Combinatorial Differential Algebra of $x^p$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2021
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913858360180736 |
|---|---|
| author | Manssour, Rida Ait El Sattelberger, Anna-Laura |
| author_facet | Manssour, Rida Ait El Sattelberger, Anna-Laura |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_03182 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Combinatorial Differential Algebra of $x^p$ Manssour, Rida Ait El Sattelberger, Anna-Laura Algebraic Geometry Symbolic Computation Combinatorics 05E40, 13P10, 12H05 (primary), 52B20 (secondary) 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. |
| title | Combinatorial Differential Algebra of $x^p$ |
| topic | Algebraic Geometry Symbolic Computation Combinatorics 05E40, 13P10, 12H05 (primary), 52B20 (secondary) |
| url | https://arxiv.org/abs/2102.03182 |