Solutions with infinite support bases of a functional equation arising from multiplication of quantum integers

Lan Nguyen
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 3, Pages 1—28
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Authors and affiliations

Lan Nguyen
Mathematics Department, University of Michigan-Ann Arbor
Ann Arbor, MI 48109, United States

Abstract

It follows from our previous works and those of Nathanson that if P is a set of primes, then the greater the cardinality of P, the less likely that there exists a sequence of polynomials, satisfying the functional equation arising from multiplication of quantum integers studied by Nathanson, which has P as its support base and which cannot be generated by quantum integers. In this paper we analyze the set of roots of the polynomials involved leading to a direct construction of a polynomial solution Γ which has infinite support base P and which cannot be generated by quantum integers. Our results demonstrate that there are more to these solutions than those provided by quantum integers. In addition, we also show that a result of Nathanson does not hold if the condition t_Γ = 1 is removed.

Keywords

• Diophantine equation
• Quantum integer
• q-series
  Sumset
• Polynomial functional equation
• Cyclotomic polynomial

AMS Classification

• 11P99
• 11C08

References

1. Borisov, A., M. Nathanson, Y.Wang, Quantum Integers and Cyclotomy, Journal of Number Theory, Vol. 109, 2004, No. 1, 120–135.
2. Nathanson, M., A Functional Equation Arising From Multiplication of Quantum Integers, Journal of Number Theory, Vol. 103, 2003, No. 2, 214–233.
3. Nguyen, L., The Grothendieck Group associated to the Collection of all Solutions of a Functional Equation Arising from Multiplication of Quantum Integers with a given Support. Proceeding of Combinatorics and Additive Number Theory, 2011 (to appear).
4. Nguyen, L., On the Solutions of a Functional Equation Arising from Multiplication of Quantum Integers, Journal of Number Theory Vol. 130, 2010, No. 6, 1292–1347.
5. Nguyen, L., On the Classification of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers. Uniform Distribution Theory Journal, 2012 (to appear).
6. Nguyen, L., On the Support Base of a Functional Equation Arising from Multiplication of Quantum Integers, Journal of Number Theory, Vol. 130, 2010, Issue 6, 1348–1373.
7. Nguyen, L., Extension of Supports of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers. (in preparation)