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The $α$-representation for the Tait coloring and for the characteristic polynomial of matroid

Eduard Lerner

TL;DR

The paper develops an α-representation framework over finite fields of odd characteristic to express the characteristic polynomial χ_M(q) of regular matroids and the Tait-coloring counts for cubic planar graphs. It employs a Fourier-analytic approach over F_q, multidimensional Gaussian sums, and determinant-based invariants (s, s', L(M; α)) to derive explicit sums of Legendre symbols that evaluate χ_M(q) and χ_{M^ot}(q) at q. Key contributions include a unified α-representation for regular matroids, a dual-matroid variant, a generalization to representable (nonregular) matroids, and a concrete α-based formula for the number of Tait colorings via the matrix of faces FM(α), with illustrative examples. The work connects matroid theory, Laplace–Kirchhoff-type matrices, and classical results like Heawood’s theorem, offering a computational framework with potential ties to the Four Colors Theorem. Overall, it broadens the applicability of α-representation techniques to combinatorial invariants and planar graph colorings through finite-field arithmetic and Fourier analysis.

Abstract

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.

The $α$-representation for the Tait coloring and for the characteristic polynomial of matroid

TL;DR

The paper develops an α-representation framework over finite fields of odd characteristic to express the characteristic polynomial χ_M(q) of regular matroids and the Tait-coloring counts for cubic planar graphs. It employs a Fourier-analytic approach over F_q, multidimensional Gaussian sums, and determinant-based invariants (s, s', L(M; α)) to derive explicit sums of Legendre symbols that evaluate χ_M(q) and χ_{M^ot}(q) at q. Key contributions include a unified α-representation for regular matroids, a dual-matroid variant, a generalization to representable (nonregular) matroids, and a concrete α-based formula for the number of Tait colorings via the matrix of faces FM(α), with illustrative examples. The work connects matroid theory, Laplace–Kirchhoff-type matrices, and classical results like Heawood’s theorem, offering a computational framework with potential ties to the Four Colors Theorem. Overall, it broadens the applicability of α-representation techniques to combinatorial invariants and planar graph colorings through finite-field arithmetic and Fourier analysis.

Abstract

Consider a finite field , , where is an odd number. Let be a regular matroid; denote by the family of its bases, , where , . Let a subset in have the maximal cardinality and satisfy the condition , while . Let us represent the value of the characteristic polynomial of the matroid at the point as the linear combination of Legendre symbols with respect to , whose coefficients are modulo equal to . 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 -representation of vacuum Feynman amplitudes in the case of a finite field, which has inspired the Kontsevich conjecture (1997). The -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.

Paper Structure

This paper contains 17 sections, 14 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The historical background and the goal of the paper
  3. Denotations and statements of main results
  4. Examples of calculations by formulas given in theo-rems \ref{['thm1']} and \ref{['thm2']}
  5. Auxiliary assertions necessary for proving the-o-rems \ref{['thm1']} and \ref{['thm2']}
  6. Remark on the set $A^*$
  7. Another statement of Theorem \ref{['thm1']}
  8. A more general variant of Theorem \ref{['thm1']}
  9. Theorem \ref{['th:main2']} as a corollary of Theorem \ref{['thm3']} and some refinements with respect to the choice of $A^*$ and $A^*_\perp$
  10. Flows and an analog of the Laplace--Kirchhoff matrix
  11. The Heawood theorem
  12. Multidimensional Gaussian sums over the field ${\mathbb F}_q$
  13. The $\alpha$-representation
  14. The Fourier transform over the field ${\mathbb F}_q$ and its properties
  15. Proof of Theorem \ref{['thm2']}
  16. ...and 2 more sections

Key Result

Theorem 1

Put $q=p^d$, where $p$ is an odd prime number. The following formula is valid for any regular matroid without loops: here the sum is calculated over everywhere nonzero values $\alpha$ in $\mathbb F_q^E$.

Figures (1)

  • Figure 1: We will calculate the number of proper colorings of vertices and edges of this graph in 3 colors with the help of theorems \ref{['thm1']} and \ref{['thm2']}, correspondingly.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Corollary 8
  • Proposition 9
  • Proposition 10: reiner1, Exercise 11 (c)
  • ...and 4 more