Computing the Delta Set of an Affine Semigroup: A Status Report
Host
Department of Applied Mathematics
Speaker
Christopher O'Neill
Department of Mathematics, University of California at Davis
https://www.math.ucdavis.edu/~coneill/research.html
Description
An affine semigroup \(S\) is a subset of \(\mathbb Z_{\geq 0}^k\) that is closed under vector addition, and a factorization of \(a\in S\) is an expression of \(a\) as a sum of generators of \(S\). The delta set of \(a\) is a set of positive integers determined by the ``missing factorization lengths" of \(a\), and the delta set of \(S\) is the union of the delta sets of its elements. Though the delta set of any affine semigroup is finite, its definition as an infinite union makes explicit computation difficult. In this talk, we explore algebraic and geometric properties of the delta set, and survey the history of its computation for affine semigroups. The results presented here span the last 20 years, ranging from a first algorithm for a small class of semigroups that is impractical for even basic examples, to recent joint work with Garcia-Sanchez and Webb expressing the delta set of any affine semigroup in terms of Groebner bases, and include results from numerous undergraduate research projects.
Event Topic
Nonlinear Algebra and Statistics (NLASTATS)