Matroidal Polynomials Have Rational Singularities

We introduce the class of matroidal polynomials. They are (possibly inhomogeneous) polynomials attached to a matroid and axiomatized by Deletion-Contraction identities. When homogeneous they are matroid support polynomials--polynomials whose monomial support are the bases of a matroid. We show that matroidal polynomials have rational singularities, provided the underlying matroid is connected of rank at least two. The proof is jet-theoretic and involves controlling the dimension of certain jet loci.

By similar methods we show flag matroidal polynomials (attached to a flag of matroids) and Feynman integrands (attached to a Feynman diagram) have rational singularities. When the Feynman diagram satisfies general kinematics, this proves: the Feynman integral in Lee-Pomeransky form is the Mellin transformation of a polynomial with rational singularities.

Our goal is to explain most of these concepts as well as the matrimony between matroids and jets.

Joint with Uli Walther.