k of degree d the following holds. If P is of nc-rank >r then 2d E ˜ −s keq(P )kUd = | v∈V eq(P (h1,...,hd))| < q . (2) Let r,d > 0. For any finite field k of size q, any vector space V over k, any polynomial P : V → k of degree d, if E ˜ −r | h1,...,hd eq(P (h1,...,hd)| < q for some polynomial, then P is of partition rank >r. Proof. Part (1) was proved for partition rank in increasing generality in [8, 14, 4]. The most general version of the first part can be found at the survey [9] (Theorem 8.0.1). Part (2) for was observed in [19], [23]. Now the Theorem follows from Lemma 2.5. Remark 3.5. The dependence of r on s in (1) is polynomial, and double expo- nential in d. This was shown for d = 2, 3 in [10], for d = 4 in [21] and in full generality in [26] (A similar bound but with a weak dependence in |k| was proved independently in [12]). Remark 3.6. By repeated applications of Cauchy-Schwartz we have that E | x∈V eq(P (x))|≤keq(P )kUd . Remark 3.7. (1) We use Claim 1.2 in our proof of Theorems 2.9 and 2.11. (2) Let P : V → k be a polynomial of degree d and rank R, and let W ⊂ V be a subspace of codimension s. Then the rank of P|W is ≥ R − s. Proof. We may assume that V = An and W ⊂ V consists of vectors (xi), 1 ≤ i ≤ n such that xi = 0 for i ≤ s. If the rank of P|W is < R −s we can find polynomials Qj, Rj, with j ≤ r, fro some r < R−s of degrees Now we can prove Proposition 3.1. ¯b ¯ ¯ Proof. The number of points on XP¯ = {x : P (x)= b} is given by c −c q eq ai(Pi(x) − bi) . c i=1 ! a¯X∈k Xx∈V X By Theorem 3.4 and Remark 3.6, for any s> 0 we can choose r so that for any a¯ =6 0 we have c −s eq ai(Pi(x) − bi) < q |V |. ! x∈V i=1 X X 3.2. Proof of Theorem 2.9. Let ǫ > 0 we say that a property holds for ǫ a.e.s ∈ S if it holds for all but (1 − ǫ)|S| of the elements in S. Theorem 3.8. For any d¯ there exists C = C(d¯) such that for any s,m > 0 there ¯ exists an effective bound r = r(s, d) such that if P¯ = (Pi)1≤i≤c is a collection of n polynomials on V = k with deg Pi = di and the nc-rank P¯ is >r then: m (1) For any collection of polynomials R¯ = (Ri)1≤i≤c, with Ri : k → k of m n degree di, there exist an affine map w : k → k such that P¯(w(x)) = ¯ R(x). Furthermore, if we denote by nR¯ the number of such affine maps, ¯ ¯ −s+mC then for any R1, R2 as above |1 − nR¯1 /nR¯2 | < q . (2) If P¯ is homogeneous, then for any homogeneous collection R¯ =(Ri)1≤i≤c, m m n Ri : k → k of degree di, there exist a linear map w : k → k such that ¯ ¯ P (w(x)) = R(x). Furthermore, if we denote by nR¯ the number of such ¯ ¯ −s+mC linear maps, then for any R1, R2 as above |1 − nR¯1 /nR¯2 | < q . Proof. We provide two proofs of the Theorem: one which is valid only in the case where char(k) > d, but is more algebraic in nature, and the other relies on multiple applications of Cauchy-Schwartz. We start with the proof in the case where char(k) > d. Since the proof for general d involves many indices, we first prove the case when c = 1 and d =2 so as to make the argument clear. We are given P (t) = 1≤i≤j≤n aijtitj + 1≤i≤n aiti + a of rank r. Note that for any linear form l(t)= n c t we have that P (t)+ l(t) is of rank ≥ r. P i=1 i i P P 10 DAVIDKAZHDANANDTAMARZIEGLER Denote w(x)=(w1(x)+s1,...,wn(x)+sn), where the wi are linear forms. We can write i i j j i i P (w(x)) = aij(w (x)+ s )(w (x)+ s )+ ai(w (x)+ s )+ a 1≤i≤j≤n 1≤i≤n X m m X i j i = aij wkwl xkxl + ai wkxk 1≤i≤j≤n k,l=1 1≤i≤n k=1 X m X X X i j j i i j i + aij(s wk + s wk)xk + aijs s + ais + a, 1≤i≤j≤n 1≤i≤j≤n 1≤i≤n X Xk=1 X X which we can write as i j i j i j 2 aij(wkwl + wl wk)xkxl + aijwl wl xl 1≤i≤j≤n 1≤i≤j≤n 1≤Xk i j i j i j j i i aij(wkwl + wl wk) aij(s wk + s wk)+ aiwk (1≤i≤j≤n ) (1≤i≤j≤n 1≤i≤n ) X 1≤k i j i i j aijs s + ais aijwl wl (1≤i≤j≤n 1≤i≤n ) (1≤i≤j≤n ) [ X X [ X 1≤l≤m Proof. We need to show that any non-trivial linear combination i j i j i j bkl aij(wkwl + wl wk) + bll aijwl wl "1≤i≤j≤n # "1≤i≤j≤n # 1≤Xk b11P (w1)+ lw2,...,wm,s(w1) 1 n where wj =(wj ,...,wj ), and lw2,...,wm,s is affine in w1, so as a polynomial in w1 this is of rank ≥ r and thus also of rank ≥ r as a polynomial in w. Similarly in the case where bll =6 0, for some 1 ≤ l ≤ m. Suppose b12 =6 0. We can write the above as 1 2 (∗) b12Q(w1,w2)+ lw3,...,wm,s(w1)+ lw3,...,wm,s(w2) PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 11 2 i where Q : V → k, and Q(t, t)=2P (t), and lw3,...,wm,s : V → k are affine maps. Thus restricted to the subspace W where w1 = w2 we get that (∗) is of rank ≥ r and thus of rank ≥ r − 1 on W . Similarly if bkl =6 0 for some k m n If P is homogeneous P (t) = 1≤i≤j≤n aijtitj of rank r, and w : k → k a linear map, we can write P m i j i j P (w(x)) = aijw (x)w (x)= aij wkwl xkxl 1≤i≤j≤n 1≤i≤j≤n X X k,lX=1 which we can write as i j i j i j 2 aij(wkwl + wl wk)xkxl + aijwl wl xl 1≤i≤j≤n 1≤i≤j≤n 1≤k For d> 2 we perform a similar computation. To simplify the notation we carry it out in the case P, R are homogeneous; the non-homogeneous case is similar. In this case it suffices to use linear maps. Denote In = {I = (i1,...,id):1 ≤ n d i1 ≤ ... ≤ id ≤ n}, and for t ∈ k denote by tI = j=1 tij . Let P be a degree d n polynomial P (t)= aI tI on k of rank r. I∈In Q Let w : km → kn Pbe a linear map. We can write d m d ij ij P (w(x)) = aI w (x) = aI wlj xl1 ...xld . I∈I j=1 ! I∈I j=1 ! Xn Y Xn l1,...,lXd=1 Y We rewrite this as m d d i a w j x ...x = a wij x ...x I lj l1 ld I σj l1 ld I∈I j=1 ! I∈I j=1 ! l1,...,lXd=1 Xn Y lX∈Im Xn σX∈S(l) Y where S(l) is the set of permutations of l1,...ld. Once again, our aim is to show that the collection of coefficients for all possible monomials in the variables xj, 1 ≤ j ≤ m, is of rank ≥ r: Claim 3.10. The collection below is of rank ≥ r − d + 1: d a wij I σj I∈In σ∈S(l) j=1 X X Y l∈Im 12 DAVIDKAZHDANANDTAMARZIEGLER Proof. Consider a linear combination of the above collection with coefficient cl =06 for some l ∈ Im. Consider d ij Q(wl1 ,...,wld )= aI wσj . j=1 IX∈In σX∈S(l) Y Then if ∆ P (y) = P (y + w) − P (y) then Q(w ,...,w ) = ∆ ... ∆ P (y). w l1 ld wl1 wld Now we can write the given linear combination as clQ(wl1 ,...,wld )+T (wl1 ,...,wld ) where T is a function of lower degree in wl1 ,...,wld (it also depends on the wt for t =6 li, for 1 ≤ i ≤ d). Since (char(k),d) = 1, by Remark 3.7(2) we have that the rank of Q is the same as that of P , and thus the rank of the collection is the same as rank of P . s When c > 1,we are given Ps(t) = I∈Is aI tI , 1 ≤ s ≤ c, of rank r, where Id (n) is the set of ordered tuples I = (i1,...,id ) with 1 ≤ i1 ≤ ... ≤ id ≤ n, s P s s and tI = ti1 ...tids . Note that for any polynomials ls(t) of degrees s i1 ids aI w ...w lσ(1) lσ(ds) σ∈Sds I∈Ids (n) X X 1≤s≤c,(l1,...,lds )∈Is(m) d is of rank >r. Write [1,c]= f=2 Cf where Cf = {s : ds = f}. s We need to show that for any f =2,...,d if B =(b )s∈C ,(l ,...,l )∈I (m) S (l1,...,lds ) f 1 ds ds is not 0,¯ then i bs aswi1 ...w f (l1,...,lf ) I lσ(1) lσ(f) s∈C σ∈S Xf (l1X,...,lf ) Xf I∈IXf (n) is of rank > r. Suppose (bs ) =6 0.¯ Then restricted to the subspace (l1,...,lf ) s∈Cf wl1 = ... = wlf we can write the above as s b(l1,...,lf )(f!)Ps(wl1 )+ R(w) s∈C Xf PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 13 1 n where wj =(wj ,...,wj ), and R(w) is of lower degree in wl1 , so as a polynomial in wl1 this is of rank >r and thus also of rank >r as a polynomial in w. C(d¯) Now (1), (2) follow from Proposition 3.1, since |Im| = m . We give an alternative proof that is valid for all characteristic. Following the computations above it suffices to show the following Claim: Claim 3.11. For any t > 0 there exists r = r(t, d¯) such that if the nc-rank of P¯ is > r then for any polynomial Q of degree 2 ≤ f ≤ c that is a non trivial combination of the polynomials in the collection i aswi1 ...w f I lσ(1) lσ(f) σ∈Sf I∈If (n) X X s∈Cf ,(l1,...,lf )∈If (m) −t we have that |Eeq(Q)| < q . Proof. Let Q be of degree f, and write i Q = bs aswi1 ...w f (l1,...,lf ) I lσ(1) lσ(f) (l1X,...,lf ) sX∈Cf σX∈Sf I∈IXf (n) where ¯b = (bs ) =6 0 for some (l ,...,l ) ∈ I (m). Observe that (l1,...,lf ) (l1,...,lf ) s∈Cf 1 f f any (l1,...,lf ) determines a unique set of variables wl1 ,...,wlf , thus after f applications of the Cauchy-Schwarz inequality we can isolate this collection and arrive at the multilinear from associated with differentiating i bs aswi1 ...w f (l1,...,lf ) I lσ(1) lσ(f) s∈C σ∈S Xf Xf I∈IXf (n) with respect to the associated set of variables. Namely we have i |e (Q)| ≤ e bs aswi1 ...w f q U1 q (l1,...,lf ) I lσ(1) lσ(f) Uf s∈Cf σ∈Sf I∈If (n) X X X But the latter is equal to E e (∆ ... ∆ ¯b · P¯f (x)). wl1 ,...,wlf q wl1 wld (l1,...,lf ) ¯f ¯ where P =(Ps)s∈Cf . Now by Theorem 3.4 we can choose r such that if P is of rank >r then the above is < q−s. This completes the proof of Theorem 3.8. 14 DAVIDKAZHDANANDTAMARZIEGLER 3.3. Proof of Theorem 2.11. In this subsection we prove Theorem 2.11. Let V be a vector space and l : V → k be a non-constant linear function. For any subset I of k we denote WI = {v ∈ V|l(v) ∈ I} so that Wb = W{b}, for b ∈ k. For X ⊂ V we write XI = X ∩ WI. Theorem 2.11 follows from the following proposition: ¯ Proposition 3.12. Fix d = {di} and m,s > 0. Let d := maxi di. There exists an effective bound r = r(d,¯ s, m) such that for any finite field k, any k-vector space V ¯ V −s and P ∈ Pd¯( ) of nc-rank >r the following holds. For any b ∈ k and q -almost any affine m-dimensional subspace L ⊂ Xb there exists an (m + 1)-dimensional affine subspace M ⊂ X containing L such that M ∩ X0 =6 ∅. ′ Proof. We fix d and define d = min(d +1, q). Let M0 = {a0,...,ad} ⊂ k be a ′ subset of d distinct points. To simplify notations we assume that a0 = 0. Claim 3.13. Let Q(x) be a polynomial of degree ≤ d such that Q|M0 ≡ 0. Then Q(a) = 0 for all a ∈ k. Proof. Since any polynomial of degree ≤ d in one variable vanishing at d+1 point is equal to 0, the claim is true if q ≥ d +1. On the other hand if d ≥ q then there is nothing to prove. m+1 Let J(d) be the subset of [0,d] of tuples t = (t1,...,tm+1) such that 0 ≤ m+1 m+1 tm+1 ≤ ... ≤ t1. Let T := {at =(at1 ,...,atm+1 ): t ∈ J(d)} ⊂ k . Claim 3.14. Let Q(x1,...,xm+1) be a polynomial of degree ≤ d such that Q|T m+1 ≡ 0. Then Q = 0. Proof. Our proof is by induction in m and for a fixed m by induction in d. Consider first the case when m = 1. We will write x, y instead of x1, x2. We 2 have T = {(at1 , at2 ):0 ≤ t2 ≤ t1 ≤ d}. a b We prove the claim by induction in d. Let Q = a,b qa,bx y , with a + b ≤ d. 0 a The restriction of Q to the line {y = 0} is equal to Q (x) = a≤d qa,0x . Since 0 0 P′ Q 2 ≡ 0 we see that Q = 0. So Q(x, y) = yQ (x, y). So, by the inductive |T P assumption, we have Q′ ≡ 0. Assume now the validity of Claim for polynomials in m variables and for poly- nomials in m+1 variables degree ≤ d−1. Let Q(x1,...,xm+1) be a polynomial of degree ≤ d such that Q|T m+1 ≡ 0. Let R be the restriction of Q on the subspace of points (x1,...,xm+1) such that xm+1 = 0. Since R|T m ≡ 0 it follows from the in- ′ ′ ductive assumption that R ≡ 0 and therefore Q = Q xm+1 where deg(Q )= d−1. ′ ′ Since Q|T ≡ 0 we see from the inductive assumption that Q ≡ 0. Denote I(d) the set of indexes I(d)= {t := (t1,...,tm+1) ∈ J(d):1 ≤ tm+1} . PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 15 m An affine m-dimensional subspace in Xb is parametrized as {x+ i=1 siyi; si ∈ k}, such that for all 2 ≤ e ≤ d, all P e ∈ P¯e we have j P m e (∗) Pj x + siyi =0, l(x)= b, l(yi)=0. i=1 ! X Let Y be the set of (x, y¯) satisfying (∗). We need to show that almost every (x, y¯) ∈ Y we can find z such that for all e ¯e 2 ≤ e ≤ d, all Pj ∈ P we have m e ∀s,s1,...,sm ∈ k, Pj x + siyi + sz =0, l(z)= −b, i=1 ! X or alternatively m m e ∀s,s1,...,sm ∈ k, Pj x + siyi + sz =0, l x + siyi + sz = (1−s)b. i=1 ! i=1 ! e X X By Claim 3.14, since Pj is of degree e, we can reduce this system to m m e Pj x + ati yi + am+1z =0, l x + ati yi + atm+1 z = (1−atm+1 )b, t ∈ I(e). i=1 ! i=1 ! X X Denote I = e∈[d] |I(e)|. Fix (x, y¯) ∈ Y and estimate the number of solutions A(x, y¯) to the above system of equations, which is given by P q−2I z e∈[d],c¯e :te∈I(e),h :t∈I(d) X te X t m m e e e ¯ e e eq c¯t · P x + ati yi + atm+1 z + ht l x + ati yi + atm+1 z +(atm+1 − 1)b . te i=1 t i=1 eX∈[d] X X X X e e ¯ Supposec ¯te = 0 for all t , but h =6 0, and recall that l(x)= b, l(yi)=0. We get m eq ht l x + ati yi + atm+1 z +(atm+1 − 1)b z t i=1 ! !! X X X = eq ht(atm+1 l(z)+ atm+1 b) . z t ! X X Now if t htatm+1 l(z) 6≡ 0 then the sum is 0. Otherwise also t htatm+1 b =0 so that the sum is |V |. P e P Now supposec ¯t0 =6 0 for some t0, and let e be the largest degree for which this holds. Let m e e e ¯ e e T (x, y, z)= c¯t · P x + ati yi + atm+1 z + Q(x, y, z) te i=1 X X 16 DAVIDKAZHDANANDTAMARZIEGLER where Q is of degree < e. We estimate E E 2 Bt0 = x,y¯∈V | zeq(T (x, y, z))| . Lemma 3.15. For any functions ft : V → C, kftk∞ ≤ 1, t ∈ I(d), we have m m ′ E ′ f x + a y + a z f¯ x + a y + a z + a z ≤kf k . x,y,z,z¯ t ti i tm+1 t ti i tm+1 tm+1 t0 Ud t∈I(d) i=1 i=1 Y X X Proof. To simplify the notation we prove this in the case m = 1. Without loss −1 −1 of generality a1 = 1 (make a change of variables y → a1 y, z → a1 z). We prove this by induction on d. When d = 1, I(d) = {(1, 1)}, and the claim in this case follow from the following inequality ′ ′ E ′ E ′ E x,y,z,z f1(x + y + z)f2(x + y + z + z ) = z,z f1(z)f2(z ) ≤ zfi(z) ≤kft0 kU1 . Assume d> 1. We can write the average as ¯ ′ Ex,y,z,z′ fi,j(x + aiy + ajz)fi,j(x + aiy + ajz + ajz ) (i,j)∈YI(d−1) ′ fd,j(x + ady + ajz)f¯d,j(x + ady + ajz + ajz ). 1≤Yj≤d Shifting x by ady we get ′ Ex,y,z,z′ fi,j(x +(ai − ad)y + ajz)f¯i,j(x +(ai − ad)y + ajz + ajz ) (i,j)∈YI(d−1) ¯ ′ fd,j(x + ajz)fd,j (x + ajz + ajz ). 1≤Yj≤d Applying the Cauchy-Schwarz inequality we can bound the above as ¯ ′ Ex,y,y′,z,z′ fi,j(x +(ai − ad)y + ajz)fi,j(x +(ai − ad)y + ajz + ajz ) (i,j)∈I(d−1) Y ′ f¯i,j(x +(ai − ad)y +(ai − ad)y + ajz) (i,j)∈YI(d−1) ′ ′ 1/2 fi,j(x +(ai − ad)y +(ai − ad)y + ajz + ajz ) . Shifting x by ady and rearranging we get ′ Ex,y,y′,z,z′ fi,j(x + aiy + ajz)f¯i,j(x + aiy +(ai − ad)y + ajz) (i,j)∈I(d−1) Y ′ ′ ′ 1/2 f¯i,j(x + aiy + ajz + ajz )fi,j(x + aiy +(ai − ad)y + ajz + ajz ) . (i,j)∈I(d−1) Y Now if we denote ′ gi,j,y′ (x)= fi,j(x)f¯i,j(x +(ai − ad)y ), PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 17 then by the induction hypothesis we get that the above is bounded by 1/2 E ′ ′ y kgi,j,y kUd−1 ≤kfi,j(x)kUd for any (i, j) ∈ I(d − 1). We do a similar computation for (i.j) ∈ I(d) \{I(d − 1), (d, 1)} , splitting ¯ ′ Ex,y,z,z′ fi+1,j+1(x + ai+1y + aj+1z)fi+1,j+1(x + ai+1y + aj+1z + aj+1z ) (i,j)∈YI(d−1) ′ fj,1(x + ajy + z)f¯j,1(x + ajy + z + z ), 1≤Yj≤d and shifting x by z to get ¯ ′ Ex,y,z,z′ fi+1,j+1(x − z + ai+1y + aj+1z)fi+1,j+1(x − z + ai+1y + aj+1z + aj+1z ) (i,j)∈YI(d−1) ′ fj,1(x + ajy)f¯j,1(x + ajy + z ). 1≤Yj≤d The only term left uncovered is fd,1, so we split ′ Ex,y,z,z′ fi+1,j(x + ai+1y + ajz)f¯i+1,j(x + ai+1y + ajz + ajz ) (i,j)∈YI(d−1) ′ fi,i(x + aiy + aiz)f¯i,i(x + aiy + aiz + aiz ). 1≤Yi≤d We make the change of variable z → z − y to get ¯ ′ Ex,y,z,z′ fi+1,j(x + ai+1y + aj(z − y))fi+1,j(x + ai+1y + aj(z − y)+ ajz ) (i,j)∈YI(d−1) ′ fi,i(x + aiz)f¯i,i(x + aiz + aiz ). 1≤Yi≤d Observe that the argument of fd,1 is (x +(ad − a1)y + a1z), so that the coefficient of y is not zero. Now proceed as in previous cases. e e By the Lemma 3.15 we obtain that Bt is bounded by keq(¯c e · P¯ )kU . By 0 t0 e Theorem 3.4 there exists an effective bound r = r(s,d), such that if P is of rank e e −s > r then keq(¯c e · P¯ )kU < q . It follows that we can choose r so that for t0 e q−s-almost all (x, y) ∈ Y , the number of solutions A(x, y¯) is bounded below by |V |q−4I . 18 DAVIDKAZHDANANDTAMARZIEGLER 3.4. Proof of Theorems 2.13 and 2.15. We fix m, d and v. As follows from Theorem 2.9, there exists an effective bound r = r(m,d,c) such that for any finite field Fq, any Fq-vector space V the mapκ ˜P (Fq) is surjective for any family P¯ =(Pi) of polynomials Pi ∈ Pd(V) such that rnc(P¯) ≥ r. We now consider the case when k is an algebraically closed field. n c 3.4.1. The surjectivity of κ˜P (k). We fix V = A and consider Pd(V) as a scheme defined over Z. Let T be the set of sequences (ai, bi), 1 ≤ i ≤ r, such that 0 ≤ ai, bi < d and ai + bi ≤ d. For any t = {(ai, bi)} ∈ T we denote by r V V V νt : ⊕i=1Pai ( ) ⊗ Pbi ( ) → Pd( ) the linear map given by r νi({Qi ⊗ Ri})= QiRi. i=1 X Let Z be the union of images of the maps of νt, t ∈ T . Y V c ¯ Z Let ⊂ Pd( ) be constructible subset of families P = {Pi} such that ∩LP¯ = V {0} where LP¯ ⊂ Pd( ) is the span of (Pi). So Y(k) ⊂ Pd(V)(k) consists families P¯ of polynomials of rank >r. We define R Y ¯ Y ⊂ as the constructible subset of P ∈ such that the morphismκ ˜P¯ is not surjective. Our goal is to show that R = ∅. We first consider the case when k = F¯p is the algebraic closure of Fp. Claim 3.16. R(F¯p)= ∅. Proof. Assume that R(F¯p) =6 ∅. Then there exists P¯ ∈ Y(F¯p) such that the F¯ F¯ mapκ ˜P¯( p) is not surjective. Then there exists Q ∈ Pd¯( p) which is not in F¯ ¯ Y F¯ the image ofκ ˜P¯( p). By definition there exists l ≥ 1 such that P ∈ ( q) and F l F Q ∈ Pd¯( q), q = p . But as follows from Theorem 2.9 there exists φ ∈ Affm( q) such that Q =κ ˜P¯(φ). But the existence of such affine map φ contradicts the F¯ assumption that Q is not in the image ofκ ˜P¯( p). Corollary 3.17. (1) The map κ˜P (k) is surjective for any algebraically closed field k and a polynomial P ∈ Pd(V) of nc-rank >r(m, d). (2) The map κ˜P (k) is surjective for any algebraically closed field k of charac- teristic 0 and a polynomial P ∈ Pd(V) of nc-rank >r(m, d). Proof. The part (1) follows from the completeness of the theory ACFp of alge- braically closed fields of a fixed characteristic p. To prove the part (2) one choses a non-trivial ultrafilter U on the set of primes and considers the U-ultraproduct of theories ACFp. Let l be the U-ultraproduct of fields F¯p. Let ACF be the theory of algebraically closed fields and α be the formula in ACF expressing the surjectivity of the map κP . As follows from Claim 3.16 α holds for algebraic closures of fields Fp. We fix now m,d,c and for any n ≥ 1 define by αn the following formula in ACF . PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 19 For any family P¯ = (Pi),Pi ∈ k[x1....,xn] such that rnc(P¯) ≥ r(m,d,c) the An Am c map κP¯ : Affm( ) → (Pd( )) is surjective. By the Theorem ofLo´sapplied to the formula αn we see that the map κP (l) is surjective for any family P¯ ∈ Pd(V), dim(V)= n of nc-rank >r. Since the theory ACFp of algebraically closed fields of characteristic 0 is complete and n ≥ 1 is arbitrary the corollary is proved. 