Quasidegrees is a package that enables the user to construct multigraded rings and look at the graded structure of multigraded finitely generated modules over a polynomial ring. The quasidegree set of a $\ZZ^d$-graded module $M$ is the Zariski closure in $\CC^d$ of the degrees of the nonzero homogeneous components of $M$. This package can compute the quasidegree set of a finitely generated module over a $\ZZ^d$-graded polynomial ring. This package also computes the quasidegree sets of local cohomology modules supported at the maximal irrelevant ideal of modules over a $\ZZ^d$-graded polynomial ring.
The motivation for this package comes from $A$-hypergeometric functions and the relation between the rank jumps of $A$-hypergeometric systems and the quasidegree sets of non-top local cohomology modules supported at the maximal irrelevant ideal of the associated toric ideal as described in the paper:
Laura Felicia Matusevich, Ezra Miller, and Uli Walther. Homological methods for hypergeometric families. J. Am. Math. Soc., 18(4):919-941, 2005.
This package is written when the ambient ring of the modules in question are positively graded and are presented by a monomial matrix, that is, a matrix whose entries are monomials. This is due to the algorithms depending on finding standard pairs of monomial ideals generated by rows of a presentation matrix.
Version 1.0 of this package was accepted for publication in volume 9 of The Journal of Software for Algebra and Geometry on 26 February 2019, in the article Computing quasidegrees of A-graded modules. That version can be obtained from the journal or from the Macaulay2 source code repository.
This documentation describes version 1.0 of Quasidegrees.
The source code from which this documentation is derived is in the file Quasidegrees.m2.