Finding Integer Solutions of a Rational Function

• Published: 28 August 2019

On the rational function solutions of functional equations arising from multiplication of quantum integers

Mathematische Zeitschrift volume 293,pages 903–933 (2019)Cite this article

Abstract

We prove results concerning rational function solutions of the functional equations arising from multiplication of quantum integers and thus resolve some problems raised by Melvyn Nathanson. First, we show that the rational function solutions contain more structure than the polynomial solutions in an essential way, namely the non-cyclotomic part of the former is no longer necessarily trivial, which allows us to resolve a problem on the associated Grothendieck group \(K(\Upsilon _{P})\) of the collection of all polynomial solutions \(\Upsilon _{P}\) with fields of coefficients of characteristic zero and support base P. Second, we show that, contrary to the polynomial solution case, there exists at least one (infinitely many) rational function solution, with support base P containing all primes, which are not constructible from quantum integers. Third, we show that even in the case where the non-cyclotomic part is trivial, rational function solutions are different from polynomial solutions in that there still exist infinitely many rational function solutions to these functional equations with support base P containing all primes and which are not constructible from quantum integers.

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Affiliations

1. Department of Mathematics, University of Wisconsin-Parkside, Kenosha, Wisconsin, USA

   Lan Nguyen

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Correspondence to Lan Nguyen.

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Nguyen, L. On the rational function solutions of functional equations arising from multiplication of quantum integers. Math. Z. 293, 903–933 (2019). https://doi.org/10.1007/s00209-019-02380-z

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• Received: 27 December 2015

• Accepted: 12 April 2018

• Published: 28 August 2019

• Issue Date: December 2019

• DOI : https://doi.org/10.1007/s00209-019-02380-z

Keywords

• Quantum integer
• Quantum algebra
• q-Series
• Sumset
• Polynomial functional equation
• Polynomial algebra
• Cyclotomic polynomial
• Rational function
• Grothendieck group
• Semigroup

Mathematics Subject Classification

• 11A07
• 11B13
• 11B75
• 11B83
• 11C08
• 11P99
• 11Z99

Finding Integer Solutions of a Rational Function

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