Matroids motives, and a conjecture of Kontsevich

By P. Belkale, P. Brosnan. 2000

118 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5351424ce3b902edd51575daf8b8989e6bcab151

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Citations in the corpus

The Massless Higher-Loop Two-Point Function (2008, background): “After reducing with respect to α1, we have: S [6,1] = { α2 + α3 + α4 + α2α3 + α3α4 , α2 + α3 + α4 , α3α4 + α3 + α4 , α2α4 + α3α4 + α2 + α3 + α4 , α3α4 + α2 + α3 + α4 } Reducing with respect to the variable α2 gives: S [6,1,2] = {α4 + 1 , α3 + 1 , α3α4 + α3 + α4 , α3 + α4 } .”
From polygons and symbols to polylogarithmic functions (2011, background): “[85], work of Belkale–Brosnan [86] or more recently by Brown [87] and others.”