Log-Concavity of the Grothendieck Classes of Banana Graphs and Clasped Necklaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913777479319552 |
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| author | Chen, Stephanie |
| author_facet | Chen, Stephanie |
| contents | The Grothendieck classes of melonic graphs satisfy a recursive relation and may be written as polynomials in the class of the moduli space $\mathcal{M}_{0,4}$ with nonnegative integer coefficients, conjectured to be log-concave. In this article, we investigate log-concavity and ultra-log-concavity for the Grothendieck class of banana graphs and the three families of polynomials involved in the recursive relation. We prove that all four are log-concave, establishing the specific case of banana graphs for the log-concavity conjecture. We additionally introduce the infinite family of clasped necklaces, melonic graphs obtained by replacing an edge of a $2$-banana with a string of $m$-bananas. Using the recursive relation, we explicitly compute the classes of clasped necklaces and prove that they too are log-concave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Log-Concavity of the Grothendieck Classes of Banana Graphs and Clasped Necklaces Chen, Stephanie Algebraic Geometry Combinatorics 14C15 The Grothendieck classes of melonic graphs satisfy a recursive relation and may be written as polynomials in the class of the moduli space $\mathcal{M}_{0,4}$ with nonnegative integer coefficients, conjectured to be log-concave. In this article, we investigate log-concavity and ultra-log-concavity for the Grothendieck class of banana graphs and the three families of polynomials involved in the recursive relation. We prove that all four are log-concave, establishing the specific case of banana graphs for the log-concavity conjecture. We additionally introduce the infinite family of clasped necklaces, melonic graphs obtained by replacing an edge of a $2$-banana with a string of $m$-bananas. Using the recursive relation, we explicitly compute the classes of clasped necklaces and prove that they too are log-concave. |
| title | Log-Concavity of the Grothendieck Classes of Banana Graphs and Clasped Necklaces |
| topic | Algebraic Geometry Combinatorics 14C15 |
| url | https://arxiv.org/abs/2503.16296 |