On one example and one counterexample in counting rational points on graph hypersurfaces

Abstract

In this paper we present a concrete counterexample to the conjecture of Kontsevich about the polynomial countability of graph hypersurfaces. In contrast to this, we show that the "wheel with spokes" graphs $W{S}_n$ are polynomially countable.

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• Algebraic Structures in local QFT
• Chern classes of graph hypersurfaces and deletion-contraction
• Matroid connectivity and singularities of configuration hypersurfaces
• Feynman integrals and periods in configuration spaces
• Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications
• Arithmetic of Potts model hypersurfaces
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• On $c_2$ invariants of 4-regular Feynman graphs
• Motives of graph hypersurfaces with torus operations