Algebraically constructible functions on a real algebraic set are sums of signs of polynomials on this set. The representation theorem gives an e¨ective criterion to characterize these functions. On the other hand results on spaces of orderings can be used to bound the minimal number of polynomials needed to describe a given algebraically constructible function. Brought to you by | University of California Authenticated Download Date | 6/10/15 4:23 AM Si B est un anneau de valuation de K, on dit que B est compatible avec s e si son ide Âal maximal m B est convexe pour s, i.e. si pour tous f e B et g e m B ve Âri®ant 0`f`g pour s, on a f e m B . Dans ce cas, s induit sur le corps re Âsiduel k B de B un ordre unique s de ®ni de la manie Áre suivante: si f e Bnm B , alors s f s f (ou Á f est la classe de f dans k B ). On dit alors que s se spe Âcialise en s, et on note s 3 s. Si F est un e Âventail de K, et si B est compatible avec tous les ordres de F , alors les spe Âcialisations des ordres de F forment un e Âventail F sur k B . Le the Âore Áme de trivialisation Bonnard, Fonctions alge Âbriquement constructibles62 Brought to you by |
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.
We present a characterization of sums of signs of global analytic functions on a real analytic manifold M of dimension two. Unlike the algebraic case, obstructions at infinity are not relevant: a function is a sum of signs on M if and only if this is true on each compact subset of M . This characterization gives a necessary and sufficient condition for an analytically constructible function, i.e. a linear combination with integer coefficients of Euler characteristic of fibres of proper analytic morphisms, to be such a sum of signs.
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