Un critère pour reconnaître les fonctions algébriquement constructibles

Abstract

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 |

Description of algebraically constructible functions

“…By Example 2.1, in dimension one we have NðjÞ ¼ MðjÞ, and we can calculate NðjÞ in an e¤ective way for any dimension of V . This bound was already in [7], where its sharpness is proved if k is even. (For odd values of k, this bound can be improved a bit to get a sharp one, see [7].)…”

“…This bound was already in [7], where its sharpness is proved if k is even. (For odd values of k, this bound can be improved a bit to get a sharp one, see [7].) This proves the inequality d.…”

“…We use the notion of walls of the function j. This notion has been used by Acquistapace, Andradas, Broglia and Vélez to study basicness ( [2]) and separation ([1]) of semi-algebraic sets, and by the author to characterize algebraically constructible functions ( [7]). We recall the definition.…”

“…We consider the algebraically constructible function q W j on W defined in [7]. We only recall here the generic definition in our non-singular case.…”

“…As in [7], we identify the algebraically constructible function j ¼ P r j¼1 sgn P j considered generically on V , with the signaturej j of the form h f 1 ; . .…”