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A criteria for the recognition of algebraically constructible functions. (Un critère pour reconnaître les fonctions algébriquement constructibles.) (French) Zbl 0959.14035
Let $$V$$ be an algebraic subset of $$\mathbb{R}^n$$. A function $$\phi: V\to \mathbb{Z}$$ is said to be constructible if $$\phi$$ is constant on each member of some finite partition of $$V$$ in semialgebraic subsets. A typical constructible function on $$V$$ is a sum of signs of finitely many polynomials on $$V$$. In fact, these constructible functions are called algebraically constructible [C. McCrory and A. Parusiński, Ann. Sci. Éc. Norm. Supér., IV. Sér. 30, No. 4, 527-552 (1997; Zbl 0913.14018)]. The question is to characterize those constructible functions that are algebraically constructible. Due to general results on real spaces [see the book by C. Andradas, L. Broecker and J. M. Ruiz: “Constructible sets in real geometry” (1996; Zbl 0873.14044)], one has a theoretical criterion for a constructible function to be algebraically constructible. This is a criterion using fans, and seems to be hard to handle in practice.
In the first part of the paper under review, the author establishes a nice inductive criterion for a constructible function on a nonsingular compact algebraic subset to be algebraically constructible (theorems 2 and 3). With this criterion at hand, it becomes considerably easier to decide whether or not a constructible function is algebraically constructible.
In the second part, the author establishes an explicit upper bound on the number of polynomials that are needed to write a given algebraically constructible function as a sum of signs of polynomials. She also shows that her bound is sharp in case $$V=\mathbb{R}^n$$.

##### MSC:
 14P05 Real algebraic sets 14P10 Semialgebraic sets and related spaces
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