A certificate for semidefinite relaxations in computing positive dimensional real varieties

TL;DR

This paper introduces a certificate based on moment matrix coranks for computing Pommaret bases that serve as Groebner bases for ideals with positive dimensional real varieties, ensuring termination of the algorithm.

Contribution

It presents a novel certificate condition for the termination of moment relaxation algorithms in computing Pommaret bases for positive dimensional real varieties.

Findings

The certificate condition is satisfiable in generic delta-regular coordinate systems.

The method computes a Pommaret basis that is also a Groebner basis of an ideal J.

The approach nests between the original ideal and its real radical.

Abstract

For an ideal I with a positive dimensional real variety, based on moment relaxations, we study how to compute a Pommaret basis which is simultaneously a Groebner basis of an ideal J generated by the kernel of a truncated moment matrix and nesting between I and its real radical ideal. We provide a certificate consisting of a condition on coranks of moment matrices for terminating the algorithm. For a generic delta-regular coordinate system, we prove that the condition is satisfiable in a large enough order of moment relaxations.