Systems of cubic forms in many variables 7 1 0 S. L. Rydin Myerson 2 n January 17, 2017 a J 4 1 Abstract ] We consider a system of R cubic forms in n variables,with integer T coefficients, which define a smooth complete intersection in projective N space. Provided n 25R, we prove an asymptotic formula for the ≥ . number of integerpoints in an expanding box at which these forms si- h t multaneously vanish. In particular we can handle systems of forms in a 2 O(R)variables,previousworkhavingrequiredthatn R . Onecon- m ≫ jectures that n 6R+1 should be sufficient. We reduce the problem [ ≥ to an upper bound for the number of solutions to a certain auxiliary 1 inequality. To prove this bound we adapt a method of Davenport. v 1 0 Contents 9 3 0 1 Introduction 2 . 1.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 0 1.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 1.3 Reduction to an auxiliary inequality . . . . . . . . . . . . . . 3 1 : 1.4 Outline of remaining steps . . . . . . . . . . . . . . . . . . . . 5 v 1.5 Structure of this paper . . . . . . . . . . . . . . . . . . . . . . 6 i X 1.6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 r a 2 The eigenvalues of the Hessian matrix H (x) 6 c 3 Intermission: Eigenvalues and minors 9 4 Counting points in the sets K (E ,...,E ) 11 k 1 k+1 5 Small values of a trilinear form 16 6 Constructing singular points on V(c) 20 7 Acknowledgements 21 1 1 Introduction 1.1 Main result Let c (x),...,c (x) be homogeneous cubic forms in n variables x ,...,x , 1 R 1 n with integer coefficients. We treat the simultaneous Diophantine equations c (x) = 0, ..., c (x) = 0 1 R andthecorrespondingprojectivevarietyinPn−1,whichwecallV(c ,...,c ). Q 1 R We assume throughout that the c generate the ideal of V(c ,...,c ), and i 1 R are linearly independent. The cubic case of a classic result of Birch gives us: Theorem 1.1 (Birch [2]). Let B be a box in Rn, contained in the box R [ 1,1] , and having sides of length at most 1 which are parallel to the co- − ordinate axes. For each P 1, write ≥ N (P) = # x Zn : x/P B, c (x) = 0,...,c (x) = 0 . c1,...,cR { ∈ ∈ 1 R } If the variety V(c ,...,c ) is a smooth complete intersection, and we have 1 R n 8R2+9R, (1.1) ≥ then for some I, S 0 and all P 1, the bound ≥ ≥ N (P) = ISPn−3R+O(Pn−3R−δ) (1.2) c1,...,cR holds, where the implicit constant depends only on the forms c , and δ is a i positive real number depending only on d and R. If the variety V(c ,...,c ) 1 R has a smooth point over Q for each prime p, and a real point whose homo- p geneous co-ordinates lie in B, then S and I are positive. In particular this follows from Theorem 1 of Birch [2], on inserting the bound dimV∗ R 1 for the dimension of the variety V∗ occurring in ≤ − that result. This bound follows from Lemma 3.1 of Browning and Heath- Brown [4] whenever V(c ,...,c ) is a smooth complete intersection. See 1 R [17, Lemma 1.1] for details. We sharpen (1.1) as soon as R 3. In 1.3 we prove: ≥ § Theorem 1.2. In Theorem 1.1 we may replace (1.1) with the condition n 25R. (1.3) ≥ For example whenR = 3 and V(c ,c ,c ) is a smooth complete intersec- 1 2 3 tion, Theorem 1.2 applies when n 75, whereas Birch’s theorem requires ≥ n 