Numerical Evidence for a Conjecture in Real Algebraic Geometry

Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, at deformation and cheater's ho-motopy. The numerical diiculties are overcome if we work in the true synthetic spirit of the Schubert calculus by selecting the numerically most favorable equations to represent the geometric problem. Since a well-conditioned polynomial system allows perturbations on the input data without destroying the reality of the solutions we obtain not just one instance, but a whole manifold of systems that satisfy the conjecture. Also an instance that involves totally positive matrices has been veriied. The optimality of the solving procedure is a promising rst step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory. 1. Introduction Solving a polynomial system numerically means that approximations are computed to all isolated solutions of the system. Having an approximate zero (as in 6]) implies that Newton's method doubles its accuracy in each step. Homotopy continuation methods provide paths to all isolated approximate zeros. The references 26], 22] and 9] treat polynomial homotopies respectively from within the elds of engineering, numerical analysis and computational algebraic geometry. Path-following methods are described in 1, 2]. Optimal homotopies for solving polynomial systems arising in the Schubert calculus of enumerative geometry were proposed by Birk Huber, Frank Sottile and Bernd Sturmfels in 18]. These homotopies are optimal in the sense that every path leads to a solution when applied to a generic problem instance, whereas the standard homotopies force to trace many diverging solution paths. The type of polynomial system that needs to be solved is presented in the next section, followed by a survey on standard root-counting methods. Thereafter come implementational aspects for the homotopies and a derivation of the equations in the pole placement problem. A report on the main numerical diiculties and solutions is given in the sixth section. The last part of the paper contains a short description of the freely available software package PHC developed by the author. Execution times are listed illustrating the performance of the methods.