1 research outputs found
Solving parametric systems of polynomial equations over the reals through Hermite matrices
We design a new algorithm for solving parametric systems having finitely many
complex solutions for generic values of the parameters. More precisely, let with and
, be the algebraic set
defined by and be the projection . Under the
assumptions that admits finitely many complex roots for generic values of
and that the ideal generated by is radical, we solve the following
problem. On input , we compute semi-algebraic formulas defining
semi-algebraic subsets of the -space such that
is dense in and the number of real points in
is invariant when varies over each .
This algorithm exploits properties of some well chosen monomial bases in the
algebra where is the ideal generated by in
and the specialization property of the so-called Hermite
matrices. This allows us to obtain compact representations of the sets by
means of semi-algebraic formulas encoding the signature of a symmetric matrix.
When satisfies extra genericity assumptions, we derive complexity bounds on
the number of arithmetic operations in and the degree of the
output polynomials. Let be the maximal degree of the 's and , we prove that, on a generic , one can compute
those semi-algebraic formulas with operations in and that the polynomials involved
have degree bounded by .
We report on practical experiments which illustrate the efficiency of our
algorithm on generic systems and systems from applications. It allows us to
solve problems which are out of reach of the state-of-the-art