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\begin{document}
\title{The $C_{ab}$ Curve}
\thanks{The latest version of this note
can be downloaded from \texttt{http:\slash \slash
tsk-www.{\allowbreak}ss.{\allowbreak}titech.{\allowbreak}ac.{\allowbreak}jp\slash
\~{ }ryutaroh\slash cab.html}}
\author{Ryutaroh Matsumoto}
\address{Sakaniwa Lab., Dept. of Electrical and
Electronic Engineering,
Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8552 JAPAN}
\email{ryutaroh@ss.titech.ac.jp}
\urladdr{http://tsk-www.ss.titech.ac.jp/\~{ }ryutaroh/}
\subjclass{Primary 14H99; Secondary 14H05, 94B27, 11T71}
\date{December 17, 1998}
\begin{abstract}
We characterize the defining equation of
a plane algebraic curve with exactly one rational place $Q$
at infinity,
then give a basis of $L(mQ)$ with pairwise distinct pole orders
at $Q$.
The defining equation can be regarded as a generalization of
the Weierstrass form of a hyperelliptic curve.
\end{abstract}
\maketitle
In this informal note I give an English proof of the results
of the $C_{ab}$ curve
found by Miura \cite{miura92,miurathesis,miura98}.
Throughout in this note, $K$ denotes a perfect field,
and $a,b$ denote relatively prime positive integers.
For a place $Q$ of an algebraic function field, we define
\[
L(\infty Q) = \bigcup_{i=1}^\infty L(iQ),
\]
and $v_Q$ denotes the discrete valuation at $Q$.
We say a place $Q$ \emph{$K$-rational} if
the degree of $Q$ is one.

\begin{theorem}\cite[Theorem 5.17 and Lemma 5.30]{miurathesis},
\cite[Appendix B and Lemma, p.1416]{miura98}
Let $\bar{K}$ be the algebraic closure of $K$,
$\chi \subset \bar{K}^2$ be a possibly reducible affine algebraic
set defined over $K$, $x,y$ the coordinate of the affine plane
$\bar{K}^2$, and
$a,b$ relatively prime positive integers.
The following $2$ conditions are equivalent.
\begin{enumerate}
\item $\chi$ is an absolutely irreducible
algebraic curve with exactly one $K$-rational place $Q$ at infinity,
and the pole divisors of $x$ and $y$ are $a Q$ and $b Q$
respectively.
\item $\chi$ is defined by a bivariate polynomial of form
\begin{equation}
\alpha_{b,0} x^b + \alpha_{0,a} y^a +
\sum_{i a + j b < ab} \alpha_{i,j} x^i y^j, \label{eq:cab}
\end{equation}
where $\alpha_{i,j} \in K$ for all $i,j$ and
$\alpha_{b,0}, \alpha_{0,a}$ are nonzero.
\end{enumerate}
\end{theorem}

\begin{proof}
($1 \Rightarrow 2$) Let $X,Y$ be variables over $K$,
$F(X,Y) \in K[X,Y]$ the defining equation of $\chi$,
and $x, y \in K[X,Y] / F(X,Y)$ the elements represented by $X,Y$.

Consider the minimal polynomial $G(x,Y)$ of $y$ over the subfield
$K(x)$. Since $[K(x,y) : K(x)] = a$ \cite[Theorem I.4.11]{bn:stichtenoth},
the degree of $G(x,Y)$ is $a$. The integral closure of $K[x]$ in $K(x,y)$ is
$L(\infty Q)$ \cite[Theorem III.2.6]{bn:stichtenoth}, and $y \in L(\infty Q)$.
Thus $y$ is integral over $K[x]$, and $G(x,Y) \in K[x,Y]$.
Write $G(X,Y)$ as
\[
\sum_{X,Y} \beta_{i,j} X^i Y^j.
\]

If $v_Q(x^i y^j) = -ab$ and both $i$ and $j$ are nonnegative,
then $(i,j)$ is either $(b,0)$ or $(0,a)$.
By the strict triangle inequality of a discrete
valuation \cite[Lemma I.1.10]{bn:stichtenoth},
for every term $\beta_{i,j} x^i y^j$ in $G(x,y)$,
$v_Q(x^i y^j) \geq -ab$, and
the coefficient $\beta_{b,0}$ of the term $\beta_{b,0} x^b$ is
nonzero.
Therefore if $\beta_{i,j} x^i y^j$ is a term in $G(x,y)$,
then the exponent $(i,j)$ is either $(b,0)$ or $(0,a)$, or
$ai + bj < ab$.

