Computing Real Radicals by Moment Optimization

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. CCS CONCEPTS• Theory of computation → Semidefinite programming; • Mathematics of computing → Grobner bases and other special bases; • Computing methodologies → Hybrid symbolic-numeric methods.