  Configuration polynomials generalize the classical Kirchhoff polynomial
defined by a graph. Their study sheds light on certain polynomials appearing in
Feynman integrands. Contact equivalence provides a way to study the associated
configuration hypersurface. In the contact equivalence class of any
configuration polynomial we identify a polynomial with minimal number of
variables; it is a configuration polynomial. This minimal number is bounded by
$r+1\choose 2$, where $r$ is the rank of the underlying matroid. We show that
the number of equivalence classes is finite exactly up to rank $3$ and list
explicit normal forms for these classes.
