On the number of solutions of multivariable polynomial systems

Systems of multivariable polynomial equations play a key role in many important control and stability problems. In this paper we derive analytical expressions for the number of solutions as a function of the polynomial coefficients. The new results are obtained by combining standard results from elimination theory with the properties of inner determinants. For problems where one of the variables is constrained to a given interval, the number of solutions can be expressed in terms of Sturm's sequences. Tests for system solvability, and the uniqueness of a solution are also presented. A numerical example illustrates that the new results also provide a promising noniterative approach for solving systems of polynomial equations.