Hilbert functions of multigraded algebras, mixed multiplicities of ideals and their applications

TL;DR

This survey reviews key results on Hilbert functions of multigraded algebras and mixed multiplicities of ideals, highlighting their applications in complex geometry, algebraic geometry, and polytope theory.

Contribution

It compiles and discusses major findings on Hilbert functions and mixed multiplicities, emphasizing their diverse applications across multiple mathematical fields.

Findings

Connections between Hilbert functions and mixed multiplicities elucidated

Applications to Milnor number computations for hypersurfaces

Insights into multiplicities of blowup algebras and mixed volumes

Abstract

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.