Potentials of a Frobenius like structure

Abstract

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

The paper generalizes arXiv:1608.08423. Theorem 2.1 in the new paper is a matroid version of the linear algebra result theorem 1.3 in the old paper. The main changes between old and new paper are in chapter 2. The new paper has 13 pages. It is accepted for publication in the Glasgow Mathematical Journal