Computing the Resolution Regularity of Bi-Homogeneous Ideals

Co-author(s): Nasibeh Aramideh and Amir Hashemi

Reference: Journal of Symbolic Computation, 103 (2021) 141-156

Description: An important result about Pommaret bases asserts that their degree is just the Castelnuovo-Mumford regularity of the ideal generated by them. In this paper, we are concerned with an extension of this result to bi-homogeneous ideals. Here the resolution regularity is defined as a two-dimensional vector. We introduce the new concept of an x-Pommaret basis and show how it allows us to determine one component of this vector (this other one can be determined by swapping the role of the variables). As a by-product, we define the notion of an x-quasi-stable monomial ideal and show that an ideal possesses an x-Pommaret basis, if and only if its leading ideal is x-quasi-stable.

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