Finally we have to show that $F(X,Y)$ is a constant multiple of $G(X,Y)$,
and it is enough to show that $G(X,Y)$ generates the kernel of the
canonical homomorphism $\varphi : K[X,Y] \rightarrow K[x,y]$.
If $H(X,Y) \in \ker \varphi \setminus \{ 0 \}$,
then the degree of $H(X,Y)$ in $Y$
is at least $a$, because $G(x,Y)$ is the minimal polynomial of $y$
over $K(x)$. Thus $\{ G(X,Y) \}$ is a Gr\"{o}bner basis of
$\ker \varphi$ with respect to the lexicographic monomial order $Y > X$
\cite[Definition 5, Section 2.5]{bn:cox}, and
a Gr\"{o}bner basis generates the ideal
\cite[Corollary 6, Section 2.5]{bn:cox}.

($2 \Rightarrow 1$)
Let $F(X,Y)$ be the polynomial of (\ref{eq:cab}),
and $x,y$ the elements in $K[X,Y]/F(X,Y)$ represented by $X,Y$.
By the theory of Gr\"{o}bner bases
\cite[Proposition 4, Section 5.3]{bn:cox},
we see that
\begin{equation}
\{ x^i y^j \mid 0 \leq i, \; 0 \leq j \leq a-1 \} \label{eq:basis}
\end{equation}
is a basis of $K[x,y]$ as a $K$-linear space,
where $\{ F(X,Y) \}$ is viewed as a Gr\"{o}bner basis
with respect to the lexicographic monomial order $Y>X$.

Any element $f$ in $K[x,y]$ can be written uniquely as a polynomial
\begin{equation}
\sum_{i,j} \beta_{i,j} x^i y^j, \label{eq:remainder}
\end{equation}
where each monomial $x^i y^j$ belongs to the basis (\ref{eq:basis}) and
$\beta_{i,j} \in K$.
We define a function $o$ from $K[x,y]$ to $\mathbb{Z} \cup \{ - \infty \}$
to be
\begin{eqnarray*}
o(0) & = & - \infty, \\
o(f) & = & \max \{ a k + b l \mid x^k y^l \mbox{ is a monomial of $f$
written as (\ref{eq:remainder}).} \},
\end{eqnarray*}
where $f$ is nonzero. Then for $f,g \in K[x,y]$, $o(f) = -\infty$
iff $f = 0$ and $o(fg) = o(f) + o(g)$, where the sum of $- \infty$
and an integer is $- \infty$.

Now we can prove the absolute irreducibility of the polynomial
(\ref{eq:cab}). The following discussion is based on
\cite[Proposition 12]{hoholdt97}.
Suppose that $fg = 0$ for $f, g \in K[x,y]$. Then $o(fg) = - \infty$,
which implies $o(f) = - \infty$ or $o(g) = -\infty$. Thus
$K[x,y]$ is an integral domain. This argument is valid
if $K$ is replaced by its algebraic closure. So the polynomial
(\ref{eq:cab}) is absolutely irreducible.

Next we will show that $\chi$ has exactly one place at infinity.
Note that $K(x,y)/K$ is an algebraic function field with
the full constant field $K$.
We define a function $v$ from $K(x,y)$ to $\mathbb{Z} \cup \{
\infty \}$ such that for nonzero $f/g \in K(x,y)$,
$v(f/g) = o(g) - o(f)$ and $v(0) = \infty$.
Then $v$ satisfies the axiom of the discrete valuation
\cite[Definition I.1.9]{bn:stichtenoth}.
Let $Q$ be the place of $K(x,y)/K$ corresponding to $v$,
and $P_\infty$ be the pole of $x$ in the rational function field $K(x)$.
Then $Q$ lies over $P_\infty$, and the ramification index of $Q$
over $P_\infty$ is $a$, which equals to the extension degree
$[K(x,y) : K(x)]$. Thus $P_\infty$ is totally ramified,
$Q$ is $K$-rational,
and
the pole divisor of $x$ in $K(x,y)$ is $a Q$.
A similar argument shows that the pole divisor of $y$ is $b Q$.
Since $K[x,y] \subseteq L(\infty Q)$,
the set of places at infinity is $\{ Q \}$.
\end{proof}

\begin{definition}
A plane curve defined by a polynomial of the form (\ref{eq:cab})
is said to be a \emph{$C_{ab}$ curve}.
\end{definition}

\begin{corollary}\cite[Theorem 5.17]{miurathesis},
\cite[Appendix B]{miura98}
Let $F(X,Y) \in K[X,Y]$ be a polynomial of the form (\ref{eq:cab}),
$Q$ a unique place at infinity of the $C_{ab}$ curve defined by
$F(X,Y)$. Then
\[
\{ X^i Y^j \bmod F(X,Y) \mid 0 \leq i, \; 0 \leq j \leq a-1 \}
\]
is a $K$-basis of $K[X,Y]/F(X,Y)$ and elements in the basis
have pairwise distinct discrete valuations at $Q$.
If the $C_{ab}$ curve is nonsingular,
then $K[X,Y]/F(X,Y) = L(\infty Q)$ and a basis of $L(mQ)$ is
\[
\{ X^i Y^j \bmod F(X,Y) \mid 0 \leq i, \; 0 \leq j \leq a-1,\;
a i + b j \leq m \},
\]
for a nonnegative integer $m$.
\end{corollary}
The previous corollary overlaps with
\cite[Proposition 13 and 14]{saints95}.

