Computing real radicals by moment optimization

TL;DR

This paper introduces a novel convex moment optimization-based algorithm for computing the real radical of polynomial ideals, providing effective stopping criteria and real decomposition verification methods.

Contribution

The paper presents a new algorithm leveraging moment optimization to compute real radicals, including a stopping criterion and a real irreducible decomposition check.

Findings

Successfully computes real radicals for polynomial ideals.

Provides an effective stopping criterion for the algorithm.

Demonstrates the method's effectiveness through examples.

Abstract

We present a new algorithm for computing the real radical of an ideal and, more generally, the-radical of, which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of generates the real radical of. We give an effective, general stopping criterion on the degree to detect when the prime ideals lying over the annihilator are real and compute the real radical as the intersection of real prime ideals lying over. The method involves several ingredients, that exploit the properties of generic positive moment sequences. A new efficient algorithm is proposed to compute a graded basis of the annihilator of a truncated positive linear functional. We propose a new algorithm to check that an irreducible decomposition of an algebraic variety is real, using a generic real projection to reduce to the hypersurface case. There we apply the Sign Changing Criterion, effectively performed with an exact MOP. Finally we illustrate our approach in some examples.