Bernstein's theorem in affine space

Abstract

The stable mixed volume of the Newton polytopes of a polynomial system is defined and shown to equal (generically) the number of zeros in affine space Cn. This result refines earlier bounds by Rojas, Li, and Wang [5], [7], [8]. The homotopies in [4], [9], and [10] extend naturally to a computation of all isolated zeros in Cn. Our object of study is a system F = (f1 , . . . , fn) of polynomial equations of the form fi = Σq∈Ai ci,q,·xq where ci,q ∈ C* and xq = xq11⋯xqnn. (1) Here Ai is a finite subset of Nn, called the support of fi, and Qi = conv(Ai) is the Newton polytope of fi. The mixed volume M(A1 , . . . , An) is the coefficient of l1l2;⋯ ln in the homogeneous polynomial Vol(l1Q1 +⋯+ lnQn), where Vol is the Euclidean volume, and Q1,+⋯+Qn := {x1, +⋯+ xn ∈ Rn: xi ∈ Qi for i = 1 , . . . , n} (2) denotes the Minkowski sum of polytopes [2]. The following toric root count is well known.