Solving polynomial systems: an LMI-based approach

This paper considers the problem of computing the real solutions of systems of polynomial equalities and inequalities, and proposes a new approach based on convex linear matrix inequality (LMI) optimizations. In particular, the original polynomial systems is converted into an equivalent one whose number of solutions of the equality part that do not satisfy the inequalities (infeasible equality solutions) is reduced by introducing suitable auxiliary polynomials. Moreover, the solutions of this system can be computed by finding vectors with given polynomial structure in suitable linear spaces, operation that can be easily performed if the dimension of these linear spaces is not large. Examples show that the number of infeasible equality solutions can be drastically reduced, hence allowing for an easier and more accurate computation of the results