UCL Discovery

## On problems related to multiple solutions of Pell's equation and continued fractions over function fields

Kalaydzhieva, Nikoleta Dianova; (2020) On problems related to multiple solutions of Pell's equation and continued fractions over function fields. Doctoral thesis (Ph.D), UCL (University College London).

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## Abstract

We study old problems, connected to the theory of continued fractions, with a new twist: changing the setting from the real numbers to the field of formal Laurent series in 1/t. In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers √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∈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 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≤ 3, and all new'' quadratic rational factors when n≤ 6. Another application of continued fractions arises from the theory of rational approximations to real irrational numbers. There, if we truncate the continued fraction expansion of $\alpha\in\Ree$, the resulting rational number best'' approximates it. This consequence remains true when we replace real numbers by formal Laurent series in $1/t$. In the framework of power series over the rational numbers, we define the Lagrange spectrum, related to Diophantine approximation of irrationals, and the Markov spectrum, related to elements represented by indefinite binary quadratic forms. We compute both spectra, by showing they equal sets whose elements are quantities attached to doubly infinite sequences of non-constant polynomials. Moreover, we prove that Lagrange and Markov spectra coincide and exhibit no gaps, contrary to what happens over the real numbers.