THE HEIGHT OF THE MIXED SPARSE RESULTANT

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1 THE HEIGHT OF THE MIXED SPARSE RESULTANT Abstract. We present an upper bound for the height of the mixed sparse resultant, defined as the logarithm of the maximum modulus of its coefficients. We obtain a similar estimate for its Mahler measure. Let A 0,..., A n Z n be finite sets of integer vectors and let Res A0,...,A n Z[U 0,..., U n ] be the associated mixed sparse resultant or (A 0,..., A n -resultant which is a polynomial in n + groups U i := {U i a ; a A i } of m i := #A i variables each. We refer to [Stu94] and [CLO98, Chapter 7] for the definitions and basic facts. This resultant is widely used as a tool for polynomial equation solving, a fact that has sparked a lot interest in its computation, see e.g. [CLO98, Sec. 7.6], [EM99], [D An02], [JKSS04], while it is also studied from a more theoretical point of view because of its connections with toric varieties and hypergeometric functions, see e.g. [GKZ94], [CDS98]. We assume for the sequel that the family of supports A 0,..., A n is essential (see [Stu94, Sec. ] which does not represent any loss of generality, by [Stu94, Cor..]. Set A := (A 0,..., A n, and let L A Z n denote the Z-module affinely spanned by the pointwise sum n A i. This is a subgroup of Z n of finite index [Z n : L A ] := #(Z n /L A because we assumed that the family A is essential. Also set Q i := Conv(A i R n for the convex hull of A i for i = 0,... n. We note by MV the mixed volume function as defined in e.g. [CLO98, Sec. 7.4]: this is normalized so that for a polytope P R n, the mixed volume MV(P,..., P equals n! times its Euclidean volume Vol R n(p. We also set Vol(P := MV(P,..., P = n! Vol R n(p. Under this notation and assumption, the resultant is a multihomogeneous polynomial of degree deg Ui ( ResA0,...,A n = [Z n : L A ] MV(Q 0,..., Q i, Q i+,..., Q n > 0 with respect to each group of variables U i, see [PS93, Cor. 2.4]. The absolute height of a polynomial g = a c a x a C[x,..., x n ] is defined as H(g := max{ c a ; a N n }. Hereby we will be mainly concerned with its (logarithmic height: h(g := log H(g = log max{ c a ; a N n }. Date: April 5, Mathematics Subject Classification. Primary G50; Secondary 3P99. Key words and phrases. Mixed sparse resultant, height of polynomials, Mahler measure. Supported by a Marie Curie post-doctoral fellowship of the European Community Program Improving Human Research Potential and the Socio-economic Knowledge Base, contract n o HPMFCT

2 2 The main result of this paper is the following upper bound for the height of the resultant: Theorem.. h ( Res A0,...,A n [Z n : L A ] MV(Q 0,..., Q i, Q i+,..., Q n log(#a i. We write for short Res A := Res A0,...,A n and MV i (A := [Z n :L A ] MV(Q 0,..., Q i, Q i+,..., Q n for i = 0,..., n. The previous result can thus be rephrased as H ( Res A n (#A i MV i(a. This improves our previous bound for the unmixed case [Som02, Cor. 2.5] and extends it to the general case. We remark that the obtained upper bound is polynomial in the size of the input family of supports A and in the mixed volumes MV i (A, and hence it represents a truly substantial improvement over all previous general estimates. These are the ones which follow either from the Canny-Emiris type formulas (Inequality (4 in the appendix, see also [KPS0, Prop..7] or [Roj00, Thm. 23] or from direct application of the unmixed case (see the inequality (3 below for k =. We also consider the Mahler measure, which is another usual notion for the size of a n-variate polynomial. The Mahler measure of g C[x,..., x n ] is defined as m(g := log g dµ n, S n where S C is the unit circle and dµ is the Haar measure over S of total mass. This can be compared with the height: in our case ( MV i (A log(m i m(res A h(res A MV i (A log(m i by [KPS0, Lem..]. We refer to [KPS0, Sec...] for an account on some of the notions of height of complex polynomials: just note that the height h(g here coincides with log g in that reference. We obtain the same estimate as before for the Mahler measure of the resultant. Theorem.2. m ( Res A0,...,A n [Z n : L A ] MV(Q 0,..., Q i, Q i+,..., Q n log(#a i. Note that this improves by a factor of 2 the estimate which would derive from direct application of Theorem. and the inequalities ( above. Both estimates are a consequence of the following: Lemma.3. Let f 0 C A 0,..., f n C An. Then log ResA0,...,A n (f 0,..., f n [Z n MV(Q 0,..., Q i, Q i+,..., Q n log f i, : L A ] where f i := a A i f i a denotes the l -norm of the vector f i = (f i a ; a A i.