3.4.2. The computation of the dimensions of fibers of κ˜(k). Let Homaf (W, V) be the variety of affine maps from W to V and let T ⊂ Y be be the subscheme Am −1 of polynomials P such that there exists Q ∈ Pd( ) such that dim(κP (Q)) =6 dim(Homaf (W, V)) − dim(Pd(W)). We want to prove that T = ∅. The same arguments as before show that it is sufficient to prove that −1 W V W dim(κP (Q)) = dim(Homaf ( , )) − dim(Pd( )) m for all finite fields k = Fq and Q ∈ Pd(A )(k). Let w := dim(Homaf (W, V)) − dim(Pd(W)). As follows from [22] that there exists constant A(n, d),l > 0 such −1 F w w−1/2 lm that ||κP (Q)( qm )| − q | ≤ A(n, d)q for any q = p . This implies that −1 log (|κ (Q)(km)|) dim(κ−1(Q)) = lim q P , P m→∞ lm F where km = qml is the extension of degree l. Now the equality −1 dim(κP (Q)) = w follows from Theorem 2.9. Part (3) of Theorem 2.13 follows now from Theorem 23.1 in [25], and part (4) follows from the part (3). The derivation of Theorem 2.15 from Theorem 2.11 is completely analogous. 3.5. Proof of Theorem 1.9. Proof of Theorem 1.9. Let G be a subfunctor of Fd such that r(P ),P ∈ G(W ) is not bounded above. We want to show that G(W )= Fd(W ) for any finite-dimensional k-vector space W . Let m = dim(W ) and choose a polynomial P ∈ G(V ), where V is a k-vector space V such that rnc(P ) ≥ r(m, d), where r(m, d) is as in the Corollary 3.17. Then for any polynomial Q on W of degree d, there exist an affine map φ : W → V ⋆ such that Q = φ (P ). We see that G(W )= Fd(W ). 4. Extending weakly polynomial functions from high rank varieties 4.1. Introduction. We fix d,a,c ≥ 1 and a field k such that |k| > ad and that there exists a root of unity β ∈ k of order m > ad. A field is admissible if it satisfies these conditions. 20 DAVIDKAZHDANANDTAMARZIEGLER Definition 4.1. Let V be a k-vector space, and let X ⊂ V . We say that a function f : X → k is weakly polynomial of degree ≤ a if restrictions f|L to affine subspaces L ⊂ X are polynomials of degree ≤ a. Remark 4.2. If |k| > a it suffices to check this on 2-dimensional subspaces (see [15], Theorem 1). Namely a function is weakly polynomial of degree ≤ a if the restriction f|L to 2-dimensional affine subspace L ⊂ X is a polynomial of degree ≤ a. The goal of this section is to show in the case when k is admissible field which is either finite or algebraically closed then any weakly polynomial function f on a subvariety X ⊂ V of a sufficiently high rank extends to a polynomial F of degree ≤ a on V . The main difficulty is in the case when a ≥ d when an extension F of f is not unique. To state our result properly we introduce the following definition: X V k Definition 4.3. An algebraic k-subvariety ⊂ satisfies the condition ⋆a if any weakly polynomial function of degree ≤ a on X is a restriction of a polynomial function of degree ≤ a on V . The following example demonstrates the existence of cubic surfaces X ⊂ A2 k F which do not have the property ⋆1 for any field k =6 2. 2 Example 4.4. Let V = k , Q = xy(x − y). Then X = X0 ∪ X1 ∪ X2 where X0 = {v ∈ V |x = 0},X1 = {v ∈ V |y = 0},X2 = {v ∈ V |x = y}. The function f : X → k such that f(x, 0) = f(0,y)=0, f(x, x) = x is weakly linear but one can not extend it to a linear function on V . The main result of this section is that high rank hypersurfaces over admissible k fields satisfy ⋆a. Theorem 4.5. There exists an r = r(a, d) such that for any admissible field k which is either finite or algebraically closed, any hypersurface X of degree d and k nc-rank ≥ r in a k-vector space satisfies ⋆a. The result extends without difficulty to complete intersections X ⊂ V of bounded degree and codimension, and high rank (see Definition 2.1). Theorem 4.6. For any c> 0, there exists an effective bound r = r(a,d,c) such that for any admissible field k, which is either finite or algebraically closed, a k-vector space V, any subvariety X ⊂ V of codimension c, degree d and nc-rank k ≥ r satisfies ⋆a. Our proof of Theorem 4.6 consists of two steps. We first construct for any d hypersurfaces Xn ⊂ Vn over Z of degree d and arbitrary high rank such that for any admissible field k and any c the subset c c Xn(k) ⊂ Vn(k) satisfying the conditions of Theorem 4.6. This result is purely algebraic. In the second step we show how to derive the general case of Theorem 4.6 from this special case. PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 21 Remark 4.7. The case a |Ew∈V ψ(µ(w))| = t< 1. Now we observe that 1 1 1 1 1 2 d µ˜(u ,...,u ,...,u ,...,u )| l = u u ··· u 1 d 1 d {uj =0,l6=j} 1 2 d Denote d l U = {((u1)1,..., (un)1,..., (u1)d,..., (un)d) ∈ V :(ui)j =0,l =6 j, i ∈ [n]} Then restricted to U we have n ˜ 1 2 d Pn((u1)1,..., (un)1,..., (u1)d,..., (un)d)|U = µ((ui)1, (ui)2,..., (ui)d), i=1 X so that E ˜ E ˜ E n n u∈V d ψ(Pn(u)) ≤ u∈U ψ(Pn(u)) = | w∈W ψ(µ(w))| = t . ǫ It follows by 3.4 that P˜n is of rank > n for some ǫ> 0. Definition 4.10. (1) For any set X we denote by k[X] the space of k-valued functions on X. 22 DAVIDKAZHDANANDTAMARZIEGLER w (2) For a subset X of a vector space V , we denote by Pa (X) ⊂ k[X] the subspace of weakly polynomial functions of degree ≤ a. w (3) We denote