Polynomials Attached to Feynman Diagrams Have Rational Singularities

Given a matroid on ground set E we define and consider a large class of polynomials in C[E], the polynomial ring whose variables correspond to elements of the ground set, assembled via the matroid structure. We show that all of these polynomials have rational singularities. The method involves jet schemes and Mustata's characterization of rationality in terms of irreducibility of jet spaces.

Within our class of polynomials are three important members: matroid basis polynomials; configuration polynomials; polynomials attached to Feynman diagrams. The last is constructed out of the former two in a straightforward way. In the world of Quantum Field Theory and Feynman diagrams, one instance of our rationality result is the following: given a Feynman diagram with standard simplifying assumptions, the Feynman integral is the Mellin transformation of a polynomial with rational singularities.