Combinatorial Differential Algebra of x^p
share
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 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 Gröbner 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.
READ FULL TEXT