Some relational structures with polynomial growth and their associated algebras. I: Quasi-polynomiality of the profile

Maurice Pouzet¹ ; Nicolas Marc Thiéry²

¹Université Claude-Bernard, Lyon1
²Univ Paris-Sud, Laboratoire de Mathématiques d'Orsay, Orsay, F-91405; CNRS, Orsay, F-91405

The electronic journal of combinatorics, Tome 20 (2013) no. 2

Résumé

The profile of a relational structure $R$ is the function $\phi_R$ which counts for every integer $n$ the number $\phi_R(n)$, possibly infinite, of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\phi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $KA(R)$, introduced by P. J. Cameron. In this paper we give a closer look at this association, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompass well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. We prove that $\phi_R$ is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on $R$, $\phi_R$ is eventually a quasi-polynomial when it is bounded by some polynomial.

Détail

DOI : 10.37236/2193

Classification : 05E99
Mots-clés : relational structure, profile, graded algebra, Hilbert function, Hilbert series, polynomial growth