by Pa(X) ⊂ Pa (X) the subspace of functions f : X → k which are restrictions of polynomial functions on V of degree ≤ a. d n (4) As before we define W = A , Vn := W and denote by µ the product map µ : W → A given by d µ(a1,...,ad)= as. s=1 Y We write elements of Vn in the form v =(w1,...,wn), 1 ≤ i ≤ n, wi ∈ W. It is clear that Theorem 4.6 is equivalent to the following statement. w Theorem 4.11. Let k be an admissible field, then Pa (Xn)= Pa(Xn). We fix n and write X instead of Xn, and V instead of Vn. The proof of the part (2) of Theorem 4.8 uses the existence of a large group of symmetries of X, the existence of a linear subspace L ⊂ V of large dimension and the existence of the subroup ∆ ⊂ k⋆, ∆ ∼= Z/mZ for m>ad. Proof. We start the proof of Theorem 4.11 with the following result. N Claim 4.12. Let Q be a polynomial of degree ≤ ad on k such that Q|∆N ≡ 0. Then Q = 0. Proof. The proof is by induction on N. If N = 1 then Q = Q(x) is polynomial such that Q(δ)=0 for δ ∈ ∆. Since |∆| > ad we see that Q = 0. Assume that the result is known for N ′ = N − 1. Let Q be a polynomial of de- N gree ≤ ad on k such that Q|∆N ≡ 0. By induction we see that Q(δ, x2,...,xs) ≡ 0 for all δ ∈ ∆. Then for any x2,...,xs the polynomial x → Q(x, x2,...,xs) van- ishes for all δ ∈ ∆. Therefore Q(x, x2,...,xs) = 0 for all x ∈ k. Definition 4.13. n (1) Γ := (Sd) . The group Γ acts naturally on X. n n (2) L := {(c1,...,cn) ∈ k | i=1 ci =0}. n n (3) L∆ = (∆) ∩ L ⊂ k . (4) For c ∈ k we write w(c)P := (c, 1,..., 1) ∈ W . κ : L ֒→ X ⊂ V is the linear map given by (5) κ(c1,...,cn) := (w(c1),...,w(cn)) and write κγ := γ ◦ κ, γ ∈ Γ. (6) For any function f : X → k,γ ∈ Γ define a function hγ,f : L → k by hγ,f := f ◦ κγ . d d n (7) T1 := {(u1,...,ud) ∈ (∆) | j=1 uj =1} and T := T1 . Q PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 23 We denote by ζi : T1 ֒→ T, 1 ≤ i ≤ n the imbedding onto the i-th (8) component. ′ ′ (9) For any j, j , 1 ≤ j =6 j ≤ d we denote by φj,j′ : ∆ → T1 the morphism −1 such that φj,j′ (u)=(xl(u), 1 ≤ l ≤ d) where xj(u)= u, xj′ (u)= u and ′ xl(u)=1 for l =6 j, j . ⋆ (10) We denote by Θ1 the group of homomorphisms χ : T1 → k . n (11) Θ = (Θ1) . ′ ′ ⋆ (12) For χ ∈ Θ1, j, j , 1 ≤ j =6 j ≤ d we define a homomorphism χj,j′ : ∆ → k ∼ by χj,j′ := χ ◦ φj,j′. Since ∆ = Z/mZ there exists unique αj,j′ (χ) ∈ α ′ (χ) (−m/2, m/2] such that χj,j′ (u)= u j,j for any u ∈ ∆. adm (13) Θ1 := {χ ∈ Θ1 : |αj,j′(χ)| ≤ a}. adm,+ adm ′ (14) Θ1 := {χ ∈ Θ1 : αj,j′ (χ) ≥ 0,j w w Proof. Since f ∈ Pa (X) we have hγ,f ∈ Pa (L). Since L is linear space we see that hγ,f is a polynomial of degree ≤ a. Claim 4.16. (1) The subset Θadm of Θ is Γ-invariant. (2) For any θ ∈ Θadm there exists γ ∈ Γ such that θ ◦ γ ∈ Θadm,+. Proof. Clear. w¯ Definition 4.17. We denote by Pa (X) the space of functions f such that hγ,f is a polynomial of degree ≤ a on L for all γ ∈ Γ. w¯ The group T acts naturally on X, and on the spaces Pa (X) and Pa(X), and we have direct sum decompositions w w θ Pa (X)= ⊕θ∈ΘPa (X) and θ Pa(X)= ⊕θ∈ΘPa(X) . Lemma 4.18. Let Z ⊂ V be a homogeneous k-subvariety of degree d and let f : Z → k a polynomial function of degree ad which is a weakly polynomial function on Z of degree ≤ a. Then it is a restriction of polynomial function on V (k) of degree ≤ a. 24 DAVIDKAZHDANANDTAMARZIEGLER Proof. Lemma 4.18 follows inductively from the following claim. Claim 4.19. Let f : Z → k be a polynomial function of degree ≤ a which is a weakly polynomial of degree < a. Then f is a polynomial of degree < a. Proof. We can write f as a sum f = Q + f ′ where f’ < a and Q is homogeneous of degree a. Since f is weakly polynomial of degree < a the function Q is also weakly polynomial of degree < a. It is sufficient to show that Q ≡ 0. Choose z ∈ Z. To show that Q(z) = 0 consider the function g on k,g(t) = Q(tz). Since Z is homogeneous tz ∈ Z. Since Q is homogeneous of degree a we have g(t)= cta. On the other hand, since Q is weakly polynomial of degree < a we see that g(t) is a polynomial of degree < a. Since a < q we see that g ≡ 0. So Q(z)= g(1) = 0. Corollary 4.20. Let f : X → k be a weakly polynomial of degree < a on X which is a restriction of polynomial function of degree ≤ ad on V . Then f is a restriction of polynomial function of degree ≤ a on V . w θ It is clear that for a proof Theorem 4.11 it suffices to show that Pa (X) = θ Pa(X) for any θ ∈ Θ. This equality follows now immediately from the following statement. Fix θ ∈ Θ. Proposition 4.21. For any weakly polynomial function f : X → k of degree ≤ a satisfying the equation f(tx) = θ(t)f(x), t ∈ T, x ∈ X there exists a polynomial P on V of degree ≤ ad such that f = P|X . We start the proof of Proposition 4.21 with a set of definitions. Let f : X → k be a function such that f(tx) = θ(t)f(x), t ∈ T, x ∈ X, and such that hγ,f are polynomial functions on L of degree ≤ a for all γ ∈ Γ. Definition 4.22. (1) We write h, hγ : L → k instead of hId,f and hγ,f . (2) ν : V → L is the map given by ν(w1,...,wn) := (µ(w1),...,µ(wn)). (3) W 0 := {w =(a1,...,ad) ∈ W |ai ∈ ∆ for i ≥ 2} ⊂ W . (4) X0 := (W 0)n ∩ X. (5) For w = (a1,...,ad) ∈ W we define I(w) := {i, 1 ≤ i ≤ d|ai 6∈ ∆} and write z(w) := max(|I(w)| − 1, 0). (6) For x =(w1,...,wn) we write z(x)= j z(wj). (7) Y := {x ∈ X|z(x) ≤ s},s ≥ 0. s P Claim 4.23. 