Combinatorial Differential Algebra of $x^p$

Abstract

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 a certain subscheme of the scheme of $n$-jets as a $\mathbb{C}$-vector space is a polynomial of degree $n+1$, namely the Ehrhart 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\"{o}bner 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.