Standard pairs for monomial ideals in semigroup rings

Standard pairs of monomial ideals over non-normal affine semigroups in SageMath

“…The goal of this article is to present the SageMath library stdPairs.spyx, which systematizes computations for monomial ideals in affine semigroup rings. The algorithms implemented here are based on the notion of standard pairs, introduced for monomial ideals in polynomial rings by [8], and generalized to the semigroup ring case in [6]. Standard pairs are a combinatorial structure that contains information on primary and irreducible decompositions of monomial ideals, as well as multiplicities.…”

“…Standard pairs are a combinatorial structure that contains information on primary and irreducible decompositions of monomial ideals, as well as multiplicities. One of the main contributions of [6] is that standard pairs and the associated algebraic concepts can be effectively computed over affine semigroup rings.…”

“…The SageMath library stdPairs.spyx implements the algorithms of [6] to calculate standard pairs for monomial ideals in any pointed (non-normal) affine semigroup ring. This library can be regarded as a generalization of standardPairs function in Macaulay2 implemented by [4].…”

“…A pair is a tuple (a, F) of an element a in NA and a face F of NA [6]. A proper pair of an ideal I is a pair (a, F) such that a + NF ⊆ std(I).…”

“…Moreover, stdPairs.spyx provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from [6]. We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.1.1.…”