Counting Real Zeros

This paper presents an n-dimensional generalization of Hermite's theorem for counting real roots of a polynomial using quadratic forms. We solve the problem of counting the number of real solutions of a system of polynomial equations within an algebraic polyhedron in n-dimensional space, where the polynomials are taken to have rational coefficients. Our algorithm is purely symbolic, which means that it may be used to implement infinite-precision algorithms for arithmetic in the real-algebraic subset of the real numbers. We present algorithms for doing this as an application of the general theory. Our algorithms are based on resultant theory, both because this theory provides insights into the algorithms, and because it makes possible a comparatively clear complexity analysis which shows the algorithms to be worst-case optimal, i.e., singly exponential in the degree of the polynomials.