Matroids, motives and conjecture of Kontsevich

TL;DR

This paper investigates Kontsevich's conjecture on the polynomial nature of the number of zeros of Kirchhoff polynomials over finite fields, revealing its falsehood through matroid theory and universality results.

Contribution

It disproves Kontsevich's conjecture by connecting Kirchhoff polynomials to matroid representation schemes and demonstrates their universality in scheme arithmetic.

Findings

Kontsevich's conjecture is false.

Kirchhoff polynomial schemes relate to matroid representation spaces.

These schemes generate all arithmetic of schemes of finite type over integers.

Abstract

Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of Feynman amplitudes. Motivated by work of D. Kreimer and D. J. Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of a Kirchhoff polynomial over the field with q elements is always a polynomial function of q. We show that this conjecture is false by relating the schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that these schemes essentially generate all arithmetic of schemes of finite type over the integers.