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

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.