We link $n$-jets of the affine monomial scheme defined by $x^p$ to the stable set polytope of some perfect graph. We prove that, as $p$ varies, the dimension of the coordinate ring of the scheme of $n$-jets as a $\mathbb{C}$-vector space is a polynomial of degree $n+1,$ namely the Erhart polynomial of the stable set polytope of that graph. One main ingredient for our proof is a result of Zobnin who determined a differential Gr\"obner basis of the differential ideal generated by $x^p.$ We generalize Zobnin's result to the bivariate case. We study $(m,n)$-jets, a higher-dimensional analog of jets, and relate them to regular unimodular triangulations of the $m\times n$-rectangle.

中文翻译：

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$ x ^ p $的组合微分代数

我们将由$ x ^ p $定义的仿射单项式方案的$ n $ -jets链接到某个完美图的稳定集多边形。我们证明，随着$ p $的变化，作为$ \ mathbb {C} $-向量空间的$ n $ -jets方案的坐标环的维数是度为nn + 1的多项式，即该图的稳定集多边形的Erhart多项式。我们证明的一个主要成分是Zobnin的结果，他确定了$ x ^ p。$产生的微分理想的微分Gr \“观察者基础。我们将Zobnin的结果推广到双变量情况。我们研究$（m，n）$ -jets，喷气机的高维类似物，并将它们与$ m \乘以n $矩形的规则单模三角剖分相关。