Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Continued fractions in function fields: polynomial analogues of
  McMullen's and Zaremba's conjectures   We examine the polynomial analogues of McMullen's and Zaremba's conjectures
on continued fractions with bounded partial quotients. It has already been
proved by Blackburn that if the base field is infinite, then the polynomial
analogue of Zaremba's conjecture holds; we will prove this again with a
different method and examine some known results for finite base fields.
Translating to the polynomial setting a result of Mercat, we will prove that
the polynomial analogue of McMullen's conjecture holds over infinite algebraic
extensions of finite fields and that, over finite fields, it would be a
consequence of the polynomial analogue of Zaremba's conjecture. We will then
prove that the polynomial analogue of McMullen's conjecture holds over
uncountable base fields, over $\overline{\mathbb Q}$ (thanks to the theory of
reduction of a formal Laurent series modulo a prime) and over number fields.
For this purpose, we will examine the connection between the continued
fractions of polynomial multiples of $\sqrt D$ and pullbacks of generalized
Jacobians of the hyperelliptic curve $U^2=D(T)$.

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The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: stat
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The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int