ALGEBRAICALLY CONSTRUCTIBLE FUNCTIONS AND SIGNS OF POLYNOMIALS

Let W be a real algebraic set. We show that the following families of integer-value functions on W coincide: (i) the functions of the form w --> chi(N-w), where X-w are the fibres of a regular morphism f : X -- > W of real algebraic sets, (ii) tile functions of the form w --> chi( S-w), where X-w are the fibres of a proper regular morphism f: X --> W of real algebraic sets, (iii) the finite sums of signs of polynomials on W. Such functions are called algebraically constructible on W. Usi ng their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the li nk operator anti specialization.