4 research outputs found
A prolongation-projection algorithm for computing the finite real variety of an ideal
We provide a real algebraic symbolic-numeric algorithm for computing the real
variety of an ideal , assuming it is finite while may not
be. Our approach uses sets of linear functionals on , vanishing on a
given set of polynomials generating and their prolongations up to a given
degree, as well as on polynomials of the real radical ideal of , 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.Comment: revised versio
A prolongation-projection algorithm for computing the finite real variety of an ideal
We provide a real algebraic symbolic-numeric algorithm for computing the real variety of an ideal , assuming it is finite while may not be. Our approach uses sets of linear functionals on , vanishing on a given set of polynomials generating and their prolongations up to a given degree, as well as on polynomials of the real radical ideal of , 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