99. ≥ The “square-root cancellation” heuristic suggests that in place of (1.1) the condition n 6R + 1 should suffice, see for example the discussion ≥ around formula (1.5) in Browning [3]. By handling systems of forms in O(R) variables we come within a constant factor of this conjecture. Our strategy is an extension of our previous work [17]. In forthcoming papers we further generalise this approach to treat systems of R forms with degree d 2, with rational or real coefficients. ≥ 2 1.2 Related work We begin with the case when the forms c (x) are diagonal. i In the case of a single diagonal form c, Baker [1] proves that V(c) has a rational point whenever n 7. ≥ Bru¨dern and Wooley [8, 7, 11] treat diagonal systems in n 6R + 1 ≥ variables. Thisisthebestvalueofnpossiblewiththeclassicalcirclemethod. In particular they prove the Hasse principle for V(c ,...,c ) whenever the 1 R c are diagonal, V(c ,...,c ) is smooth and n 6R+1. They also prove an i 1 R ≥ asymptotic formula of the type (1.2) whenever n 6R+3 holds, or when ≥ R = 2 and n 14 holds [5, 6, 9]. In the case R = 2 they prove a Hasse ≥ principle for certain pairs of diagonal cubics in as few as 11 variables [10]. Returning to the case of general (not necessarily diagonal) forms, we consider the case R = 1. Let c be a cubic form. Hooley [16] proves that if n = 8,thevarietyV(F)issmooth,andtheboxB doesnotcontainapointat which the Hessian determinant of F vanishes, then the asymptotic formula (1.2) holds. In this work he assumes a Riemann hypothesis for a certain modified Hasse-Weil L-function. When n = 9 he proves the same result unconditionally, the weaker error term o(Pn−3) in place of the O(Pn−3−δ) from (1.2). Heath-Brown [15] proves that if n 14 then V(c) always has a ≥ rational point, regardless of whether it is singular. In the case R = 2, Dietmann and Wooley [14] have shown that V(c ,c ) 1 2 always has a rational point when n 827, whether or not it is smooth. ≥ In the general case R 1, Schmidt [18] shows that V(c ,...,c ) always 1 R ≥ has a rational point if n (10R)5. Recent work of Dietmann [13] improves ≥ this condition to n 400 000R4. ≥ 1.3 Reduction to an auxiliary inequality To prove Theorem 1.2 we will use Theorem 1.3 from the author’s previous work [17]. This will reduce the problem to proving an upper bound for the number of solutions to the following auxiliary inequality. Definition 1.1. For any k 1 and t Rk, we write t = max t for ≥ ∈ k k∞ i| i| the supremum norm. When c(x) is a real cubic form in n variables with real coefficients, we define a symmetric matrix 1 ∂3c(x) H (x)= (1.4) c c (cid:18)∂x ∂x (cid:19) k k∞ i j 1≤i,j≤n where c = 1max ∂3c(x) . Thus H (x) is the Hessian of the k k∞ 6 i,j,k∈{1,...,n} ∂xixjxk c cubic form c(x)/ c , which ha(cid:12)s been n(cid:12)ormalised so that 1 is the absolute k k∞ (cid:12) (cid:12) value of its largest coefficient. For each B 1 we put Naux(B) for the c ≥ number of pairs of vectors (x,y) (Zn)2 with ∈ x , y B, H (x)y < B. k k∞ k k∞ ≤ k c k∞ 3 We show that this definition of Naux(B)above agrees with the one given c in [17, Definition 1.1]. There we consider a degree d polynomial f and a system of multilinear forms m(f)(x(1),...,x(d−1)), and when d = 3 and f(x)= c(x), we see that m(f)(x(1),x(2)) = c H (x(1))x(2). k k∞ c It follows that the definition above agrees with Definition 1.1 from [17]. The case d = 3 of Theorem 1.3 in [17] therefore states that: Theorem 1.3. Let the counting function N (P) be as in Theorem 1.1. c1,...,cR Suppose that for some C 1 and C > 3R, we have 0 ≥ Naux(B) C B2n−8C (1.5) β·c ≤ 0 for all β RR and B 1, where we write β c for β c + +β c . Then 1 1 R R ∈ ≥ · ··· for some I, S 0 we have ≥ N (P) = ISPn−dR+O(Pn−dR−δ) c1,...,cR for all P 1, where the implicit constant depends at most on C , C and the 0 ≥ c ,and the positive constant δ depends atmost onC,dand R. The constants i I and S are positive under the same conditions as in Theorem 1.1. We give the following bound for the counting function Naux(B). The c proof occupies the bulk of this paper and is completed in 6. § Proposition 1.1. We call a set of real cubic forms in n variables a closed K cone if (i) for all c and λ 0 we have λc K, and (ii) is closed in ∈ K ≥ ∈ K the real linear space of cubic forms in n variables. Let be a closed cone as above, and let Naux(B) be as in Definition 1.1. c K If we set σ = 1+ max dimSingV(c), (1.6) K c∈K\{0} so that σ 0,...,n 1 , then for all ǫ > 0, c and B 1 we have K ∈ { − } ∈ K ≥ Naux(B) Bn+σK+ǫ. (1.7) c K,ǫ ≪ We will outline the proof after deducing Theorem 1.2. Proof of Theorem 1.2. Suppose that (1.3) holds. We claim that for all B ≥ 1, ǫ > 0 and β RR we have ∈ Naux(B) Bn+R−1+ǫ (1.8) β·c ≪c1,...,cR,ǫ where β c is as in Theorem 1.3. If we set C = (n R+ 1)/8 and let C · − 2 0 besufficiently largeintermsoftheformsc ,wecanthenapplyTheorem1.3. i For (1.8) implies (1.5) on setting ǫ = 1 in (1.8). Moreover we have C > 3R, 2 4 by (1.3). So the hypotheses of Theorem 1.3 are satisfied, and Theorem 1.2 follows. Setting = β c :β RR in Proposition 1.1, we see that (1.8) follows K { · ∈ } from (1.7) unless σ > R 1 holds. Suppose for a contradiction that we K − have σ > R 1. K − By the definition (1.6) there must be β RR 0 with ∈ \{ } dimSingV(β c) R 1. (1.9) · ≥ − We may assume that V(c ,...,c )= V(c ,...,c ,β c) holds, after per- 1 R 1 R−1 · muting the c if necessary. Since V(c ,...,c ) is a smooth complete inter- i 1 R section, we then have V(c ,...,c ) SingV(β c) SingV(c ,...,c ), 1 R−1 1 R ∩ · ⊂ and so dimSingV(c ,...,c ) > dimSingV(β c) R holds. Thus (1.9) 1 R · − implies that V(c ,...,c ) is singular, which is false by assumption. 1 R 1.4 Outline of remaining steps To prove Proposition 1.1 we adapt the argument used to prove Lemma 3 in Davenport [12], and subsequently a somewhat more general result in 5 of § Schmidt [18]. These authors consider the counting function defined by Naux-eq(B) = # (x,y) (Zn)2 : x , y B, H (x)y = 0 c { ∈ k k∞ k k∞ ≤ c } forsome cubicformcwithinteger coefficients. Davenport proves thateither aux-eq N (B) is small, or there is a large rational linear space on which c c vanishes. In order to briefly sketch his line of reasoning, we define some additional notation. Definition1.2. Let H (x) = max H (x) andletλ (x),...,λ (x) k c k∞ i,j| c ij| c,1 c,n be the eigenvalues of the real symmetric matrix H (x), listed with multi- c plicity and in order of decreasing absolute