Intersection multiplicity, Milnor number and Bernstein's theorem

Abstract

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which completes works of Kushnirenko [Kus76] and Wall [Wal99]. Given a fixed collection of n convex integral polytopes in R n , we also give an explicit characterization of systems of n polynomials supported at these polytopes which have the maximum number (counted with multiplicity) of isolated zeroes on n , or more generally, on the complement of a union of coordinate subspaces of n ; this completes the program (undertaken by many authors including Khovanskii [Kho78], Huber and Sturmfels [HS97], Rojas [Roj99]) of the extension to n of Bernstein's theorem [Ber75] on number of solutions of n polynomials on ( * ) n . Our solutions to these two problems are connected by the computation of the intersection multiplicity at the origin of n hypersurfaces determined by n generic polynomials.

On the Multiplicity of Isolated Roots of Sparse Polynomial Systems

“…Our formulas for the multiplicity of the origin can be seen as a generalization of those in [12], in the sense that the only hypotheses on the supports we make are the necessary ones, proved in [10, Proposition 6], so that the origin is an isolated zero of a generic system with the given supports. An earlier approach from [18] to compute the multiplicity of the origin under no further assumptions on the supports leads to a formula which, unlike ours, is not symmetric in the input polynomials, as already stated by the author. Furthermore, in this paper we give formulas for the multiplicity of arbitrary isolated affine zeroes of a generic sparse system.…”

“…In this sense, we may speak of µ A as the multiplicity of 0 as an isolated root of a generic sparse system supported on A. Explicit conditions on the coefficients satisfying mult A (c) = µ A are given in [18,Theorem 4.12]. Therefore, in this section, we will focus on the computation of the multiplicity of the origin as a common zero of a generic polynomial system supported on A = (A 1 , .…”

“…A generalization of this result to the mixed case under the same assumption can be found in [12], where the multiplicity of the origin is expressed in terms of mixed covolumes. Recently, in [18] a formula for the intersection multiplicity at the origin of the hypersurfaces defined in C n by n generic polynomials with fixed Newton diagrams is proved.…”