具有多项式增长的关系结构及其相关代数I：轮廓的拟多项式性
Some relational structures with polynomial growth and their associated algebras I: Quasi-polynomiality of the profile
摘要：
关系结构$R$的轮廓是函数$φ_R$，它对每个整数$n$计算在$n$ -元素子集上诱导的$R$的子结构的数目$φ_R(n)$，可能是无限的，同构子结构被识别。如果$φ_R$只取有限值，则这是与$R$相关联的分次代数的希尔伯特函数，该分次代数是由P. J. Cameron引入的年龄代数。在本文中，我们仔细看看这个协会，特别是当关系结构$R$承认一个有限的单态分解。这种设置仍然包括研究充分的分次交换代数，如有限置换群的不变环，或准对称多项式的环。我们证明了$φ_R$最终是一个拟多项式，这支持了猜想，在温和的假设下，$R$，$φ_R$最终是一个拟多项式时，它是有界的一些多项式。
英文摘要：
The profile of a relational structure $R$ is the function $φ_R$ which counts for every integer $n$ the number $φ_R(n)$, possibly infinite, of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $φ_R$ 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 $φ_R$ is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on $R$, $φ_R$ is eventually a quasi-polynomial when it is bounded by some polynomial.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
