AFFINE PLANE CURVES WITH ONE PLACE AT INFINITY

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This implies that ${\mathit{g}}_{\mathit{k}}$ has the pole of order ${\overline{\mathit{\delta }}}_{\mathit{k}}$ on ${\mathit{E}}_{\mathit{\sigma }}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$. On the other hand,

by Lemma 1, ${\mathit{g}}_{\mathit{k}\mathrm{+}\mathrm{1}}$ has the pole of order ${\mathit{q}}_{\mathit{k}}{\overline{\mathit{\delta }}}_{\mathit{k}}$ on ${\mathit{E}}_{\mathit{\sigma }}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$. Hence, ${\mathit{E}}_{\mathit{\sigma }}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$ is

neither the zero nor the pole of $\mathrm{\Phi }\mathrm{=}\frac{{\mathit{g}}_{\mathit{k}\mathrm{+}\mathrm{1}}}{{\mathit{g}}_{\mathit{k}}^{{\mathit{q}}_{\mathit{k}}}}$. Further, (I) is holomorphic in a

neighborhood of $\mathit{Q}$ and $\mathrm{\Phi }\mathrm{\left(}\mathit{Q}\mathrm{\right)}\mathrm{=}\mathrm{0}$. Therefore, (I is not constant on ${\mathit{E}}_{\mathit{\sigma }}^{\mathrm{(}}$').

Now, set $\mathit{\psi }\mathrm{=}{\mathit{g}}_{\mathit{k}\mathrm{+}\mathrm{1}}\mathrm{-}{\mathit{g}}_{\mathit{k}}^{\mathit{q}\mathit{k}}$ . Then,

$\frac{\mathit{\psi }}{{\mathit{g}}_{\mathit{k}}^{\mathit{q}\mathit{k}}}\mathrm{=}\mathrm{\Phi }\mathrm{-}\mathrm{1}$

is also a non-constant function on ${\mathit{E}}_{\mathit{\sigma }}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$. Therefore, $\mathit{\psi }$ has also the pole of

order ${\mathit{q}}_{\mathit{k}}{\overline{\mathit{\delta }}}_{\mathit{k}}$ on ${\mathit{E}}_{\mathit{\sigma }}^{\mathrm{\left(}\mathit{\sigma }\mathrm{\right)}}$. On the other hand, since

${\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}\mathit{\psi }\mathrm{\right)}\mathrm{<}{\mathit{n}}_{\mathit{k}\mathrm{+}\mathrm{1}}\mathrm{=}{\mathit{n}}_{\mathit{k}}{\mathit{q}}_{\mathit{k}}$, ${\mathit{n}}_{\mathit{k}}\mathrm{=}{\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{g}}_{\mathit{k}}\mathrm{\right)}$,

by the division of $\mathit{\psi }$ by ${\mathit{g}}_{\mathit{k}}^{{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathrm{1}}$, we get

$\mathit{\psi }\mathrm{=}{\mathit{c}}_{\mathrm{1}}{\mathit{g}}_{\mathit{k}}^{{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathit{\psi }}_{\mathrm{1}}$

with ${\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{c}}_{\mathrm{1}}\mathrm{\right)}\mathrm{<}{\mathit{n}}_{\mathit{k}}\mathrm{,}$ ${\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{\psi }}_{\mathrm{1}}\mathrm{\right)}\mathrm{<}{\mathit{n}}_{\mathit{k}}\mathrm{(}{\mathit{q}}_{\mathit{k}}$ - 1 $\mathrm{)}$. Dividing ${\mathit{\psi }}_{\mathit{i}\mathrm{-}\mathrm{1}}$ by ${\mathit{g}}_{\mathit{k}}^{{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathit{\iota }}$

successively for $\mathrm{i}\mathrm{=}\mathrm{2}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}$ ${\mathit{q}}_{\mathit{k}}\mathrm{-}\mathrm{1}$, we get

${\mathit{\psi }}_{\mathit{i}\mathrm{-}\mathrm{1}}\mathrm{=}{\mathit{c}}_{\mathit{i}}{\mathit{g}}_{\mathit{k}}^{{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathit{i}}\mathrm{+}{\mathit{\psi }}_{\mathit{i}}$,

where ${\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{c}}_{\mathrm{1}}\mathrm{\right)}\mathrm{<}{\mathit{n}}_{\mathit{k}}\mathrm{,}$ ${\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{\psi }}_{\mathit{i}}\mathrm{\right)}\mathrm{<}{\mathit{n}}_{\mathit{k}}\mathrm{(}{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathrm{i}\mathrm{)}$. Thus, setting ${\mathit{c}}_{{\mathit{q}}_{\mathit{k}}}\mathrm{=}{\mathit{\psi }}_{{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathrm{1}}$, we

get

$\mathit{\psi }\mathrm{=}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{1}}^{{\mathit{q}}_{\mathit{k}}}{\mathit{c}}_{\mathit{i}}{\mathit{g}}_{\mathit{k}}^{{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathit{i}}$.

Here, we have

${\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{c}}_{\mathit{i}}\mathrm{\right)}\mathrm{<}{\mathit{n}}_{\mathit{k}}\mathrm{=}{\mathit{n}}_{\mathit{k}\mathrm{-}\mathrm{1}}{\mathit{q}}_{\mathit{k}\mathrm{-}\mathrm{1}}$, ${\mathit{n}}_{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{=}{\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{g}}_{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{\right)}$.

In the same way, dividing ${\mathit{c}}_{\mathit{i}}$ and its rests by ${\mathit{g}}_{\mathit{k}\mathrm{-}\mathrm{1}}^{{\mathit{q}}_{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{-}\mathrm{1}}\mathrm{,}$ ${\mathit{g}}_{\mathit{k}\mathrm{-}\mathrm{1}}^{{\mathit{q}}_{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{-}\mathrm{2}}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}$ ${\mathit{g}}_{\mathit{k}\mathrm{-}\mathrm{1}}$

successively, we get

${\mathit{c}}_{\mathit{i}}\mathrm{=}{\displaystyle \sum }_{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathit{q}\mathit{k}\mathrm{-}\mathrm{1}}{\mathit{c}}_{\mathit{i}\mathit{j}}{\mathit{g}}_{\mathit{k}\mathrm{-}\mathrm{1}}^{{\mathit{q}}_{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{-}\mathit{j}}$

with ${\mathrm{deg}}_{\mathit{y}}\mathrm{\left(}{\mathit{c}}_{\mathit{i}\mathit{j}}\mathrm{\right)}\mathrm{<}{\mathit{n}}_{\mathit{k}\mathrm{-}\mathrm{1}}$. Thus, we have

$\mathit{\psi }\mathrm{=}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{q}\mathit{k}}{\displaystyle \sum }_{\mathit{j}\mathrm{=}\mathrm{1}}^{\mathit{q}\mathit{k}\mathrm{-}\mathrm{1}}{\mathit{c}}_{\mathit{i}\mathit{j}}{\mathit{g}}_{\mathit{k}}^{{\mathit{q}}_{\mathit{k}}\mathrm{-}\mathit{i}}{\mathit{g}}_{\mathit{k}\mathrm{-}\mathrm{1}}^{{\mathit{q}}_{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{-}\mathit{j}}$.