Moment Matrices, Border Bases and Real Radical Computation

TL;DR

This paper introduces new algorithms combining moment matrices and border bases to efficiently compute the radical and real radical of polynomial ideals, directly obtaining quotient structures without prior algebraic methods.

Contribution

It presents a unified approach that integrates border basis algorithms with moment matrix methods to compute radicals and real radicals directly from polynomial systems.

Findings

Efficient algorithms for real radical computation using moment matrices.

Direct computation of quotient structures of radical ideals.

Unified framework combining border bases and semi-definite programming.

Abstract

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of [17] are efficient and numerically stable for computing complex roots, algorithms based on moment matrices [12] allow the incorporation of additional polynomials, e.g., to re- strict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Gr\"obner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.