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.

quantum integer, generalized quantum integer, q-series, sumset, polynomial functional equation, cyclotomic polynomial.

Received: November 28, 2023; Accepted: February 6, 2024; Published: October 7, 2024

How to cite this article: Lan Nguyen, An equivalence relation arising from multiplication of quantum integers, JP Journal of Algebra, Number Theory and Applications 63(6) (2024), 517-586. https://doi.org/10.17654/0972555524031

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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