ON THE HILBERT QUASI - POLYNOMIALS FOR NON - STANDARD GRADED RINGS

Abstract The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x1, . . . , xk ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring graduation is non-standard, then its Hilbert function is not definitely equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[x1, . . . , xk ]/I

On the Hilbert quasi-polynomials for non-standard graded rings

The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x1, . . . , xk ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi- polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[x1 , . . . , xk]/I

Graded Betti numbers of powers of ideals

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive \ZZ-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of \ZZ-grading, \ZZ^2 can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(\mu, t)$, \dim_k \left(\tor_i^S(I^t, k)_{\mu} \right) is equal to one of these polynomials in $(\mu, t)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a \ZZ^d-graded algebra, for a positive grading.Comment: 20 page

Some relational structures with polynomial growth and their associated algebras II: Finite generation

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.Comment: 27 pages; submitte

The structure of the inverse system of Gorenstein k-algebras

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given.Comment: 19 pages, to appear in Advances in Mathematic