We use Z-gradings to study d-dimensional monomial ideals. The Koszul functor is employed to interpret the quasidegrees of local cohomology in terms of the geometry of distractions and to explicitly compute the multiplicities of exponents. These multigraded techniques originate from the study of hypergeometric systems of differential equations.

We study the nonlinear generalization of classical multiview variety, which is a fundamental concept in computer vision. In this paper, we take first comprehensive step to develop analogue varieties. To end, introduce multigraded version saturated special fiber ring. By applying tool, are able compute multidegrees several families

The purpose of this paper is to give an explicit description of the irreducible decomposition of the multigraded Sn-module of coinvariants of Sn×Sn. Many of the results presented can be extended to S n , and analogous questions for other finite reflection group.

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 numbers of complex analytic hypersurfaces with isolated singularity, multiplicities of blowup algebras and mixed volumes of polytopes.

Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a monomial ideal of S. It is well known that the Poincaré series of k over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.