Definition:Continued Fraction Expansion of Laurent Series

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Definition

Let $k$ be a field.

Let $k((t^{-1}))$ be the field of formal Laurent series in the variable $t^{-1}$.


Irrational Laurent series

Let $f \in k((t^{-1}))$ be an irrational formal Laurent series.


The continued fraction expansion of $f$ is the infinite continued fraction $(\lfloor \alpha_n \rfloor)_{n\geq0}$ where $\alpha_n$ is recursively defined as:

$\alpha_n = \displaystyle \begin{cases} f & : n = 0 \\ \dfrac 1 {f - \lfloor f \rfloor} & : n \geq 1 \end{cases}$

where $\lfloor \cdot \rfloor$ denotes the polynomial part.


Rational Laurent series