0 (1) Y0 = {γ(W ),γ ∈ Γ}. (2) For any x ∈ X0 there exist unique t(x) ∈ T such that x = t(x)κ(ν(x)). (3) f(x)= θ(t(x))f(κ(ν(x))) for any x ∈ X0. PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 25 (4) For any γ ∈ Γ, l ∈ L, we have ν(γ(l)) = l where γ : X → X is as in Definition 4.13. Proof. Clear. Lemma 4.24. Let f be a function on X satisfying the conditions of Proposition 4.21 and such that f|Y0 ≡ 0. Then f ≡ 0. Proof. It is clear that it is sufficient to prove the following statement Claim 4.25. Let f be a function on X satisfying the conditions of Proposition 4.21 and such that f|Ys ≡ 0,s ≥ 0. Then f|Ys+1 ≡ 0. Proof. We want to show that f(x) = 0 for all x = (wj) ∈ Ys+1. Since the restriction of f on any line is a polynomial of degree ≤ ad it is clear that for a proof of the equality f(x) = 0 it is sufficient to prove the following geometric statement. Claim 4.26. There exists a line N ⊂ X containing x and such that |N ∩Ys| > ad. 1 d Proof. Let x = (wj),wj = (xj ,...,xj ), 1 ≤ j ≤ n. We start with the following observation. i Claim 4.27. For any x ∈ X \ Y0, x = {xj} there exists j0 such that either there i0 i1 i i0 exist (i0, i1), 1 ≤ i0 =6 i1 ≤ d such that xj0 =0, xj0 6∈ ∆ or i xj0 =6 0 and xj0 6∈ ∆ for some i0, 1 ≤ i0 ≤ d. Q Proof. Clear. i0 Consider first the case when xj0 = 0forsomepair(i0, j0), 1 ≤ i0 ≤ d, 1 ≤ j0 ≤ n i1 and xj0 6∈ ∆ for some i1 =6 i0, 1 ≤ i0 ≤ d. Let α : k → X be the map given by i i i i1 α(c)= xj(c) where xj(c) = xj if (i, j) =6 (i1, j0) and xj0 (c) = c. By construction i1 x ∈ N := Im(α) and xj0 (∆) ⊂ Ys for c ∈ ∆. Since |∆| = q >ad we see that the line N satisfies the conditions of Claim 4.26. i i0 So we may assume the existence of j0 such i xj0 =6 0 and xj0 6∈ ∆ for some i0, 1 ≤ i0 ≤ d. To simplify notations we may and will assume that j0 = i0 = 1. Q i Since x ∈ X there exists j1, 2 ≤ j ≤ n such that i xj1 =6 0. We may assume i that j1 = 2. It is clear that either x2 ∈ ∆ for all i, 1 ≤ i ≤ d or there exists i1 Q i1, 1 ≤ i1 ≤ d such that x2 6∈ ∆ in which case we may and will assume that i1 = 1. d i d i Let a := i=2 x1, b := i=2 x2 and β : k → X be the map given by β(c) := xi (c) where x1(c) = −bc, x2(c) = ac − n d xi and xi (c) = xi otherwise. j Q1 Q1 j=3 i=1 j j j Let a := d xi , b := d xi and β : k → X be the map given by β(c) := xi (c) i=2 1 i=2 2 P Q j where x1(c)= −bc, x1(c)= ac − b−1 n d xi and xi (c)= xi otherwise. Let 1 Q 2Q j=3 i=1 j j j N = Im(β). Then x ∈ N and |N ∩ Y | = q >ad. Ps Q 26 DAVIDKAZHDANANDTAMARZIEGLER w¯ θ adm Lemma 4.28. Pa (X) = {0} for any θ 6∈ Θ . Proof. As follows from Lemma 4.24 it is sufficient to show that hγ (f) ≡ 0 for any adm w¯ θ adm ′ θ 6∈ Θ ,γ ∈ Γ and f ∈ Pa (X) . Since θ 6∈ Θ there exist i, j, j , 1 ≤ i ≤ ′ ′ n, 1 ≤ j, j ≤ d, such that |αj,j′(χi)| ≥ a. Choose s ∈ Sd such s(j)=1,s(j ) = 2, .and denote bys ˜ ∈ Γ the image of s under the imbedding Sd ֒→ Γ as the i-factor After the replacing f → f ◦ s,˜ θ → θ ◦ s we may assume that |α1,2(χi)| > a. The functions hγ and hγ◦s are weakly polynomial functions of degrees ≤ a on the linear space L. Therefore hγ and hγ◦s are polynomial functions of degrees ≤ a. α1,2(χi) For any l ∈ L such that li ∈ ∆ we have hγ◦s(l)= li hγ (l). Since α1,2(χi) > a and α1,2(χi) ≤ m/2, this is only possible if hγ = 0. Corollary 4.29. As follows from Claim 4.16 it is sufficient to prove Proposition 4.21 for θ ∈ Θadm,+. Definition 4.30. Let P : V → k be the polynomial given by n d 1,j j α (χi) j P (v)= (xi ) h(ν(v)), v =(xi ). i=1 j=1 Y Y Lemma 4.31. deg(P ) ≤ ad. Proof. Let b = deg(h). It is sufficient to show that for any sequencee ¯ = n 1,e(i) (e(i))1≤i≤n, e(i) ∈ [1,d], we have i=1 α (χi)+ b ≤ a. Suppose there existse ¯ such that n α1,e(i)(χ )+b > a. Since θ ∈ Θadm, there P i=1 i n exists a subset I of [1, n] such that a< α1,e(i)(χ )+ b ≤ 2a. P i∈I i Let γ = (σi)1≤i≤n, σi ∈ Sd, be such that e(i) = σi(1) for i ∈ I and σi = Id if ⋆ P i 6∈ I. Consider hγ := κγ (f). On one hand it is a polynomial of degree ≤ a on L. 1,e(i) α (χi) 1,e(i) On the other hγ (l) = h(l) i∈I li . The inequalities a < i∈I α (χi)+ b ≤ 2a imply that h ≡ 0. Q P ¯ ¯ By construction P|L ≡ f|L. Let f := f − P . Then f is weakly polynomial function of degree ≤ ad on X vanishing on L such that f¯(tx) = θ(t)f(x) for ¯ t ∈ T, x ∈ X. As follows from Claim 4.23 we have f|X0 ≡ 0. It is clear that for a proof the part (1) of Proposition 4.21 it is sufficient to show that f¯(x)=0 for all x ∈ X. By Lemma 4.24 it suffices to prove the following lemma: ¯ Lemma 4.32. f|Y0 ≡ 0. 0 Proof. Since Y0 = ΓX = T Lγ , it suffices to show that f|Lγ ≡ 0 for all γ ∈ Γ. Let hγ : L → k be given by hγ (l)= f¯(γ(l)). We have to show that hγ ≡ 0. Since hγ is a polynomial of degree ≤ ad it follows from Claim 4.12 that it is sufficient to show that the restriction of hγ on L∆ vanishes. But for any l ∈ L∆ we have ′ ′ γ(l)= tl , t ∈ T,l ∈ L. Since f¯(tx)= θ(t)f(x) we see that hγ(l) = 0. PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 27 This completes the proof of Theorem 4.11. 4.4. Proof of Theorem 4.6. Proposition 4.33. There exists an effective bound r = r(d,¯ a) such that if k is an admissible field which is either finite or algebraically closed, W ⊂ V is a ¯ hyperplane then any weakly polynomial function on XP¯, rnc(P ) ≥ r of degree ≤ a vanishing on XP¯ ∩ W is a restriction of a polynomial on V of degree ≤ a. As an immediate corollary we obtain: Corollary 4.34. There exists an effective bound r = r(d,¯ a) such that the follow- ing holds for all admissible fields k which are either finite or algebraically closed. Let X = {v ∈ V|Pi(v)=0} ⊂ V be a subvariety of degree ≤ d and W ⊂ V an affine subspace such that nc-rank of P¯|W ≥ r. Let f be a weakly polynomial function on X of degree ≤ a such that f|X∩W extends to a polynomial on W of degree ≤ a. Then there exists an extension F of f to a polynomial on V of degree ≤ a. Proof. Consider first the case when W ⊂ V is a hyperplane. By the assumption there exists an extension R of the restriction f|W . Choose a linear projection s : V → W and f ′ : X → k by f ′(x) = f(x) − R(s(x)). Then f ′ is weakly ′ polynomial function on X of degree ≤ a such that f|X∩W ≡ 0. As follows from Proposition 4.33 there exists an extension of f ′ to a polynomial F ′ on V of degree ≤ a. But then F := F ′ + R ◦ s is an extension F of f to a polynomial on V of degree ≤ a. In the case when the codimension of W is > 1 we choose a flag F = {W0 = W ⊂ W1 ···⊂ Wdim(V )−dim(W ) = V }, dim(Wi) = dim(W )+ i, and extend f by induction in i, 1 ≤ i ≤ dim(V ) − dim(W ) to a polynomial F on V . Remark 4.35. The choice of F depends on a choice of flag F and on choices of projections used in the inductive arguments. 