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

TL;DR

This paper provides a counterexample to Kontsevich's conjecture on polynomial countability of graph hypersurfaces and demonstrates that certain wheel with spokes graphs are polynomially countable.

Contribution

It presents a specific counterexample disproving the conjecture and confirms polynomial countability for a class of graphs, advancing understanding of graph hypersurfaces.

Findings

Counterexample to Kontsevich's conjecture

Wheel with spokes graphs are polynomially countable

Disproves universal polynomial countability of 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 $WS_n$ are polynomially countable.