Abstract
A unit u in a commutative ring with unity R is called exceptional if \(u-1\) is also a unit. We introduce the notion of a polynomial version of this (abbreviated as \(f\hbox {-exunits}\)) for any \(f(X) \in \mathbb {Z}[X]\). We find the number of representations of a non-zero element of \(\mathbb {Z}/n\mathbb {Z}\) as a sum of two f-exunits for an infinite family of polynomials f of each degree \(\ge 1\). We also derive the exact formulae for certain infinite families of linear and quadratic polynomials. This generalizes a result proved by Sander (J Number Theory 159:1–6, 2016).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dolžan, D.: The sums of exceptional units in a finite ring. Arch. Math. (Basel) 112, 581–586 (2019)
Hu, S., Sha, M.: On the additive and multiplicative structures of the exceptional units in finite commutative rings. Publ. Math. Debr. 94, 369–380 (2019)
Houriet, J.: Exceptional units and Euclidean number fields. Arch. Math. (Basel) 88, 425–433 (2007)
Knox, M.L., McDonald, T., Mitchell, P.: Evaluationally relatively prime polynomials. Notes Number Theory Discrete Math. 21, 36–41 (2015)
Lenstra, H.W.: Euclidean number fields of large degree. Invent. Math. 38, 237–254 (1977)
Louboutin, S.R.: Non-Galois cubic number fields with exceptional units. Publ. Math. Debr. 91, 153–170 (2017)
Miguel, C.: On the sumsets of exceptional units in a finite commutative ring. Monatsh. Math. 186, 315–320 (2018)
Miguel, C.: Sums of exceptional units in finite commutative rings. Acta Arith. 188, 317–324 (2019)
Nagell, T.: Sur un type particulier d’unités algébriques. Ark. Mat. 8, 163–184 (1969)
Niklasch, G.: Counting exceptional units. Collect. Math. 48(1–2), 195–207 (1997)
Niklasch, G., Smart, N.P.: Exceptional units in a family of quartic number fields. Math. Comput. 67, 759–772 (1998)
Poulakis, D.: Integer points on algebraic curves with exceptional units. J. Austral. Math. Soc. Ser. A 63, 145–164 (1997)
Sander, J.W.: On the addition of units and nonunits mod m. J. Number Theory 129, 2260–2266 (2009)
Sander, J.W.: Sums of exceptional units in residue class rings. J. Number Theory 159, 1–6 (2016)
Silverman, J.H.: Exceptional units and numbers of small Mahler measure. Exp. Math. 4, 69–83 (1995)
Silverman, J.H.: Small Salem numbers, exceptional units, and Lehmer’s conjecture. Rocky Mt. J. Math. 26, 1099–1114 (1996)
Stewart, C.L.: Exceptional units and cyclic resultants. Acta Arith. 155, 407–418 (2012)
Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83, 2nd edn, pp. 487. Springer, New York (1997)
Xu, G.: On solving a generalized Chinese remainder theorem in the presence of remainder errors. In: Geometry, Algebra, Number Theory, and Their Information Technology Applications. Springer Proceedings in Mathematics and Statistics, vol. 251, pp. 461–476 (2018)
Yang, Q.H., Zhao, Q.Q.: On the sumsets of exceptional units in \(\mathbb{Z}_n\). Monatsh. Math. 182, 489–493 (2017)
Acknowledgements
We came across the MathScinet review (MR3412708) of [14] by Prof. B. Sury and this motivated us to coin the notion of \(f\hbox {-exunits}\). We thank Prof. R. Thangadurai for his constant encouragement and support throughout the project. We are grateful to him for going through the manuscript meticulously several times and clearing up many doubts. This work was completed when the first author was visiting Harish-Chandra Research Institute and he acknowledges the hospitality provided by the institute. We gratefully thank the referee for the detailed comments that enhanced the readability of the paper.
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
Anand, Chattopadhyay, J. & Roy, B. On sums of polynomial-type exceptional units in \(\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}\). Arch. Math. 114, 271–283 (2020). https://doi.org/10.1007/s00013-019-01413-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-019-01413-7