Counterexamples to a conjecture of Kontsevich.

Prakash Belkale (University of Utah)

(Joint with Patrick Brosnan, Max Planck Institute)

Motivated by the relationship between Feynmann amplitudes and multiple
zeta values, Kontsevich introduced a polynomial $P_G$ associated to
every graph $G$ and conjectured that the number of zeros of $P_G$ over
${\mathbf F}_q$ is a polynomial in $q$. We show that this conjecture is
false by showing its relation to the representability problem of matroids.
We show that if $X$ is any scheme over $Z$, then we can express the
function $q \rightarrow X(\mathbf F}_q$ as a linear combination with
$Q(q)$ coefficients of similiar functions of suitable $P_G$'s.