Let k be an arbitrary field, P= P km 1× k⋯× k P kmp be a multiprojective space over k, and
X⊆ P be a closed subscheme of P. We provide necessary and sufficient conditions for the …

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert
series, and this is later applied to study the projective degrees of rational maps. We use a …

We study several properties of multihomogeneous prime ideals. We show that the
multigraded generic initial ideal of a prime has very special properties, for instance, its …

The main results of this paper interpret mixed volumes of lattice polytopes as mixed
multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In …

This paper studies mixed multiplicities of an arbitrary standard bigraded algebra and mixed
multiplicities of two ideals I, J in a local ring (A, m), where I is an m-primary ideal and J an …

Given a bigradedk-algebraS=⊕(u, ν) S (u, ν),(u, ν) εℕ× ℕ,(ka field), one attaches to it the so-
called diagonal subalgebraSδ=⊕(u, u) S (u, u). This notion generalizes the concept of …

We describe the eventual behaviour of the Hilbert function of a set of distinct points in P
n1×⋯× P nk. As a consequence of this result, we show that the Hilbert function of a set of …

This paper is a survey on major results on Hilbert functions of multigraded algebras and
mixed multiplicities of ideals, including their applications to the computation of Milnor …

Let S be an abelian semigroup, written additively. Let A be a finite subset of S. We denote
the cardinality of A by| A|. For any positive integer h, the sumset hA is the set of all sums of h …

This paper investigates Hilbert polynomials of bigraded algebras which are generated by
elements of bidegrees (1,0),(d_1,1),...,(d_r,1), where d_1,...,d_r are non-negative integers …