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On Hilbert quasi-polynomial functions of finitely generated bigraded modules over non standard bigraded rings

Henri DICHI
Université Blaise Pascal, Labo de Maths, Compl. Sci.
Clermont Ferrand II Cézeaux 63177 Aubière France

Daouda SANGARÉ
Labo de Mathématiques et Informatique, Université Nangui Abrogoua,
22 BP 1709 Abidjan 22 Abidjan, Côte d'Ivoire

Mathematics Subject Classification: [2010] : 13A02, 13A15, 13A30, 13B25, 13D40, 13F20, 13F25, 13H15
Keywords : non standard bigraded module, quasi-polynomial function in two variables, Hilbert two variables functions, Poincaré series, multiplicities

Abstract:

In this paper we prove, as main results, that the Hilbert function of a finitely generated bigraded module over a non standard bigraded ring is a quasi- polynomial function in two variables if the ring $A_{0,0}$ is artinian and that if this function has nonnegative degree , then where $dim_{Kr}\ M$ is the Krull dimension of the module . We show also that, if $A_{0,0}$ is artinian and local, then the cumulative function $H^{\ast }(M,-,-)$ associated with the Hilbert function is a quasi- polynomial function whose degree $d^{\ast }(M)$ satisfies the equality MATH