Numerators in Parametric Representations of Feynman Diagrams

The parametric representation has been used since a long time for the evaluation of Feynman diagrams. As a dimension independent intermediate representation, it allows a clear description of singularities. Recently, it has become a choice tool for the investigation of the type of transcendent numbersappearing in the evaluation of Feynman diagrams. The inclusion of numerators has however stagnated since the ground work of Nakanishi. I here show howto greatly simplify the formulas through the use of Dodgson identities. In the massless case in particular, reduction to the completion to a vacuum graph allows for a strong reduction of the maximal power of the Symanzik polynomial in the denominator.