4.5. Proof of Proposition 4.33. The key tool in our proof of this proposition is a testing result from [18] which roughly says that any weakly polynomial func- tion of degree a on X that is ”almost” weakly polynomial of degree < a, namely it is a polynomial of degree < a on almost all affine subspaces, is weakly poly- nomial of degree < a. This part does not require X to be of high rank. We use the assumption that X is of high rank to show (see Theorem 2.11) that almost any isotropic affine plane is contained in an isotropic three affine dimensional subspace that is not contained in l−1{0}. We start by stating the testing result from [18]. In [15] (Theorem 1) the following description of degree ≤ polynomials is given: 28 DAVIDKAZHDANANDTAMARZIEGLER Proposition 4.36. Let P : V → k. Then P is a polynomial of degree ≤ a if and a+1 only if the restriction of P to any affine subspace of dimension l = ⌈ q−q/p ⌉ is a polynomial of degree ≤ a. Note that when a < q then l ≤ 2. In [15] the above criterion is used for polynomial testing over general finite fields. In [18] (Corollary 1.13) it is shown how the arguments in [15] can be adapted to polynomial testing within a subvariety variety X ⊂ V (high rank is not required). Proposition 4.37 (Subspace splining on X). For any a,d,c > 0 there exists an A = A(d,c,a) > 0, depending polynomially on c,d and exponentially on a, such that the following holds. Let X ⊂ V (k) be a complete intersection of degree d, codimension c. Then any weakly polynomial function f of degree a such that −A a the restriction of f to q -a.e l-dimensional affine subspace, l = ⌈ q−q/p ⌉, is a polynomial of degree < a, is weakly polynomial of degree < a. Proof of Proposition 4.33. Let V be a vector space and l : V → k be a non- constant affine function. For any subset I of k we denote WI = {v ∈ V|l(v) ∈ I}, so that Wb = W{b}, for b ∈ k. For a hypersurface X ∈ V we write XI = X ∩ WI. Lemma 4.38. For any finite subset S ⊂ k, any weakly polynomial function f of degree a on X such that f|XS ≡ 0, and any b ∈ k \ S, there exists a polynomial Q of degree ≤ a on V such that Q|XS ≡ 0 and (Q − f)|Xb ≡ 0. Proof. We start with the following result. Claim 4.39. Under the assumptions of Lemma 4.38, the restriction f|Xb is a weakly polynomial function of degree ≤ a −|S|. Proof. Since |k| > a it suffices to show that for any plane L ⊂ Xb the restriction f|L is a polynomial of degree ≤ a −|S|. We first do the case when the field k is finite. Since by the part (4) of Theorem 2.13 the variety X is a complete intersection it follows from Proposition 4.37, there is a constant A = A(d, a) such that it suffices to check the restriction fL on −A q -almost any affine plane L ⊂ Xb is a polynomial of degree ≤ a −|S|. As follows from Proposition 3.12, for any s> 0 there is an r = r(d,s) such that −s if X is of nc-rank > r then for q -almost any affine plane L ⊂ Xb there exists an affine 3-dim subspace M ⊂ X containing L and such that M ∩ X0 =6 ∅. Then M ∩ Xt =6 ∅ for any t ∈ k. Since f is a weakly polynomial function of degree a, its restriction to M is a polynomial R of degree ≤ a. Since the restriction of R −1 ′ to l (S) ∩ M is identically zero, we see that R = R s∈S(l − s). Since l|L ≡ b, ′ we see that the restriction f|L is equal to R , which is a is a polynomial of degree ≤ a −|S|. Q PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 29 Now we show that this claim implies Lemma 4.38. Indeed, assume that f|Xb is a weakly polynomial function of degree ≤ a −|S|. It follows from the inductive assumption on a, that there exists a polynomial Q′ of degree ≤ a −|S| on V , ′ ′ Q Qs∈S(l−s) such that f|Xb = Q . Let Q := . Then (f − Q)XS∪{b} ≡ 0. |Xb Qs∈S(b−s) For algebraically closed fields we follow the same argument replacing Proposi- tion 4.37 with Proposition A.6, and Proposition 3.12, with Theorem 2.15. Proposition 4.33 follows from Lemma 4.38 by induction. Now we can prove Theorem 4.5: Proof of Theorem 4.5 assuming Theorems 4.8 and 1.9, and Corollary 4.34. Letr ˜ be from Corollary 4.34 and r = r(a, d¯) := ρ(dim(W ),d) from Theorem 1.9. As follows from Theorem 4.8 the subvarieties Xn are of nc-rank ≥ r˜ for n ≥ dr˜. Let X ⊂ V be a subvariety of nc-rank ≥ r. By Theorem 1.9 there exists a W V X W X X k linear map φ : → such that n = {w ∈ |φ(w) ∈ }. Since n satisfies ⋆a, X k Corollary 4.34 implies that satisfies ⋆a. 5. Nullstellensatz Let k be a field and V be a finite dimensional k-vector space. We denote by V the corresponding k-scheme, and by P(V ) the algebra of polynomial functions on V defined over k. For a finite collection P¯ =(P1,...,Pc) of polynomials on V we denote by J(P¯) X the ideal in P(V ) generated by these polynomials, and by P¯ the subscheme of V defined by this ideal. Given a polynomial R ∈ P(V), we would like to find out whether it belongs to the ideal J(P¯). It is clear that the following condition is necessary for the inclusion R ∈ J(P¯). (N) R(x) = 0 for all k-points x ∈ XP¯(k). Proposition 5.1 (Nullstellensatz). Suppose that the field k is algebraically closed X and the scheme P¯ is reduced. Then any polynomial R satisfying the condition (N) lies in J(P¯). F X We will show that the analogous result hold for k = q if P¯ is of high nc-rank. ¯ c From now on we fix a degree vector d = (d1,...,dc) and write D := i=1 di. c We denote by P ¯(V) the space of d¯-families of polynomials P¯ = (Pi) on V d i=1Q such that deg(Pi) ≤ di. Theorem 5.2. There exists and an effective bound r(d¯) > 0 such that for any finite field k = Fq with q > aD, any family P¯ of degrees d¯ and nc-rank ≥ r(d¯) the following holds. Any polynomial Q on V of degree a vanishing on XP¯ belongs to the ideal J(P¯). 