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.
翻译:我们把用美元来定义的方形单体图的美元-jets与某个完美图形的稳定的多面体相链接。我们证明,由于美元的差异,美元-jets方案坐标环作为$\mathbb{C}美元-矢量空间的维度是一个多等量的一元+1美元,即该图稳定多面体的Erhart多面体的Erhart多面体。我们证据中的一个主要成分是佐布宁,他确定了由$x+p. 美元产生的差异理想的差度 Gr\"obner基数。我们将Zobnin的结果概括为双向情况。我们研究的是美元(m)n)$-jets,这是一架高维的喷气式模拟,并将它们与美元正值的正值正反角的普通单面三角测量有关。