Generalized Samuel Multiplicities of Monomial Ideals and Volumes

Abstract

We describe conjecturally the generalized Samuel multiplicities $c_0,\dots ,c_{d-1}$ of a monomial ideal $I\subset K[x_1,\dots ,x_d]$ in terms of its Newton polyhedron $\text{NP}(I).$ More precisely, we conjecture that c_i equals the sum of the normalized $(d-i)$-volumes of pyramids over the projections of the $(d-i-1)$-dimensional compact faces of $\text{NP}(I)$ along the infinite-directions of i-unbounded facets in which they are contained. For c₀ proofs are known (Guibert, Jeffries and Montaño) and for $c_{d-1}$ a proof is given.

Keywords:

• monomial ideal
• generalized Samuel multiplicity
• Newton polyhedron

2010 Mathematics Subject Classification:

• Primary 13H15
• Secondary 13F20
• 52B20

Acknowledgments

We are grateful to the referees, whose comments improved the presentation.