On the Euler characteristic of fibres of real polynomial maps

Abstract

Let Y be a real algebraic subset of R and F : Y → R be a polynomial map. We show that there exist real polynomial functions g1, . . . , gs on R such that the Euler characteristic of fibres of F is the sum of signs of gi. The purpose of this paper is to give a new, self-contained and elementary proof of the following result. Theorem. Let Y be a real algebraic subset of R and F : Y → R be a polynomial map. Then there exist real polynomials g1(y), . . . , gs(y) on R n such that the Euler characteristic of fibres of F is the sum of signs of gi, that is χ(F−1(y)) = sgn g1(y) + . . .+ sgn gs(y). Our proof is based on a classical and elementary result expressing the number of real roots of a real polynomial of one variable as the signature of an associated quadratic form known already to Hermite [He1, He2] and Sylvester [Syl], see also [B], [BW], [BCR, p. 97]. In the proof we use a modern generalized version of this result presented in [PRS] (note that we need only a one variable case of [PRS], that is precisely [BR, Proposition p. 18]). Our original proof of the theorem [PS] used different means such as the theory of local topological degree of polynomial mappings, Grobner bases and the Eisenbud-Levine Theorem and was not so explicit as the one presented below. 1991 Mathematics Subject Classification: Primary 14P05, 14P25. Received by the editors: January 23, 1997; in the revised form: November 15, 1997. The paper is in final form and no version of it will be published elsewhere.