Description of algebraically constructible functions

Abstract

Abstract. The algebraically constructible functions on a real algebraic set are the sums of signs of polynomials on this set. We prove a formula giving the minimal number of polynomials needed to write generically a given algebraically constructible function as a sum of signs. We also prove a characterization of the polynomials appearing in a generic presentation of the function with the minimal number of polynomials. Both results are e¤ective.

Semi-algebraic geometry with rational continuous functions

“…The goal of this section is to bound in terms of d the number of polynomial functions needed in such representation. This is connected to the work of I. Bonnard ([6] and [7]) that concerns the representation of general algebraically constructible functions as sums of signs of polynomial functions. However, the author cautions the reader that most of the results of this text concern specifically algebraically constructible functions that are signs of regulous functions and depend strongly of the nice properties verified by the regulous functions.…”

“…In the fourth and sixth sections, we investigate on the number of polynomial functions needed in the representation of Theorem A. This is connected to the work of I. Bonnard in [6] and [7]. We also study the case where the sign of a regulous function is the sign of a polynomial function.…”