Numerically validating the completeness of the real solution set of a system of polynomial equations
This work addresses the verification of completeness for real solution sets, a known bottleneck in polynomial system solving, but the approach is incremental and limited to specific algebraic geometry contexts.
The paper presents a method combining numerical algebraic geometry and sums of squares programming to verify whether a computed set of real solutions to a polynomial system is complete. Examples demonstrate the approach for systems with finitely and infinitely many real solutions, including one with inequalities.
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares programming to test whether a given set is "complete" with respect to the real solution set. Specifically, we test whether the Zariski closure of that given set is indeed equal to the solution set of the real radical of the ideal generated by the given polynomials. Examples with finitely and infinitely many real solutions are provided, along with an example having polynomial inequalities.