Bounds for the Multiplicity of Gorenstein algebras

TL;DR

This paper establishes upper bounds on the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras using Boij-S"oderberg theory, extending previous results to a broader class of algebras.

Contribution

It introduces a novel approach employing Boij-S"oderberg theory to derive bounds for Gorenstein algebras' multiplicity, generalizing earlier quasi-pure bounds.

Findings

Derived explicit upper bounds for multiplicity

Extended bounds to general Gorenstein algebras

Validated bounds align with known results in special cases

Abstract

We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij-S\"oderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2, (1998)].