Some Relational Structures with Polynomial Growth and their Associated Algebras I: Quasi-Polynomiality of the Profile.
ELECTRONIC JOURNAL OF COMBINATORICS(2013)
摘要
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.
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关键词
Relational structure,profile,graded algebra,Hilbert function,Hilbert series,polynomial growth
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