Modular absolute decomposition of equidimensional polynomial ideals

TL;DR

This paper introduces a modular algorithm for analyzing the absolute primary decomposition of equidimensional polynomial ideals over rationals, providing detailed structural properties of the components.

Contribution

It presents a novel modular strategy combining elimination and colon ideals with prime integer selection to compute key properties of absolute irreducible components.

Findings

Determines the number of absolute irreducible components

Computes degrees and multiplicities of components

Calculates affine Hilbert functions of reduced components

Abstract

In this paper, we present a modular strategy which describes key properties of the absolute primary decomposition of an equidimensional polynomial ideal defined by polynomials with rational coefficients. The algorithm we design is based on the classical technique of elimination of variables and colon ideals and uses a tricky choice of prime integers to work with. Thanks to this technique, we can obtain the number of absolute irreducible components, their degree, multiplicity and also the affine Hilbert function of the reduced components (namely, their initial ideal w.r.t. a degree-compatible term ordering) .