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

- 122 Downloads

## 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.

## 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## Notes

## References

- 1.Bernard, S.L., Quirost, A.: On quantum integers and rationals. Contemp. Math.
**649**, 107–131 (2015)MathSciNetGoogle Scholar - 2.Borisov, A., Nathanson, M.B., Wang, Y.: Quantum integers and cyclotomy. J. Num. Theory
**109**(1), 120–135 (2004)MathSciNetzbMATHGoogle Scholar - 3.Nathanson, M.B.: A functional equation arising from multiplication of quantum integers. J. Num. Theory
**103**(2), 214–233 (2003)MathSciNetzbMATHGoogle Scholar - 4.Nguyen, L.: On the solutions of a functional equation arising from multiplication of quantum integers. J. Num. Theory
**130**(6), 1292–1347 (2010)MathSciNetzbMATHGoogle Scholar - 5.Nguyen, L.: On the support base of a functional equation arising from multiplication of quantum integers. J. Num. Theory
**130**(6), 1348–1373 (2010)MathSciNetzbMATHGoogle Scholar - 6.Nguyen, L.: Maximal solutions with field of coefficients of characteristic zero of a functional equation arising from multiplication of quantum integers. Int. J. Num. Theor.
**7**(1), 9–56 (2011)MathSciNetGoogle Scholar - 7.Nguyen, L.: On the polynomial and maximal solutions to a functional equation arising from multiplication of quantum integers. Notes Num. Theory Discrete Math.
**18**(4), 11–39 (2012)MathSciNetzbMATHGoogle Scholar - 8.Nguyen, L.: On the classification of solutions of a functional equation arising from multiplication of quantum integers. Uniform Distrib. Theory
**8**(2), 49–120 (2013)MathSciNetzbMATHGoogle Scholar - 9.Nguyen, L.: Solutions with infinite support bases of a functional equation arising from multiplication of quantum. Notes Num. Theory Discrete Math.
**20**(3), 1–28 (2014)zbMATHGoogle Scholar - 10.Nguyen, L.: Support extension for solutions of a functional equation arising from multiplication of quantum integers. JP J. Algebra Num. Theory Appl.
**35**(2), 81–217 (2014)Google Scholar - 11.Nguyen, L.: A complete characterization of the existence of rational functional solutions with infinite support bases. J. Algebra Appl.
**15**, 9 (2016)MathSciNetGoogle Scholar - 12.Nguyen, L.: Nathanson quantum functional equations and the non-prime semi-group support polynomial solutions. Semigroup Forum
**93**(3), 459–490 (2016)MathSciNetzbMATHGoogle Scholar - 13.Nguyen, L.: On symmetries of roots of rational functions and the classification of rational function solutions of functional equations arising from multiplication of quantum integers with prime semigroup supports. Aequ. Math.
**92**(6), 1001–1035 (2018)MathSciNetzbMATHGoogle Scholar