Abstract
Fueter and Polya proved that the only quadratic polynomials giving a bijection between N and N\(^{2}\) are the two Cantor polynomials. It is conjectured that there is no bijection from N\(^2\) onto N given by a polynomial of degree at least 3. A similar problem arises when the domain of the map is replaced by the set of integral points in some sector in R\(^2\). Rational sectors were considered by Nathanson and Stanton. Here, we study and solve the case of general irrational sectors. In fact, our method enables us also to recover the results on rational sectors and also answer a question posed by Nathanson.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cantor, G.: Ein Beitrah Zur Mannigfaltigkeitslehre. J. Reine Angew. Math. 84, 212–258 (1877)
Cantor, G.: Beiträge Zur Begrundung der Transfiniter Mengenlehre. Math. Ann. 46, 484–512 (1895)
Chowla, P.: On some polynomials which represent every natural number exactly once. Det Kongelige Norske Widenskabers Selskabs Fordhandlinger 34(Nr. 2), 8–9 (1961)
Fueter, R., Pólya, G.: Rationale Abzählung der Gitterpunkte. Vierteljschr. Naturforsch. Ges. Zürich 58, 380–386 (1923)
Gjaldbæk, K. S.: Non-injectivity of nonzero discriminant polynomials and applications to packing polynomials. In: Nathanson M. B. (ed.), Combinatorial and Additive Number Theory IV, Springer Proceedings in Mathematics & Statistics, vol. 347, pp. 195–201 (2021)
Huxley, M.N.: Area, Lattice Points and Exponential Sums. London Mathematical Society Monographs, New Series, vol. 13. Oxford Science Publications, New York (1996)
Lew, J.S., Rosenberg, A.L.: Polynomial indexing of integer lattice points, I. J. Number Theory 10, 192–214 (1978)
Lew, J.S., Rosenberg, A.L.: Polynomial indexing of integer lattice points, II. J. Number Theory 10, 215–243 (1978)
Nathanson, M.B.: Cantor polynomials for semigroup sectors. J. Algebra Appl. 13(5), 1350165 (2014)
Nathanson, M.B.: Cantor polynomials and the Fueter–Pólya theorem. Amer. Math. Monthly 123(10), 1001–1012 (2016)
Smoryński, C.: Logical Number Theory I. An Introduction. Springer, New York (1991)
Stanton, C.: Packing polynomials on sectors of \({\mathbb{R}}^2\), Integers 14, Paper No. A, vol. 67, p. 13 (2014)
Vsemirnov, M. A.: Two elementary proofs of the Fueter–Pólya theorem on pairing polynomials, Algebra i Analiz, 13(5), 1- -15 (2001). English translation: St. Petersburg Math. J., 13(5), 705–715 (2002). Errata: Algebra i Analiz, 14(5), 240 (2002). English translation: St. Petersburg Math. J., 14(5), 887 (2003)
Acknowledgements
The authors acknowledge SERB, DST, Government of India, for awarding us VJR/2017/000006 under the VAJRA Faculty Scheme which enabled the second author to visit the first author when this work was done. They are very grateful to a referee who pointed out a crucial correction to Theorem 2; it can be stated only for odd p, but this suffices. They also thank both the referees for other suggestions and for drawing attention to a recent reference. Data sharing not applicable to this article as no data sets were generated or analyzed during the current theoretical study.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sury, B., Vsemirnov, M. Packing polynomials on irrational sectors. Res. number theory 8, 39 (2022). https://doi.org/10.1007/s40993-022-00338-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-022-00338-5