Combinatorial Differential Algebra of $x^p$

TL;DR

The paper develops a combinatorial differential-algebra framework for the fat point $x^p$ by studying jets via differential ideals and Gröbner bases. It proves that the dimension of coordinate rings $ ext{dim}_{\mathbb{C}}(R_n/C_{p,n})$ (and its bivariate analogue) is a polynomial in $p$, specifically the Ehrhart polynomial of explicit polytopes $P_n$ and $P_{(m,2)}$, with degrees $n+1$ and $3(m+1)$ respectively. A key tool is a generalized differential Gröbner basis in both univariate and bivariate settings, inspired by Zobnin, together with regular unimodular triangulations that yield $T$-orderings compatible with the leading terms of $(x^p)^{(k,\ell)}$. The results connect jet-space dimensions to stable-set polytopes of perfect graphs, providing a geometric interpretation via Ehrhart theory and triangulations; the work also raises open questions about which triangulations give Gröbner bases and the exact Ehrhart correspondence in broader cases.

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 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.

Figures (5)

• Figure 1: The placing triangulation $T_{m,2}$ of the $m \times 2$ rectangle of the point configuration $[(0,0),(0,1),(0,2),(1,0),(1,1),(1,2), \ldots, (m,0),(m,1),(m,2)].$
• Figure 2: The regular placing triangulation $T_{1,n}$ of the $1 \times n$ rectangle for the point configuration $[(0,0),(0,1),\ldots,(0,n),(1,0),\ldots,(1,n)].$
• Figure 3: The four regular unimodular triangulations of the $2\times 2$ square giving rise to a Gröbner basis, the first of which is $T_{2,2}.$ Note that they all arise from $T_{2,2}$ by rotating and flipping.
• Figure 4: The four regular unimodular triangular regulations of the $3\times 2$ rectangle giving rise to a Gröbner basis, the first of which is $T_{3,2}.$
• Figure 5: The four regular unimodular triangular regulations of the $1\times 3$ rectangle that give rise to a Gröbner basis.

Theorems & Definitions (26)

• Definition 1.1
• Proposition 1.2
• proof
• Definition 1.3
• Theorem 1.4: Zobnin
• Definition 1.5
• Proposition 1.6
• proof
• Definition 1.7
• Proposition 1.8