Extremal Hilbert series

TL;DR

This paper surveys the extremal Hilbert series of factor rings of various algebraic structures, focusing on minimal and maximal series, with new results on maximal series in polynomial rings.

Contribution

It introduces new findings on maximal Hilbert series in polynomial rings and discusses numerous open problems and conjectures.

Findings

Characterization of minimal Hilbert series for generic forms

New results on maximal Hilbert series in polynomial rings

Comprehensive survey with open problems and conjectures

Abstract

Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.