Algebraic Local Cohomology with Parameters and Parametric Standard Bases for Zero-Dimensional Ideals

TL;DR

This paper introduces a method for computing algebraic local cohomology with parameters for zero-dimensional ideals, enabling decomposition of parameter space and efficient computation of parametric standard bases.

Contribution

It presents a novel computation method for local cohomology with parameters and an efficient algorithm for parametric standard bases of zero-dimensional ideals.

Findings

Decomposition of parameter space based on local cohomology structure

Algorithm computes reduced parametric standard bases

Method handles ideals with parameters effectively

Abstract

A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the structure of algebraic local cohomology classes. This decomposition informs us several properties of input ideals and the output of our algorithm completely describes the multiplicity structure of input ideals. An efficient algorithm for computing a parametric standard basis of a given zero-dimensional ideal, with respect to an arbitrary local term order, is also described as an application of the computation method. The algorithm can always output "reduced" standard basis of a given zero-dimensional ideal, even if the zero-dimensional ideal has parameters.