3 THE HEIGHT OF THE MIXED SPARSE RESULTANT 3 Let g = a c a x a C[x,..., x n ]. Then for a N n we have that g(z,..., z n c a = z a + zn an+ dµ n S n by Cauchy s formula and so h(g sup { log g(ξ ; ξ S n }. Thus Theorem. is a consequence of this inequality applied to g := Res A, together with Lemma.3. On the other hand, Theorem.2 follows from Lemma.3 by a straightforward estimation of the integral in the definition of the Mahler measure. Proof of Lemma.3. Let k N. Then let k A i Z n denote the pointwise sum of k copies of A i, and set k A := (k A 0,..., k A n. It is easy to verify that k A is also essential, L k A = L A and Conv(k A i = k Q i. We identify each f i C A i with the corresponding Laurent polynomial f i = a A i f i a x a, and we set f k i C ka i for the vector which corresponds to the k-th power of f i. By the factorization formula for resultants [PS93, Prop. 7.] we get that and so (2 Res ka (f k 0,..., f k n = Res A (f 0,..., f n kn+ k n+ log ResA (f 0,..., f n h(res ka + MV i (k A log fi k h(res ka + k n+ MV i (A log f i. The first inequality follows from the straightforward estimate G(u 0,..., u n H(G n u i d i for a multihomogeneous polynomial G of degree d i in each group of variables, applied to G := Res k A and u i := fi k. The second one follows from the linearity of the mixed volume, and the sub-additivity of the l -norm with respect to polynomial multiplication (which implies that log fi k k log f i. Now let B Z n be any finite set such that L B = Z n and such that A 0,..., A n B. Set n(k := #k B and P := Conv(B R n. Then the (unmixed resultant Res kb is a polynomial in (n + n(k variables and total degree (n + Vol(k P = (n + k n Vol(P. We have also that L kb = Z n and so we are in the hypothesis of [Som02, Cor. 2.5], which gives the height estimate h(res kb 2 (n + log ( n(k Vol(k P = 2 (n + log ( n(k k n Vol(P. We have that k A i k B for i = 0,..., n and so by [Stu94, Cor. 4.2] there exists a monomial order such that Res ka divides the initial form init (Res k B. This is a polynomial in (n + n(k variables of degree and height bounded by those of Res kb, and so h(res ka h(res kb + 2 log ( (n + n(k + (n + k n Vol(P (3 4 (n + log ( (n + n(k + k n Vol(P by the inequality h(f h(g+2 deg(g log(n+, which holds for f, g Z[x,..., x n ] such that f g (see [KPS0, Lem..2(.d] applied to f := Res ka and g := init (Res k B. Finally we set B := b + d [0, ] n R n where [0, ] denotes the unit interval of R, for some b Z n and d N such that A 0,..., A n b + d [0, ]. Then n(k =

4 4 log ( #(k b + k d [0, ] n Z n = log(k d + n = O k (log k (here the notation O k refers to the dependence on k and so h(res ka = O k (k n log k. Note that alternatively, we could have obtained this from the inequality (4 in the appendix. Toghether with the inequality (2 this implies that log ResA (f 0,..., f n MV i (A log f i from where we conclude by letting k. ( log k + O k, k Let us consider some examples. For short we set H(A := H(Res A and E(A := n (#A i MV i(a ; we also set q(a := log E(A log H(A for the quotient between the height of the resultant and the estimate from Theorem.. Example.. Sylvester resultants. For d N we let A 0 (d = A (d := {0,, 2,..., d} Z. The corresponding resultant coincides with the Sylvester resultant of two univariate polynomials of the same degree d. In this case MV 0 (d = MV (d = d and #A 0 (d = #A (d = d +, and so E(d := E ( A 0 (d, A (d = (d + 2 d. We compute the height H(d := H ( A 0 (d, A (d for 2 d 7 with the aid of Maple and we collect the results in the following comparative table: d H(d E(d 8 4, ,625 60,466,76 3,84,287,20 4,398,046,5,04 q(d Example.2. We take this example from [EM99, Example 3.5]. Let A 0 := { (0, 0, (,, (2,, (, 0 }, A := { (0,, (2, 2, (2,, (, 0 }, A 2 := { (0, 0, (0,, (,, (, 0 }. Then MV 0 = 4, MV = 3 and MV 2 = 4, so that E(A = On the other hand, we can compute the resultant using its expression in [EM99, Example 3.9] as a quotient of determinants and we obtain that H(A = 8. Hence H(A = 8, E(A = 4,94,304, q(a = For reference, the straightforward estimation of H(A via the Canny-Emiris formula (see the appendix below gives: H(A 2 82 = 4,835,703,278,458,56,698,824,704.

