##
**Matroids, motives, and a conjecture of Kontsevich.**
*(English)*
Zbl 1076.14026

Let \(G\) be a finite graph with vertex set \(V(G)\), edge set \(E(G)\), and Betti numbers \(b_0(G)\) and \(b_1(G)\). The Kirchhoff polynomial of \(G\) is a multivariate polynomial defined as \(P_G= \sum_T \prod_{e\not\in T} x_e\), where the sum is taken over all spanning trees \(T\) of \(G\), the product is over all edges of \(G\) not in \(T\), and the \(x_e\) are indeterminates, one for each edge \(e\in E(G)\). Kirchhoff polynomials of graphs first appeared in the 19th century, mainly in the analysis of electrical circuits, and they were intensively studied by G. Kirchhoff, C. Maxwell, C. Borchardt, J. Sylvester, and others, which explains the terminology. In contemporary mathematics and physics, Kirchhoff polynomials also play an important role in the evaluation of Feynman amplitudes in the following sense. Let \(V(P_G)\) be the scheme of zeros of \(P_G\) over \(\mathbb{Z}\), i.e., a hypersurface in the affine scheme \(\mathbb{A}^{E(G)}\), and let \(Y_G\) denote its complement. Feynman amplitudes are then related to period integrals on the schemes \(Y_G\), and this relates Kirchhoff polynomials not only to combinatorics, but also to arithmetic algebraic geometry and quantum field theory. Some years ago, in 1997, M. Kontsevich made a conjecture about the number of points of \(Y_G\) over a finite field, which also has attracted the attention of many leading combinatorialists ever since. Kontsevich’s conjecture states that, for every finite graph \(G\), the function \(\# Y_G(\mathbb{F}_q)\) from the set of all prime powers to \(\mathbb{Z}\) is a polynomial in \(q\), i.e., \(Y_G\) is always polynomially countable. This conjecture is obviously equivalent to the conjecture that \(V(P_G)\) in a polynomially countable \(\mathbb{Z}\)-scheme.

Although Kontsevich’s conjecture has been verified for special graphs, and partial results have been obtained by combinatorialists, a complete solution of it seemed to be far away.

However, in the utmost fundamental paper under review, the authors disprove the general statement of Kontsevich’s conjecture in a stunning way. Namely, instead of exhibiting an explicit combinatorial counterexample, they show that the functions \(\#Y_G(\mathbb{F}_q)\) span a space that includes non-polynomial functions, and that the schemes \(Y_G\) are, with respect to their zeta functions, the most general schemes possible. To this end, the authors use R. Stanley’s reformulation of Kontsevich’s conjecture [cf.: R. P. Stanley, Ann. Comb. 2, No. 4, 351–363 (1998; Zbl 0927.05087)], together with his partial results in this direction, and develop then a framework of combinatorial motives that allows to interpret, and thereby to extent, Stanley’s approach in an algebro-geometric setting. Their main theorem states that a certain ring, called the ring of combinatorial motives, equals a certain module over a particular principal ideal domain \(R\), which is (over \(R\)) generated by all functions of the form \(\#Y_G(\mathbb{F}_q)\), where \(q\) runs over all prime powers. The main theorem does not only imply that Kontsevich’s conjecture is generally false, which merely appears as a by-product, but the ingredients of its proof give rise to a much more general theory. Namely, the authors show that their proof extends to the setting of the Denef-Loeser ring of geometric motives, therefore to the theory of motivic integration [cf.: J. Denef and F. Loeser, Invent. Math. 135, No. 1, 201–232 (1999; Zbl 0928.14004)], and that this generalization leads directly back to the mathematical-physical problems that originally motivated-Kontsevich’s conjecture. In this more general context, the authors obtain some partial results supporting the conjecture that certain Feynman amplitudes are rational linear combinations of multiple zeta values with respect to the schemes \(Y_G\).

Altogether, this ingenious paper represents a highly substantial contribution toward a central topic of contemporary research in combinatorics, arithmetic algebraic geometry, and mathematical physics, likewise. It offers a wealth of fundamental new ideas for further research in these areas of mathematics and physics, and therefore its value can barely be overestimated.

Although Kontsevich’s conjecture has been verified for special graphs, and partial results have been obtained by combinatorialists, a complete solution of it seemed to be far away.

However, in the utmost fundamental paper under review, the authors disprove the general statement of Kontsevich’s conjecture in a stunning way. Namely, instead of exhibiting an explicit combinatorial counterexample, they show that the functions \(\#Y_G(\mathbb{F}_q)\) span a space that includes non-polynomial functions, and that the schemes \(Y_G\) are, with respect to their zeta functions, the most general schemes possible. To this end, the authors use R. Stanley’s reformulation of Kontsevich’s conjecture [cf.: R. P. Stanley, Ann. Comb. 2, No. 4, 351–363 (1998; Zbl 0927.05087)], together with his partial results in this direction, and develop then a framework of combinatorial motives that allows to interpret, and thereby to extent, Stanley’s approach in an algebro-geometric setting. Their main theorem states that a certain ring, called the ring of combinatorial motives, equals a certain module over a particular principal ideal domain \(R\), which is (over \(R\)) generated by all functions of the form \(\#Y_G(\mathbb{F}_q)\), where \(q\) runs over all prime powers. The main theorem does not only imply that Kontsevich’s conjecture is generally false, which merely appears as a by-product, but the ingredients of its proof give rise to a much more general theory. Namely, the authors show that their proof extends to the setting of the Denef-Loeser ring of geometric motives, therefore to the theory of motivic integration [cf.: J. Denef and F. Loeser, Invent. Math. 135, No. 1, 201–232 (1999; Zbl 0928.14004)], and that this generalization leads directly back to the mathematical-physical problems that originally motivated-Kontsevich’s conjecture. In this more general context, the authors obtain some partial results supporting the conjecture that certain Feynman amplitudes are rational linear combinations of multiple zeta values with respect to the schemes \(Y_G\).

Altogether, this ingenious paper represents a highly substantial contribution toward a central topic of contemporary research in combinatorics, arithmetic algebraic geometry, and mathematical physics, likewise. It offers a wealth of fundamental new ideas for further research in these areas of mathematics and physics, and therefore its value can barely be overestimated.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

81T18 | Feynman diagrams |

05A15 | Exact enumeration problems, generating functions |

05B35 | Combinatorial aspects of matroids and geometric lattices |

05C05 | Trees |

11G25 | Varieties over finite and local fields |

### Keywords:

algebraic varieties over finite fields; zeta functions; finite graphs; trees; generating functions; matroids; motives; motivic integration; Feynman integrals
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\textit{P. Belkale} and \textit{P. Brosnan}, Duke Math. J. 116, No. 1, 147--188 (2003; Zbl 1076.14026)

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