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

Abstract

We provide a real algebraic symbolic-numeric algorithm for computing the real variety V R (I) of an ideal I ⊆ R[x], assuming V R (I) is finite (while V C (I) could be infinite). 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 R √ 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 new criterion is satisfied earlier than the previously used stopping criterion based on a rank condition for moment matrices. This algorithm is based on standard numerical linear algebra routines and semidefinite optimization and combines techniques from previous work of the authors together with an existing algorithm for the complex variety.

The Approach of Moments for Polynomial Equations

“…This class of methods was first proposed in [17] with extensions in [18,19], and is the focus of this chapter. The basic idea is to compute the real roots by working in a smaller quotient space, obtained by taking the quotient by the real radical ideal R √ I of the original ideal I, consisting of all polynomials that vanish at the set of common real roots of the original system h i = 0.…”

“…When the real variety V R (I) is finite, R[x]/ R √ I has finite dimension as a vector space, equal to |V R (I)|, and thus Ker M (Λ) is zero-dimensional with A simple geometric property of positive semidefinite matrices yields the following equivalent definition for generic elements of K. This is in fact the key tool used in [17] for computing the real radical ideal R √ I.…”

“…Such a point can be found by solving several semidefinite programs with an arbitrary SDP solver (cf. [17,Remark 4.15]), or by solving a single semidefinite program with an interior-point algorithm using a self-dual embedding technique (see, e.g., [8], [45]). Indeed consider the semidefinite program:…”

“…There is no duality gap, as λ = 1 is obviously feasible for (17). Solving the program (16) with an interior-point algorithm using a self-dual embedding technique yields 4 either a solution Λ lying in the relative interior of the optimal face (i.e., a generic element of K t ), or a certificate that (16) is infeasible thus showing V R (I) = ∅.…”

“…This leads to large moment matrices, easily touching on the limitations of current SDP solver. We refer to [17,19] for a more detailed discussion and some numerical results. In an ongoing project a more efficient, Buchbergerstyle, version of this real root finding method will be implemented based on the more general version of the flat extension theorem, which was described at the end of Section 3.2.…”