Hermite's method of separation of solutions of systems of algebraic equations and its applications (Q1208262)

The problem treated is that of finding the number of real solutions of an algebraic system \(f_ 1(x,y) = f_ 2(x,y) = 0\) satisfying an algebraic condition \(g(x,y) > 0\). A review is given of Hermite's method which permits to solve the problem using a finite number of algebraic operations on the coefficients of \(f_ 1\), \(f_ 2\) and \(g\). The method has a number of remarkable connections and applications, e.g. quadratic forms, Sturm series, the Kronecker-Poincaré index of an algebraic field, conditions for the sign-definiteness of a homogeneous higher-order polynomial of three variables. Extensions of the method to \(\mathbb{R}^ n\) are also discussed.