Toric geometry and regularization of Feynman integrals

Understand

We study multivariate Mellin transforms of Laurent polynomials by considering special toric compactifications which make their singular structure apparent.

• This gives a precise description of their convergence domain, refining results of Nilsson, Passare, Berkesch and Forsg\aa rd.
• We also reformulate the geometric sector decomposition approach of Kaneko and Ueda in terms of these compactifications.
• Specializing to the case of Feynman integrals in the parametric representation, we construct multiple such compactifications given by certain systems of subgraphs.