On circuit binomials of toric ideals of weighted oriented graphs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916424899887104 |
|---|---|
| author | Nanduri, Ramakrishna Roy, Tapas Kumar |
| author_facet | Nanduri, Ramakrishna Roy, Tapas Kumar |
| contents | In this work, we classify the circuit binomials of any weighted oriented graph $D$ and we explicitly compute the circuit binomials of $D$ in terms of the minors of the incidence matrix of $D$. We show that the circuit binomials of any weighted oriented graph $D$ are the primitive binomials corresponding to one of the classes: (i) a balanced cycle, (ii) two unbalanced cycles sharing a vertex, (iii) two unbalanced cycles connected by a path, (iv) two unbalanced cycles sharing a path. We explicitly prove a formula for the primitive binomial generator of the toric ideal $I_D$ in terms of the minors of the incidence matrix of $D$, where $D$ is as in (i), (ii), (iii) and (iv). Thus we explicitly compute all the circuit binomials $\C_D$ of any weighted oriented graph $D$. If $D$ is a weighted oriented graph which has at most two unbalanced cycles such that no two balanced cycles share a path in $D$ and no balanced cycle in $D$ shares an edge with the path which connects the two unbalanced cycles in $D$ if it exists, then we show that $I_D$ is a strongly robust circuit ideal and it has complete intersection initial ideal. For this class of ideals, we explicitly compute the Betti numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16841 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On circuit binomials of toric ideals of weighted oriented graphs Nanduri, Ramakrishna Roy, Tapas Kumar Commutative Algebra In this work, we classify the circuit binomials of any weighted oriented graph $D$ and we explicitly compute the circuit binomials of $D$ in terms of the minors of the incidence matrix of $D$. We show that the circuit binomials of any weighted oriented graph $D$ are the primitive binomials corresponding to one of the classes: (i) a balanced cycle, (ii) two unbalanced cycles sharing a vertex, (iii) two unbalanced cycles connected by a path, (iv) two unbalanced cycles sharing a path. We explicitly prove a formula for the primitive binomial generator of the toric ideal $I_D$ in terms of the minors of the incidence matrix of $D$, where $D$ is as in (i), (ii), (iii) and (iv). Thus we explicitly compute all the circuit binomials $\C_D$ of any weighted oriented graph $D$. If $D$ is a weighted oriented graph which has at most two unbalanced cycles such that no two balanced cycles share a path in $D$ and no balanced cycle in $D$ shares an edge with the path which connects the two unbalanced cycles in $D$ if it exists, then we show that $I_D$ is a strongly robust circuit ideal and it has complete intersection initial ideal. For this class of ideals, we explicitly compute the Betti numbers. |
| title | On circuit binomials of toric ideals of weighted oriented graphs |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2312.16841 |