value. Observe that λ (x) n H (x) n2 x . (1.10) | c,1 |≤ k c k∞ ≤ k k∞ Foreach i 1,...,n letD(c,i)(x)bethevector ofalli iminorsofH (x), c ∈ { } × arranged in some order. This is a vector of degree i homogeneous forms in the variables x, with real coefficients. Let J (x) be the Jacobian matrix D(c,i) (c,i) (∂∆ (x)/∂x ) . j k jk Davenport’s argument runs as follows. (1) Let σ 0,...,n 1 . Suppose that Naux-eq(B) Bn+σ for some c ∈ { − } ≫ sufficiently large implicit constant. The contribution to this count from any one vector x is at most O(Bn−rankHc(x)). So there must be an integer b in the set 0,...,n 1 such that at least Bσ+b integer { − } ≫ points x satisfy both rankH (x) = b and x B. c k k∞ ≤ 5 (2) If σ,b are as in (1), then it follows that there is an integer point x(0) such that rankH (x(0)) = b holds and the tangent space to the affine c variety D(c,b+1)(x) = 0 at the point x(0) has dimension σ + b + 1 or more. Equivalently, rankH (x(0)) = b and rankJ (x(0)) c D(c,b+1) ≤ n σ b 1 both hold. This follows from Lemma 2 of Davenport [12]. − − − (3) If there exists a vector x(0) as in (2), then it follows that there exist linear subspaces X,Y of Qn, with dimensions σ + b + 1 and n b − respectively, such that for all X X and Y,Y′ Y the equality ∈ ∈ YTH (X)Y′ = 0 holds. See Lemma 4 in Schmidt [18] or the proof of c Lemma 3 in Davenport [12]. (4) We conclude that if Naux-eq(B) Bn+σ then there are spaces X,Y as c ≫ in (3). So the space Z defined by Z = X Y is a rational linear space, ∩ with dimension at least σ + 1, such that for all Z Z the equality ∈ c(Z) = 0 holds. Our setting differs in three ways from that of Schmidt and Davenport. First, we consider the inequality H (x)y B rather than the equation k c k∞ ≤ H (x)y = 0. Second, for us the cubic form c(x) may have real coefficients. c And third, rather than concluding that c(x) has a rational linear space of zeroes, we seek to show that the variety V(c) is very singular. 1.5 Structure of this paper In 2 and the three sections 4-6 we will modify each of the four steps (1)- § §§ (4) above to accommodate the three changes described at the end of 1.4. § In the remaining section, 3, we prove some technical lemmas relating the § minors and eigenvalues of real matrices. 1.6 Notation Throughout, we let c, t , c , H (x) and Naux(B) be as in Defini- k k∞ k k∞ c c tion 1.1, and we let H (x) , λ (x), D(c,i)(x) and J (x) be as in k c k∞ c,i D(c,i) Definition 1.2. We do not require algebraic varieties to be irreducible, and we adopt the convention that dim = 1. We use Vinogradov’s notation ∅ − ≪ and big-O notation in the usual way. 2 The eigenvalues of the Hessian matrix H (x) c We show that if the counting function Naux(B) from Definition 1.1 is large, c then there are many integer points x for which the eigenvalues of H (x) lie c in some fixed dyadic ranges. This corresponds to step (1) from 1.4. § 6 Lemma 2.1. Let H be a real symmetric n n matrix and let λ ,...,λ 1 n × be the eigenvalues of the matrix H, listed with multiplicity and in order of decreasing absolute value. LetC 1and B 1, and suppose that λ CB 1 ≥ ≥ | | ≤ holds. Set N (B) = # y Zn : y B, Hy B . H { ∈ k k∞ ≤ k k∞ ≤ } Then we have Bn N (B) min . H C,n ≪ 1≤i≤n 1+ λ1 λi | ··· | Proof. The integral vectors y counted by N (B) are all contained in the H box y B, and in the ellipsoid k k∞ ≤ t Rn :tTHTHt nB2 , { ∈ ≤ } which has principal radii λ −1√nB. Hence i | | n N (B) min 1+ λ −1√nB, B H n i ≪ { | | } Yi=1 and as λ CB holds, this is i | |≤ n min 2C λ −1√nB, B . i ≤ { | | } Yi=1 It follows that n N (B) Bn min λ −1, 1 . H C,n i ≪ {| | } Yi=1 Since the inequalities λ λ hold, we deduce that 1 n | | ≥ ··· ≥ | | 1 1 1 N (B) Bnmin 1, , , ..., H C,n ≪ (cid:26) λ λ λ λ λ (cid:27) 1 1 2 1 n | | | | | ··· | Bn min . ≪ 1≤i≤n 1+ λ1 λi | ··· | Definition 2.1. Suppose that k 0,...,n and that E ,...,E R 1 k+1 ∈ { } ∈ such that the inequalities E E 1 hold. Then we define 1 k+1 ≥ ··· ≥ ≥ K (E ,...,E )to betheset of all vectors x in Rn satisfying thefollowing k 1 k+1 conditions: the inequality x B holds, and we have k k∞ ≤ 1 E < λ (x) E i c,i i 2 | |≤ whenever 1 i k holds, and we have ≤ ≤ λ (x) E c,i k+1 | |≤ whenever k+1 i n holds. ≤ ≤ 7 Corollary2.1. LetNaux(B)beasinDefinition1.1, letλ (x)andD(c,i)(x) c c,i be asinDefinition 1.2, and letK (E ,...,E )be asinDefinition 2.1. For k 1 k+1 any B 1, one of the following alternatives holds. Either ≥ Naux(B) c # Zn K (1) , (2.1) Bn(logB)n ≪n { ∩ 0 } or there is k 1,...,n 1 and there are e ,...,e N such that the 1 k ∈ { − } ∈ inequalities logB e e hold and n 1 k ≫ ≥ ··· ≥ 2e1+···+ekNaux(B) c # Zn K (2e1,...,2ek,1) , (2.2) Bn(logB)n ≪n ∩ k (cid:8) (cid:9) or there are e ,...,e N satisfying logB e e and 1 n n 1 n ∈ ≫ ≥ ··· ≥ 2e1+···+enNaux(B) c # Zn K (2e1,...,2en) . (2.3) Bn(logB)n ≪n ∩ n−1 (cid:8) (cid:9) Proof. Note that in the case that k = n, there are no values of i satisfying k+1 i n, so the last condition in the definition of K (E ,...,E ) is k 1 k+1 ≤ ≤ vacuously true and can be omitted. In particular, if k = n then (2.3) follows from (2.2), because K (2e1,...,2en,1) K (2e1,...,2en). n n−1 ⊂ So it is enough to prove that either (2.1) holds or there exist integers k and e ,...,e , satisfying the inequalities 1 k n and logB e e , 1 k n 1 n ≤ ≤ ≫ ≥ ··· ≥ such that (2.2) holds. Now the set K0(1), together with the sets Kk(2e1,...,2ek,1), partition the box x B into disjoint pieces. So, if we let k k∞ ≤ N (B)= # y Zn : y B, N (B)y B , Hc(x) { ∈ k k∞ ≤ k Hc(x) k∞ ≤ } then we have Naux(B)= N (B)+ N (B). c Hc(x) Hc(x) xX∈Zn 1≤Xk≤n xX∈Zn x∈K0(1) e1≥···≥ek≥1x∈Kk(2e1,...,2ek,1) e1≪nlogB (2.4) n The total number of terms on the right-hand side of (2.4) is O ((logB) ) n at most, so it follows that either Naux(B) N (B) c (2.5) Hc(x) ≫n (logB)n xX∈Zn x∈K0(1) holds, or else there are 1 k n and e e 1 such that 1 k ≤ ≤ ≥ ··· ≥ ≥ Naux(B) N (B) c . (2.6) Hc(x) ≫n (logB)n xX∈Zn x∈Kk(2e1,...,2ek,1) 8 If (2.5) holds then the trivial bound N (B) Bn implies (2.1). Sup- Hc(x) ≪n pose instead that (2.6) holds. By (1.10), for each real vector x the bound λ (x) B holds. So we c,1 n | | ≪ may apply Lemma 2.1 with the choice H = H (x) and some C depending c on n only. This shows that Bn N (B) . Hc(x) ≪n 2e1+···+ek Substituting this into (2.6) we see that (2.2) holds, as claimed. 