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

Maurice Pouzet (1) , Nicolas M. Thiéry (2, 3)

1 ICJ - Institut Camille Jordan
2 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
3 LMO - Laboratoire de Mathématiques d'Orsay

The profile of a relational structure R is the function φR which counts for every integer n the number φR(n), possibly infinite, of substructures of R induced on the n-element subsets, isomorphic substructures being identified. If φR takes only finite values, this is the Hilbert function of a graded algebra associated with R, the age algebra 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 φR is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on R, φR is eventually a quasi-polynomial when it is bounded by some polynomial.

Mots clés: Hilbert series; graded algebras; profile; relational structures; polynomial growth; Hilbert functions; Combinatorics

Domaines: Mathématique discrète [cs.DM]

Maurice Pouzet, Nicolas M. Thiéry. Some relational structures with polynomial growth and their associated algebras I: Quasi-polynomiality of the profile. The Electronic Journal of Combinatorics, 2013, 20 (2), 35pp. ⟨hal-01121195v2⟩