\begin{corollary}\cite{miurathesis,miura98}
Let $F/K$ is an algebraic function field with a $K$-rational place $Q$.
Then there exists a $C_{ab}$ curve defined over $K$ with the function
field $F$.
\end{corollary}
\begin{proof}
There exist elements $x,y \in F$ such that
pole divisors of $x$ and $y$ are $a Q$ and $b Q$ respectively,
and $a,b$ are relatively prime positive integers.
We claim that $F = K(x,y)$.
$[F : K(x)] = a$ and $[F : K(y)] = b$ by
\cite[Theorem I.4.11]{bn:stichtenoth},
and $[F : K(x,y)]$ divides both $[F : K(x)]$ and $[F : K(y)]$.
Thus $[F : K(x,y)] = 1$.

Consider the ring homomorphism $\varphi$:
\begin{eqnarray*}
K[X,Y] & \longrightarrow & K[x,y],\\
f(X,Y) & \longmapsto & f(x,y),
\end{eqnarray*}
where $X,Y$ are variable over $K$.
Then the plane curve defined by $\ker \varphi$ is a $C_{ab}$ curve
by Theorem 1.
\end{proof}

\begin{histnote}
The results in this note are generalizations of \cite{miura92},
and were first officially published in \cite{miurathesis}.
The results in \cite{miura98} is a subset of \cite{miurathesis},
and also contain the results in this note. In \cite{miura92}
Miura proved the following fact.
\begin{quote}
Let $\chi$ be a nonsingular
affine algebraic curve defined by a bivariate
polynomial of the form (\ref{eq:cab}).
Then $\chi$ has exactly one rational place $Q$ at infinity,
the pole divisors of $x$ and $y$ are $aQ$ and $bQ$ respectively,
and a basis of $L(mQ)$ is $\{ x^i y^j \mid 0 \leq i,\;
0 \leq j \leq a-1, \; ai + bj \leq m \}$ for a nonnegative integer $m$.
\end{quote}
In \cite{miura92} it is not clear whether the affine algebraic set
defined by the polynomial (\ref{eq:cab}) is always irreducible.
\end{histnote}

\begin{histnote}
Miura told the author that he learned the proof of
the absolute irreducibility of a polynomial
of the form (\ref{eq:cab}) from
the preprint version of Pellikaan's paper \cite{pellikaan98}.
\end{histnote}

\begin{histnote}
A subclass of $C_{ab}$ curves was treated in
\cite[pp.1007--1009]{feng94}, and called "type I of plane affine
curves."
\end{histnote}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{1}
\bibitem{bn:cox}
David Cox, John Little, and Donal O'Shea, \emph{Ideals, varieties, and
  algorithms}, second ed., Springer-Verlag, Berlin, 1996.

\bibitem{feng94}
Gui-Liang Feng and T. R. N. Rao,
\emph{A simple approach for construction of algebraic-geometric
codes from affine plane curves},
IEEE Trans. Inform. Theory \textbf{40} (1994), no.~4, 1003--1012.

\bibitem{hoholdt97}
Tom H{\o}holdt, Jacobus~H. van Lint, and Ruud Pellikaan, \emph{Order functions
  and evaluation codes}, Proc. AAECC-12, Lecture Notes in Computer Science,
  vol. 1255, Springer-Verlag, 1997, pp.~138--150.

\bibitem{miura92}
Shinji Miura, \emph{Algebraic geometric codes on certain plane curves}, Trans.
  IEICE \textbf{J75-A} (1992), no.~11, 1735--1745 (Japanese).

\bibitem{miurathesis}
\bysame, Ph.D. thesis, Univ. Tokyo, 1997 (Japanese).

\bibitem{miura98}
\bysame, \emph{Linear codes on affine algebraic curves}, Trans. IEICE
  \textbf{J81-A} (1998), no.~10, 1398--1421 (Japanese).

\bibitem{pellikaan98}
Ruud Pellikaan, \emph{On the existence of order functions},
to appear in J. Statistical Planning and
Inference (1998).

\bibitem{saints95}
Keith Saints and Chris Heegard,
\emph{Algebraic-geometric codes and multidimensional cyclic codes:
A unified theory and algorithms for decoding using {G}r\"{o}bner bases},
IEEE Trans. Inform. Theory \textbf{41} (1995), no.~6, 1733--1751.

\bibitem{bn:stichtenoth}
Henning Stichtenoth, \emph{Algebraic function fields and codes},
  Springer-Verlag, Berlin, 1993.
\end{thebibliography}
\end{document}