Combinatorial Differential Algebra of $x^p$

Rida Ait El Manssour, Anna-Laura Sattelberger

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.

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