Borel generators

TL;DR

This paper introduces Borel generators as a new approach to compute invariants of Borel ideals, simplifying calculations and revealing connections to combinatorics and geometry.

Contribution

It presents Borel generators as an efficient alternative for computing invariants and uncovers new links to Catalan numbers and pseudo-triangulations.

Findings

Simplified computation of associated primes, Hilbert series, Betti numbers

Discovery of connections between Borel ideals and Catalan numbers

Relation between Betti numbers of principal Borel ideals and pseudo-triangulations

Abstract

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinatorics involving Catalan numbers and their generalizations. We conclude with a surprising result relating the Betti numbers of certain principal Borel ideals to the number of pointed pseudo-triangulations of particular planar point sets.