AN EQUIVALENCE RELATION ARISING FROM MULTIPLICATION OF QUANTUM INTEGERS

In the paper [5], we conjecture that the relation arising from multiplication of quantum integers and the problem of finding a set of criteria for extending the support bases of solutions of a functional equation arising from multiplication of quantum integers, which is described by Nathanson in [2] as Problem 3, is an equivalence relation. We show in [5] that the answer is affirmative in the case where the fields of coefficients of these solutions are . In this paper, we prove that the answer is in fact also affirmative for all fields of characteristic zero. As a consequence of this result, we explain how it simplifies in several important ways the criteria for extension of support bases of solutions of the above functional equation.