On Weierstrass semigroup at $m$ points on curves of the type f(y) = g(x)

TL;DR

This paper determines the minimal generating set of the Weierstrass semigroup at multiple points on certain algebraic curves with a specific plane model, using the concept of discrepancy to generalize previous methods.

Contribution

It introduces a new approach using discrepancy to compute Weierstrass semigroups at multiple points on curves of the form f(y)=g(x), broadening the scope beyond specific cases.

Findings

Explicit minimal generating sets for the semigroup at multiple points.

Application of discrepancy concept to a broader class of curves.

Generalization of previous methods to new curve types.

Abstract

In this work we determine the so-called minimal generating set of the Weierstrass semigroup of certain $m$ points on curves $\mathcal{X}$ with plane model of the type $f(y) = g(x)$ over $\mathbb{F}_{q}$, where $f(T),g(T)\in \mathbb{F}_q[T]$. Our results were obtained using the concept of discrepancy, for given points $P$ and $Q$ on $\mathcal{X}$. This concept was introduced by Duursma and Park, in \cite{duursma}, and allows us to make a different and more general approach than that used to certain specific curves studied earlier.

Equations

$$H(P_1,\dots ,P_m)=\{(a_1,\dots ,a_m)\in {\mathbb{N}}_0^m\text{mbox};\exists f\in {\mathbb{F}}_q[\mathcal{X}]\text{mbox}with(f{)}_{\infty }=\sum _{i=1}^m{a}_i{P}_i\}$$

$$lub\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_t\}=(\max \{u_{1_1},\dots ,u_{t_1}\},\dots ,\max \{u_{1_m},\dots ,u_{t_m}\})\in \text{mathds}{N}_0^m.$$

$${\nabla }_i(\mathbf{n}):=\{(p_1,\dots ,p_m)\in H(P_1,\dots ,P_m)\text{mbox};p_i=n_i\}.$$

$$\Gamma (P_1,\dots ,P_m):=\{\mathbf{n}\in \text{mathds}{N}^m:\text{mbox}forsomei,1\le i\le m,\mathbf{n}\text{mbox}isminimalin{\nabla }_i(\mathbf{n})\}.$$

$$\left\{\begin{array}[]{cl}lub\{{\bf u}_{1},\ldots,{\bf u}_{m}\}\in\mathds{N}_{0}^{m}:&{\bf u}_{i}\in\Gamma(P_{1},\ldots,P_{m})\\ &\mbox{ or }(u_{i_{1}},\ldots,u_{i_{k}})\in\Gamma(P_{i_{1}},\ldots,P_{i_{k}})\\ &\mbox{ for some }\{i_{1},\ldots,i_{k}\}\subset\{1,\ldots,m\}\mbox{ such that }\\ &i_{1}<\cdots<i_{k}\mbox{ and }u_{i_{k+1}}=\cdots=u_{i_{\ell}}=0,\\ &\mbox{ where }\{i_{k+1},\ldots,i_{m}\}\subset\{1,\ldots,\ell\}\setminus\{i_{1},\ldots,i_{k}\}\end{array}\right\}\;.$$

$$a{P}_1\sim P_2+\cdots +P_{a+1},$$

$$b{P}_i\sim b{P}_j\text{mbox},foralli,j\in \{1,2,\dots ,a+1\},$$

$$t+\sum _{j=2}^m{s}_j=a+1-m,$$

$$(tb-ia)P_1+\sum _{j=2}^m(s_{j}b+i)P_j\sim \sum _{j=m+1}^{a+1}(b-i)P_j.$$

$$0<ia<tb,s_j\ge 0.$$

$$A=(b-i)(a{P}_1-P_2-\cdots -P_m)\sim \sum _{j=m+1}^{a+1}(b-i)P_j.$$

$$A^{\prime }=((a+1-m)b-ia)P_1+i(P_2+\cdots +P_m)\sim \sum _{j=m+1}^{a+1}(b-i)P_j.$$

$$(tb-ia)P_1+(sb+i)P_2+i(P_3+\cdots +P_m)\sim \sum _{j=m+1}^{a+1}(b-i)P_j.$$

$$\begin{array}[]{cl}K+P+Q-A^{\prime}&=(ab-a-b-1)P_{1}+P+Q-(a+1-m)b-ia)P_{1}\\ &-i(P_{2}+\cdots+P_{m})\\ &=(i-1)a+(m-2)b-1)P_{1}+P+Q-i(P_{2}+\cdots+P_{m}).\end{array}$$

