Potentials of a Frobenius like structure

TL;DR

This paper demonstrates the existence of potentials in Frobenius-like structures within arrangements using matroid theory, employing power series and elementary transformations to establish the proof.

Contribution

It introduces a novel approach to proving potentials in Frobenius-like structures through matroid-based decompositions and elementary transformations.

Findings

Existence of potentials of the first and second kind is proven.

A new matroid partition result is established.

The proof utilizes a power series ansatz and elementary transformations.

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.