The $α$-representation for the Tait coloring and for the characteristic polynomial of matroid
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866917967271297024 |
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| author | Lerner, Eduard |
| author_facet | Lerner, Eduard |
| contents | Consider a finite field $\mathbb F_q$, $q=p^d$, where $p$ is an odd number. Let $M=(E,r)$ be a regular matroid; denote by ${\mathcal B}$ the family of its bases, $\bar s(M;α)=\sum_{B\in {\mathcal B}}\prod_{e\not\in B} α_e$, where ${α_e\in \mathbb F_q}$, $α_e\neq 0$. Let a subset $A\equiv A(α)$ in $E$ have the maximal cardinality and satisfy the condition $\bar s(M|A;α)\neq 0$, while ${r^*}(α)=|A|-r(E)$. Let us represent the value of the characteristic polynomial of the matroid $M$ at the point $q$ as the linear combination of Legendre symbols with respect to $\bar s(M|A;α)$, whose coefficients are modulo equal to $1/q^{r^*(α)/2}$. This representation generalizes the formula for a flow polynomial of a graph which was obtained by us earlier. The latter formula is an analog of the so-called $α$-representation of vacuum Feynman amplitudes in the case of a finite field, which has inspired the Kontsevich conjecture (1997). The $α$-representation technique is also applicable for expressing the number of Tait colorings for a cubic biconnected planar graph in terms of principal minors of the matrix of faces of this graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $α$-representation for the Tait coloring and for the characteristic polynomial of matroid Lerner, Eduard Combinatorics 05B35, 05C31 Consider a finite field $\mathbb F_q$, $q=p^d$, where $p$ is an odd number. Let $M=(E,r)$ be a regular matroid; denote by ${\mathcal B}$ the family of its bases, $\bar s(M;α)=\sum_{B\in {\mathcal B}}\prod_{e\not\in B} α_e$, where ${α_e\in \mathbb F_q}$, $α_e\neq 0$. Let a subset $A\equiv A(α)$ in $E$ have the maximal cardinality and satisfy the condition $\bar s(M|A;α)\neq 0$, while ${r^*}(α)=|A|-r(E)$. Let us represent the value of the characteristic polynomial of the matroid $M$ at the point $q$ as the linear combination of Legendre symbols with respect to $\bar s(M|A;α)$, whose coefficients are modulo equal to $1/q^{r^*(α)/2}$. This representation generalizes the formula for a flow polynomial of a graph which was obtained by us earlier. The latter formula is an analog of the so-called $α$-representation of vacuum Feynman amplitudes in the case of a finite field, which has inspired the Kontsevich conjecture (1997). The $α$-representation technique is also applicable for expressing the number of Tait colorings for a cubic biconnected planar graph in terms of principal minors of the matrix of faces of this graph. |
| title | The $α$-representation for the Tait coloring and for the characteristic polynomial of matroid |
| topic | Combinatorics 05B35, 05C31 |
| url | https://arxiv.org/abs/2503.14285 |