Some Relational Structures with Polynomial Growth and their Associated Algebras I: Quasi-Polynomiality of the Profile
Keywords: Relational structure, profile, graded algebra, Hilbert function, Hilbert series, polynomial growth
AbstractThe 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.