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Fonction de Hilbert du cône fibré d'une filtration nœthérienne d'un anneau local nœthérien, un cas d'égalité des degrés des composantes du quasi-polynôme associé

Abdoulaye Assane
Université d'Abobo-Adjamé, UFR-SFA,
Laboratoire de Mathématiques
02 BP 801 Abidjan 02, CÔTE D'IVOIRE

Mathematics Subject Classifications [2010]: 13A15, 13A30, 13D40 .
Key words : Filtration, Hilbert-function, quasi-polynomial


Let MATH be a graded nœtherian ring with $\QTR{small}{A}_{0}$ artinian, $M$ MATH$M$ $_{n}$ a finitely generated graded $\QTR{small}{A}$-module. It is known from H. Dichi and D. Sangaré, Hilbert functions, Hilbert Samuel quasi-polynomials with respect to f-good filtrations, multiplicities. J. Pure Applied Algebra,138, (1999) 205-213 that the function MATH$M$ $_{n}\QTR{small}{)}$ is quasi-polynomial with associated quasi- polynomial say MATH In their paper entitled Hilbert polynomials and power of ideals, J. Herzog, J. Puthenpurakal and J.K. Verma, raised the following question : Do all the polynomials MATH

have the same degree and the same leading term? Here we give a positive answer for the degrees of the polynomials associated with the quasi polyomial function MATH,

where MATH is a nœtherian filtration on a nœtherian local ring MATH.

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