3 Intermission: Eigenvalues and minors Here we collect some elementary facts about the eigenvalues and minors of real matrices which will be needed in 4-5. §§ Lemma 3.1. For each k,ℓ N, let ∈ T = a Nk : 1 a < < a ℓ . k,ℓ 1 k { ∈ ≤ ··· ≤ } This set has ℓ members. For each k,ℓ,m N such that k min ℓ,m , k ∈ ≤ { } and each ℓ (cid:0)m(cid:1)real matrix L, define an ℓ m real matrix L[k] by × k × k (cid:0) (cid:1) (cid:0) (cid:1) L[k] = (L[k]) , L[k] = det((L ) ), ab a∈Tk,ℓ,b∈Tk,m ab aibj 1≤i,j≤k [k] so that the L are the k k minors of L. For all ℓ m matrices L, all ab × × m n matrices M and all k min ℓ,m,n we have (LM)[k] = L[k]M[k]. × ≤ { } That is, we have [k] [k] [k] (LM) = L M . (3.1) ab aw wb w∈XTk,m Proof. Let e(1),...,e(m) be the standard basis of Rm. Fix L,a,b; then each side of (3.1) is an alternating multilinear form in those k columns of M whose indices appear in the vector b. This is some k-tuple of m-vectors. If one is given the value of an alternating multilinear form at the k- tuple e(z1),...,e(zk) for each z Tk,m, one can extend by linearity and the ∈ alternating property to find its value at any k-tuple of m-vectors. In other words,itsufficestocheck (3.1) when,forsomez T ,thek k submatrix k,m ∈ × (M ) is theidentity andall other entries of M arezero. In this case zibj 1≤i,j≤k [k] both sides of (3.1) are equal to M . zb Lemma 3.2. Let M be a real m n matrix. Recall that MTM is positive × semidefinite and symmetric. Let the eigenvalues of MTM be Λ2, ,Λ2 1 ··· n in decreasing order, where the Λ are nonnegative and in decreasing order. i That is, the Λ are the singular values of M, listed in decreasing order. i 9 In particular, if M is a symmetric matrix, then the Λ are exactly the i absolute values of the eigenvalues of M, by diagonalisation. Given a natural number k with k min(m,n), let D(k) be the vector of ≤ k k minors of M, arranged in some order. Then we have: × (i) The maximum norm ∆(k) satisfies k k∞ ∆(k) Λ Λ . (3.2) k k∞ ≍m,n 1··· k (ii) There is a k-dimensional linear space V Rn such that for all v V, ⊂ ∈ Mv v Λ . (3.3) k k∞ ≫m,n k k∞ k We may take V to be a span of k standard basis vectors e(i) in Rn. (iii) ForanyC 1,eitherthere isan(n k+1)-dimensional linearsubspace ≥ − X of Rn such that MX C−1 X for all X X, (3.4) k k∞ ≤ k k∞ ∈ or there is a k-dimensional linear subspace V of Rn, spanned by stan- dard basis vectors of Rn, such that Mv C−1 v for all v V. k k∞ ≫m,n k k∞ ∈ Proof. Part (i). First we prove the result on the assumption that MTM is diagonal. Let the sets T and the matrices L[k] be as in Lemma 3.1. Since k,ℓ MTM is diagonal with diagonal entries Λ2, we have i Λ2 Λ2 = (MTM)[k] a1··· ak aa a∈XTk,n a∈XTk,n = (M[k])2, (3.5) wa a∈XTk,n w∈Tk,m by (3.1). The left-hand side of (3.5) is Λ2 Λ2, and the right-hand side is ∆(k) 2 , so this proves (3.2).≍n 1··· k ≍m,n k k∞ Let O be an n n orthogonal matrix such that OTMTMO is diagonal. × Let ∆(k) be the vector of k k minors of MO. We claim that the norms × ∆(k) and ∆(k) are of comparable size. k ek∞ k k∞ Lemma 3.1 shows that (MO)[k] = M[k]O[k], and since (OT)[k]O[k] = I[k] e and (OT)[k] = O[k] we see that O[k] is orthogonal. Hence the maximum αβ βα norm of the entries satisfies ∆(k) = (MO)[k] M[k] = ∆(k) . k k∞ k k∞ ≍m,n k k∞ k k∞ So in provinge(3.2) we may assume that MTM is diagonal. The result follows. 10