Hilbert’s 17th Problem. Quadratic Forms in Real Algebraic Geometry

Abstract

Hilbert’s 17th problem has played a major role in the development of real algebraic geometry. We first prove the following result: a nonnegative polynomial over a real closed field is a sum of squares of rational functions. We also prove a few generalizations. The second section deals with the equivariant version of Hilbert’s 17th problem. In the third section, we address the problem of representing nonnegative polynomials as sum of squares of polynomials. A theorem of Hilbert describes exactly the couples (n,m), for which every nonnegative form of degree m in n variables is a sum of squares of forms. The fourth section studies the quantitative aspects of the problem, namely, the number of squares needed. The theory of quadratic forms, especially Pfister forms, plays an important role in the solution of this question. The use of Pfister forms is also crucial in the proof, given in Section 5, of a striking result of Brocker and Scheiderer which says that a basic open semi-algebraic subset of an algebraic set of dimension d > 0 can always be defined by d inequalities.