Potentials of a Frobenius like structure and $m$ bases of a vector space

TL;DR

This paper explores the structure of Frobenius-like systems, establishing the existence of potentials and analyzing bases in vector spaces, with implications for arrangements and transformations.

Contribution

It introduces conditions for selecting multiple bases within vector sets and demonstrates their connectivity via elementary transformations.

Findings

Existence of potentials of the first and second kind in Frobenius-like structures.

Necessary and sufficient conditions for selecting multiple bases from a vector set.

Connectivity of different basis selections through 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. Surprisingly the proof is based on the study of finite sets of vectors in a finite-dimensional vector space $V$. Given a natural number $m$ and a finite set $(v_i)$ of vectors we give a necessary and sufficient condition to find in the set $(v_i)$ $m$ bases of $V$. If $m$ bases in $(v_i)$ can be selected, we define elementary transformations of such a selection and show that any two selections are connected by a sequence of elementary transformations.