论文标题
Hilbert多项式用于Finarity Matroids
Hilbert polynomials for finitary matroids
论文作者
论文摘要
我们考虑一个元组$φ=（ϕ_1，\ ldots，ϕ_m）$ $ x $上的通勤地图。我们表明，如果$φ$满足某些条件，那么对于任何有限的设置$ a \ subseteq x $，等级为$ \ {ϕ_1^{r_1} {r_1} \ cdotsDistcardcaddartadtartaddartademant，a）：a \ a \ in a \ text {and} r_1+\ cdots+r_m = t polotian $ t polotian in $多项式的多元版本）。这使我们可以轻松地恢复Khovanskii关于总和的生长的定理，经典希尔伯特多项式的存在以及Kolchin多项式的存在。我们还证明了一些新的Kolchin多项式结果，用于差异指数场和O最低场上的衍生物，以及Simplicial Complextes中Betti数量增长的新结果。
We consider a tuple $Φ= (ϕ_1,\ldots,ϕ_m)$ of commuting maps on a finitary matroid $X$. We show that if $Φ$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{ϕ_1^{r_1}\cdotsϕ_m^{r_m}(a):a \in A\text{ and }r_1+\cdots+r_m = t\}$ is eventually a polynomial in $t$ (we also give a multivariate version of the polynomial). This allows us easily recover Khovanskii's theorem on the growth of sumsets, the existence of the classical Hilbert polynomial, and the existence of the Kolchin polynomial. We also prove some new Kolchin polynomial results for differential exponential fields and derivations on o-minimal fields, as well as a new result on the growth of Betti numbers in simplicial complexes.
