Definition:Continued Fraction Expansion of Laurent Series

Definition

Let $k$ be a field.

Let $\map k {\paren {t^{-1} } }$ be the field of formal Laurent series in the variable $t^{-1}$.

Irrational Laurent series

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

The continued fraction expansion of $f$ is the infinite continued fraction $\sequence {\floor {\alpha_n} }_{n \mathop \ge 0}$ where $\alpha_n$ is recursively defined as:

$\alpha_n = \ds \begin {cases} f & : n = 0 \\ \dfrac 1 {f - \floor f} & : n \ge 1 \end {cases}$

where $\floor \cdot$ denotes the polynomial part.

Rational Laurent series

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