5 THE HEIGHT OF THE MIXED SPARSE RESULTANT 5 Example.3. We take this example from [Stu94, Example 2.]. Let A 0 := { (0, 0, (2, 2, (, 3 }, A := { (0,, (2, 0, (, 2 }, A 2 := { (3, 0, (, }. Then MV 0 = 5, MV = 7 and MV 2 = 7, so that E(A = From the explicit monomial expansion of the resultant (see [Stu94, Example 2.] we find that H(A = 4 and so H(A = 4, E(A = 68,024,448, q(a = These examples show that there is still some room for improvement over Theorem.. It is however possible that our estimate is quite sharp anyway: in spite of the large difference between H(A and E(A in the computed examples, the quotient q(a is quite small, and moreover it does not seem to grow when E(A. In any case, it would be very interesting to have an exact expression for h(res A as was remarked to me by B. Sturmfels or at least a non trivial lower bound. Note that the only information that we dispose about the exact value of the coefficients of Res A is for the extremal ones, which are equal to ± [Stu94, Cor. 3.]. Remark.4. After a first version of this paper was circulating, C. D Andrea (personal communication obtained a non trivial lower bound for the height of the Sylvester resultant, and an improvement of the upper bound for this case: in the notation of Example. above, he obtains that H(d d!. Appendix: Estimation of the height via the Canny-Emiris formula For purpose of easy reference, we establish herein the estimate for h(res A which follows from the Canny-Emiris formula and the standard estimates for the behavior of the height of polynomials under addition, multiplication and division. Assume that L A = Z n and set Q := n Q i R n. Let M 0,..., M n be a family of Canny-Emiris (square, non singular matrices for A; we refer to [CLO98, Sec. 7.6] for their precise definition. In the sequel we just describe the aspects needed for the height estimate. A family of Canny-Emiris matrices is not unique, as their construction depends on a choice of a coherent mixed subdivision of Q and of a sufficiently small and generic vector δ Q n. Set E := (Q + δ Z n. The procedure then uses the given subdivision of Q to split this set into a disjoint union E = E 0 (j E n (j, for each 0 j n. The elements in E are in bijection with the rows of M j, and to each p E i (j corresponds a row of M j with exactly m i = #A i non zero entries, which consist of the variables in U i := {U i a ; a A i }. Set D j := det(m j Z[U 0,..., U n ] \ {0}. The Canny-Emiris formula [CLO98, Ch. 7, Thm. 6.2] states that Res A = gcd(d 0,..., D n. Then D 0 is a multihomogeneous polynomial of degree N i := #E i (0 in each set of variables U i and of height bounded by n N i log(m i. We have that Res A D 0 and

6 6 so m(res A m(d 0, which combined with [KPS0, Lem..] gives ( h(res A h(d 0 + Ni + MV i (A log(#a i (4 ( 2 Ni + MV i (A log(#a i. Applied to Example.2, this gives the stated estimate: N 0 = N = 4 and N 2 = 7 (see [EM99, Example 3.5] and so the previous estimate gives H(A = In general, the estimate so obtained is much worse than that of Theorem., especially for n 0. Consider e.g. A i := {0,..., d} n Z n for i = 0,..., n. Then it is easy to show that Inequality (4 gives h(res A ( 2 ((n+ d n +(n+! d n log(d+ n = n ( 2 (n+ n +(n+! d n log(d+ while Theorem. gives h(res A n (n +! d n log(d +. Acknowledgements. I thank Carlos D Andrea for several helpful discussions, and for having communicated me his work on the height of the Sylvester resultant. I also thank Matilde Lalín and Carlos D Andrea for disproving a conjecture I made in a previous version of this paper, on the value of m(res A, References [CDS98] E. Cattani, A. Dickenstein, B. Sturmfels, Residues and resultants, J. Math. Sci. Univ. Tokyo 5 ( [CLO98] D. Cox, J. Little, D. O Shea, Using algebraic geometry, Graduate Texts in Mathematics 85, Springer, 998. [D An02] C. D Andrea, Macaulay style formulas for sparse resultants, Trans. Amer. Math. Soc. 354 ( [EM99] I. Emiris, B. Mourrain, Matrices in elimination theory, J. Symbolic Comput. 28 ( [GKZ94] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser, 994. [JKSS04] G. Jeronimo, T. Krick, J. Sabia, M. Sombra, The computational complexity of the Chow form, Found. Comput. Math. 4 ( [KPS0] T. Krick, L.M. Pardo, M. Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 09 ( [PS93] P. Pedersen, B. Sturmfels, Product formulas for resultants and Chow forms, Math. Z. 24 ( [Roj00] J.M. Rojas, Algebraic geometry over four rings and the frontier to tractability, Contemp. [Som02] Math. 270, Amer. Math. Soc. 2000, M. Sombra, Minimums successifs des variétés toriques projectives, to appear in the J. Reine Angew. Math., 25 pp.. E-print: math.nt/ [Stu94] B. Sturmfels, On the Newton polytope of the resultant, J. Algebraic Combin. 3 ( Université de Paris 7, UFR de Mathématiques, place Jussieu, 7525 Paris Cedex 05, France. address: sombra@math.jussieu.fr Équipe de Géométrie et Dynamique, 2