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

2016/08/30 by Hertling, Claus, Varchenko, Alexander
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences

paper · doi:10.48550/arxiv.1608.08423

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.

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