A complete characterization of the existence of rational functional solutions with infinite support bases

摘要

It is known that there exist polynomial solutions G with infinite support base P, of certain functional equations arising from quantum arithmetics, which cannot be constructed from quantum integers. A description of the necessary and sufficient conditions on a set of primes P for the existence of a polynomial solution, with field of coefficients of characteristic zero and support base P, which cannot be constructed from quantum integers is also known, leading to the classification of the set of polynomial solutions. In his papers on quantum arithmetics, Melvyn Nathanson raises a question concerning the classification of the possibly non-trivially broader set of solutions, namely the set of rational function solutions. It is not known at the time that the set of rational function solutions is more than just the set of ratio of polynomial solutions. However, it is now known that there are infinitely many rational function solutions G, with support base P and field of coefficients of characteristic zero, which are not ratios of polynomial solutions with the same support base, even in the purely cyclotomic case. Thus, a natural question that should be asked in order to classify the set of rational function solutions, is: If polynomial solutions are replaced by merely rational function solutions, what would the necessary and sufficient conditions be on the support base P? In this paper, we give a complete description of the necessary and sufficient conditions on the set of primes P for the existence of a rational function solution, with field of coefficients of characteristic zero and support base P, which cannot be constructed from quantum integers.