A prolongation-projection algorithm for computing the finite real variety of an ideal

TL;DR

This paper introduces a symbolic-numeric algorithm to compute the finite real solutions of an ideal, even when the complex solutions are infinite, using moment matrices and linear functionals.

Contribution

It develops a novel prolongation-projection algorithm that unifies real and complex variety computations through semidefinite programming and linear algebra techniques.

Findings

Successfully computes finite real varieties of ideals.

Unifies real and complex variety computation methods.

Uses moment matrices and linear functionals for algebraic solving.

Abstract

We provide a real algebraic symbolic-numeric algorithm for computing the real variety $V_R(I)$ of an ideal $I$, assuming it is finite while $V_C(I)$ may not be. Our approach uses sets of linear functionals on $R[X]$, vanishing on a given set of polynomials generating $I$ and their prolongations up to a given degree, as well as on polynomials of the real radical ideal of $I$, obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as stopping criterion for our algorithm. This algorithm, based on standard numerical linear algebra routines and semidefinite optimization, combines techniques from previous work of the authors together with an existing algorithm for the complex variety. This results in a unified methodology for the real and complex cases.