Poincar\'e series on good semigroup ideals

TL;DR

This paper extends the concept of Poincaré series from plane curve rings to good semigroup ideals, revealing symmetry properties and generalizing previous results to a broader algebraic context.

Contribution

It introduces a new definition of Poincaré series for good semigroup ideals and generalizes symmetry results previously known for curve-associated semigroups.

Findings

Poincaré series defined for good semigroup ideals.

Symmetry of Poincaré series for certain ideal pairs.

Generalization of Pol's result to a wider class of ideals.

Abstract

The Poincar\'e series of a ring associated to a plane curve was defined by Campillo, Delgado, and Gusein-Zade. This series, defined through the value semigroup of the curve, encodes the topological information of the curve. In this paper we extend the definition of Poincar\'e series to the class of good semigroup ideals, to which value semigroups of curves belong. Using this definition we generalize a result of Pol: under suitable assumptions, given good semigroup ideals E and K, with K canonical, the Poincar\'e series of K-E is symmetric to the Poincar\'e series of E.