Properties of solutions to Pell’s equation over the polynomial ring
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- by Nikoleta Kalaydzhieva
- Proc. Amer. Math. Soc. 150 (2022), 3771-3785
- DOI: https://doi.org/10.1090/proc/15994
- Published electronically: May 13, 2022
Abstract:
In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers $\sqrt {D}$ is the solution to Pell’s equation for $D$. It is well-known that, once an integer solution to Pell’s equation exists, we can use it to generate all other solutions $(u_n,v_n)_{n\in \mathbb {Z}}$. Our object of interest is the polynomial version of Pell’s equation, where the integers are replaced by polynomials with complex coefficients. We then investigate the factors of $v_n(t)$. In particular, we show that over the complex polynomials, there are only finitely many values of n for which $v_n(t)$ has a repeated root. Restricting our analysis to $\mathbb {Q}[t]$, we give an upper bound on the number of “new” factors of $v_n(t)$ of degree at most $N$. Furthermore, we show that all “new” linear rational factors of $v_n(t)$ can be found when $n \leq 3$, and all “new” quadratic rational factors when $n \leq 6$.References
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Bibliographic Information
- Nikoleta Kalaydzhieva
- Affiliation: Department of Mathematics, University College London, 25 Gordon Street, London, United Kingdom, WC1H 0AY
- MR Author ID: 1462490
- ORCID: 0000-0002-9393-283X
- Email: n.kalaydzhieva@ucl.ac.uk
- Received by editor(s): February 9, 2021
- Received by editor(s) in revised form: December 1, 2021
- Published electronically: May 13, 2022
- Additional Notes: This research was supported by ERC grant no. 670239
- Communicated by: Matthew A. Papanikolas
- © Copyright 2022 by the authors
- Journal: Proc. Amer. Math. Soc. 150 (2022), 3771-3785
- MSC (2020): Primary 11C08, 13P05; Secondary 11R09
- DOI: https://doi.org/10.1090/proc/15994
- MathSciNet review: 4446228