The height of the mixed sparse resultant.pdf

a r X i v : m a t h / 0 2 1 1 4 4 9 v 1 [ m a t h .A C ] 2 8 N o v 2 0 0 2 THE HEIGHT OF THE MIXED SPARSE RESULTANT MARTI?N SOMBRA 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. LetA0, . . . ,An ? Z n be finite sets of integer vectors and let ResA0,...,An ∈ Z[U0, . . . , Un] be the associated mixed sparse resultant — or (A0, . . . ,An)-resultant — which is a polynomial in n + 1 groups Ui := {Ui a ; a ∈ Ai} of mi := #Ai 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 sparked a lot interest in its computation, see e.g. [CLO98, Sec. 7.6], [EM99], [D’An02], [JKSS02], 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 A0, . . . ,An is essential (see [Stu94, Sec. 1]) which does not represent any loss of generality, by [Stu94, Cor. 1.1]. Set A := (A0, . . . ,An), and let LA ? Z n denote the Z-module affinely spanned by the pointwise sum ∑n i=0 Ai. This is a subgroup of Z n of finite index [Zn : LA] := #(Z n/LA) because we assumed that the family A is essential. Also set Qi := Conv(Ai) ? R n for the convex hull of Ai 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 ? Rn, the mixed volume MV(P, . . . , P ) equals n! times the Euclidean volume VolRn(P ). We also set Vol(P ) := MV(P, . . . , P ) = n! VolRn(P ). Under this notation and assumption, the resultant is a multihomogeneous polynomial of degree degUi ( ResA0,...,An ) = 1 [Zn : LA] MV(Q0, . . . , Qi?1, Qi+1, . . . , Qn) 0 with respect to each group of variables Ui, see