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- W4298201745 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 $(u_n,v_n)_{ninZee}$. 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 $Qee[t]$, 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 $nleq 3$, and all new quadratic rational factors when $nleq 6$." @default.
- W4298201745 created "2022-10-01" @default.
- W4298201745 creator A5012660242 @default.
- W4298201745 date "2021-01-28" @default.
- W4298201745 modified "2023-10-17" @default.
- W4298201745 title "Properties of solutions to Pell's equation over the polynomial ring" @default.
- W4298201745 doi "https://doi.org/10.48550/arxiv.2101.12180" @default.
- W4298201745 hasPublicationYear "2021" @default.
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