Algebraically constructible functions
by
Isabelle Bonnard
University of Angers

Algebraically constructible functions have been introduced recently by A. Parusinski and C. McCrory. If V is a real algebraic set, an algebraically constructible function on V is a sum of signs of polynomials on V. It induces a function on the real spectrum of the ring of polynomials on V, which is the signature of a quadratic form. In the first part, we use the representation theorem of Becker and Bröcker to find a theoretically algorithmic criterion to decide whether a function is algebraically constructible or not. In the second part, we obtain an upper bound of the minimal number of polynomials needed to describe a given algebraically constructible function. This bound is optimal in any dimension.

Date received: July 5, 1999