30 DAVIDKAZHDANANDTAMARZIEGLER Proof. Our proof is based on the following rough bound (see [11]). ¯ c F Lemma 5.3. Let P = {Pi}i=1 ⊂ q[x1,...,xn] be a family of polynomials of Y X An degrees di, 1 ≤ i ≤ c such that the variety := P¯ ⊂ is of dimension n − c. Y F n−c c Then | ( q)| ≤ q D where D := i=1 di. For the convenience we reproduceQ the proof of this result. Proof. Let F be the algebraic closure of Fq. Then Y(Fq) is the intersection of Y with hypersurfaces Yj, 1 ≤ j ≤ n defined by the equations hj(x1,...,xn)=0 q where hj(x1,...,xn)= xj − xj. Let H1,...,Hn−c be generic linear combinations of the hj with algebraically in- ′ dependent coefficients from an transcendental extension F of F and Z1, ..., Zn−c ⊂ An be the corresponding hypersurfaces. Intersect successively Y with Z1, Z2,..., Zn−c. Inductively we see that for each j ≤ n − c, each component C of the intersection Y ∩ Z1 ∩···∩ Zj has dimension n−c−j. Really passing from j to j+1 for j < n−c we have dim(C)= n−c−i> 0. So not all the functions hj vanish on C. Hence by the genericity of the choice of linear combinations {Hj} we see that Hj+1 does not vanish on C and therefore Zj+1 ∩C is of pure dimension n−c−j−1. Thus the intersection Y∩Z1 ∩···∩Zn−c n−c has dimension 0. By Bezout’s theorem we see that |Y ∩ Z1 ∩ ... Zn−c| ≤ q D. Since Y(Fq) = Y ∩ Z1 ∩···∩ Zn ⊂ X ∩ Y1 ∩···∩ Yn−c we see that |Y(Fq)| ≤ qn−cD. Now we can finish the proof of Theorem 5.2. Let R ∈ Fq[x1,...,xn] be a X F polynomial of degree a vanishing on the set P¯( q). Suppose that R does not lie Y X in the ideal generated by the Pi, 1 ≤ i ≤ c. Let := P¯. Since R vanishes on X F Y F X F P¯( q) we have ( q)= P¯( q). X Y Since P¯ is irreducible we have dim( ) = n − c − 1. As follows from 5.3 we n−c−1 have the upper bound |Y(Fq)| ≤ aDq . On the other hand as follows from Theorem 3.2 there exists an effective bound r(d¯) > 0 such that the condition ¯ ¯ X F n−c q−1 rnc(P ) ≥ r(d) implies the inequality | P¯( q)| > q q , which is a contradiction to q > aD. Appendix A. An Algebro-Geometric analogue of Proposition [18] Proposition 4.37 was proved in [18]. In this Appendix we present another approach leading to a proof of Proposition 4.37 which (after some adjustments) provides a proof of an Algebro-Geometric analogue of Proposition 4.37. We start with two auxiliary results. Definition A.1. Let A be a finite set, B ⊂ A×A, Z ⊂ A, and let p1,p2 : B → A −1 be the natural projections. For x ∈ A we define Bx := p2(p )(x) ⊂ A. Let β(x) := |Bx|/|A|. (1) B is δ-dense if β(x) ≥ δ, x ∈ A. PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES 31 (2) A subset C ⊂ A is ǫ-invariant if |Bz ∩ C|/|Bz| ≥ 1 − ǫ, z ∈ C. Claim A.2. If B is δ-dense, C ⊂ A is ǫ-invariant, and |C|/|A| < δ(1 − ǫ), then C = ∅. Proof. Let α = |A|, β = |B|,γ = |C|. Since z × Bz ∩ C ⊂ C × C, z∈C [ we have γ2 ≥ αγδ(1 − ǫ). Since γ/α < δ(1 − ǫ), we see that γ = 0. Let k be a field and Q ∈ k[x, y, z] be a homogeneous polynomial of degree a. 2 3 We denote by CQ ⊂ P the subvariety of 1-dimensional subspaces L in A , such that QL ≡ 0, and write CQ := CQ(k). Claim A.3. 2 2 (1) The subvariety CQ ⊂ P is a curve, and therefore the subset P − CQ of P2 is Zariski dense. (2) If k = Fq then |CQ| ≤ aq. 2 Proof. The part (1) is immediate. To prove (2) we fix a point a ∈ P (k)−CQ. For 2 any line M ⊂ P (k) containing a, the subset M ∩CQ ⊂ M is defined by a non-zero polynomial of degree a. So |M ∩ CQ| ≤ a. We see that |CQ| ≤ a(q + 1). Let R ∈ k[x, y, z] be a polynomial of degree a. We denote by Q the degree a 3 homogeneous part of P . We denote by L3 the variety of affine lines L in A . For any L ∈L3, we denote by L˜ the corresponding 1-dimensional linear subspace of A3. Let LR ⊂L3 be the subvariety of lines L, such that PL is of degree < a. Claim A.4. (1) L ∈LP ↔ L˜ ∈ CQ. (2) |LP |/|L| ≤ a/q. Proof. The part (1) is immediate and the part (2) follows from the part (1) and Claim A.3. X X ¯ N Let k be a field and := P¯, where P =(Pi)1≤i≤c, are polynomials on k of degrees ≤ di. We denote by A the variety of affine lines in X, by B the variety of 3-dimensional affine subspaces in X and by E ⊂ A2 the subvariety of pairs if lines L, L′ ∈ A such that there exists unique W ∈ B containing both L and L′. We have natural projections p1,p2 : E → A and the surjection j : E → B. We write A := A(k), B := B(k), E := E(k). For any subvariety Z ⊂ A we define ˜ −1 Z Z := p2(p1 ( )). Let f : X → k be a weakly polynomial function of degree ≤ a. We denote by C = Cf ⊂ A the subset of lines L such that deg(fL)= a. For W ∈ B we denote by A(W ) ⊂ A the set of lines L ∈ W , and write C(W ) := A(w) ∩ C. 32 DAVIDKAZHDANANDTAMARZIEGLER We will show that under the assumption that rnc(P¯) ≫ 1 and k is either a finite or an algebraically closed field, the assumption that C is small implies that C = ∅. We first consider the case when k = Fq. In this case the size of C is the number |C|. So the statement to prove is Proposition 4.37. Proof. We start with the following result. Claim A.5. |C(W )|/|A(w)| ≥ 1 − a/q for any W ∈ B containing some L ∈ C. Proof. Since f is weakly of degree ≤ d, its restriction fW on W is a polynomial of degree ≤ d. Since L ∈ C, the restriction fW is of degree d it follows from Claim A.4 that |C(W )|/|A(w)| ≥ 1 − a/q. As follows from Lemma A.5 in [18] for any d¯ = (d1,...,dc) there exists t(d¯) −t(d) such that the subset B ⊂ A × A is q -dense for any X = XP¯, where P is a polynomial of degree d. So it follows from Claim A.2 that in the case when A(d, L, a)= t(d) + 1 and a/q ≤ 1/2, we have C = ∅. We consider now the case when k is algebraically closed. In this case we measure the size of C in terms of the codimension ∆(f) of the algebraic closure Z of C ∈ A. Proposition A.6. For any a there exists a constant C(d,¯ a) such that the fol- lowing holds. Any weakly polynomial function f on X of degree ≤ a such that ∆(f) ≥ C(d,¯ a) is actually of degree References [1] Ananyan, T., Hochster, M. Strength conditions, small subalgebras, and Stillman bounds in degree ≤ 4. arXiv:1810.00413 [2] Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D. Testing Low-Degree Poly- nomials over GF (2). 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