Unit Form Recognition By Mutations: Application Of Mutations In The Search Of Positive Roots

The quadratic forms are algebraic expressions that have important role in different areas of computer science, mathematics, physics, statistics and others. We deal with quadratic forms that are widely used in the study of algebras with finite representation type, called unit forms. Especially, we study unit forms q : Z(n)-> Z by means of their associated edge-bipartite signed (multi)graphs without loops. A big challenge in this field is to develop efficient algorithms to recognize the type of a given unit form. Actually, most algorithms can demand a long time on this task and becomes infeasible. Our aim is to present a new strategy, do the unit form recognition through sequences of mutations in the unit form related exchange matrix P. For that, we show that the positive roots of the unit form are in the c-matrix of P. We also present an algorithm to recognize the type of a given unit form, implemented using BFS (Breadth First Search) and using powerful techniques to simplify this process. (C) 2020 Elsevier B.V. All rights reserved.