$$(f_{j})=\left\{\begin{array}[]{ll}-aP_{1}+(P_{2}+\cdots+P_{a+1}),&j=2\\ b(P_{j}-P_{1}),&j>2\end{array}\right.$$

$$S_m=\left\{(tb-ia,s_{2}b+i,\dots ,s_{m}b+i)\text{mbox};t+\sum _{j=2}^m{s}_j=a+1-m\text{mbox},0<ia<tb\text{mbox},s_j\ge 0\right\}.$$

$$y^q+y=x^{q^r+1},$$

$$q{P}_1\sim P_2+\cdots +P_{q+1},$$

$$(q^r+1)P_i\sim (q^r+1)P_j\text{mbox},foralli,j\in \{1,2,\dots ,q+1\},$$

$$\begin{array}[]{llll}(126,0,0,126)&+&i(-5,1,1,1)\;,&i=1,\ldots,25\;,\\ (126,0,126,0)&+&i(-5,1,1,1)\;,&i=1,\ldots,25\;,\\ (126,126,0,0)&+&i(-5,1,1,1)\;,&i=1,\ldots,25\;,\\ (252,0,0,0)&+&i(-5,1,1,1)\;,&i=1,\ldots,50\;.\end{array}$$

$$\begin{array}[]{lcl}\Gamma(P_{1},P_{2})&=&\{(23,1),(18,2),(13,3),(8,4),(3,5),(16,8),\\ &&(11,9),(6,10),(1,11),(9,15),(4,16),(2,22)\}\;.\\ &&\\ \Gamma(P_{1},P_{2},P_{3})&=&\{(2,8,8),(2,15,1),(2,0,15),(9,8,1),(9,1,8),\\ &&(4,9,2),(4,2,9),(16,1,1),(11,2,2),(6,3,3),(1,4,4)\}\;.\\ &&\\ \Gamma(P_{1},P_{2},P_{3},P_{4})&=&\{(2,8,1,1),(2,1,8,1),(2,1,1,8),(9,1,1,1),(4,2,2,2)\}\;.\\ &&\\ \Gamma(P_{1},P_{2},P_{3},P_{4},P_{5})&=&\{(2,1,1,1,1)\}\;.\end{array}$$

Full text

1. Introduction

Let $\mathcal{X}$ be a nonsingular, projective, geometrically irreducible curve of genus $g\geq 1$ defined over a finite field $\mathbb{F}_{q}$, let $\mathbb{F}_{q}[\mathcal{X}]$ be the field of rational functions and $Div(\mathcal{X})$ be the set of divisors on $\mathcal{X}$. For $f\in\mathbb{F}_{q}[\mathcal{X}]$, the divisor of $f$ will be denoted by $(f)$ and the divisor of poles of $f$ by $(f)_{\infty}$. As follows, we denote $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, where $\mathbb{N}$ is the set of positive integers. Let $P_{1},\ldots,P_{m}$ be distinct rational points on $\mathcal{X}$. The set

[TABLE]

is called the Weierstrass semigroup at the points $P_{1},\ldots,P_{m}$. An element in $\mathbb{N}_{0}^{m}\setminus H(P_{1},\ldots,P_{m})$ is called gap and the set $G(P_{1},\ldots,P_{m})=\mathbb{N}_{0}^{m}\setminus H(P_{1},\ldots,P_{m})$ is called gap set of $P_{1},\ldots,P_{m}$.

It is not difficult to see that the set $H(P_{1},\ldots,P_{m})$ is a semigroup. The case $m=1$ has been studied for decades with its relationship to coding theory, see e.g. [5], [6] and [17]. An important fact about this case is that the cardinality of $G(P_{1})$ is $g$. The case $m=2$ started to be studied by Kim, in [9], where several properties were presented. Others relevant papers in the case $m=2$ are [2], [7], [8] and [14]. For $m>2$, this semigroup has been determined for some specific curves as Hermitian and Norm-trace curves, see [12] and [13]. With increasing interest in this semigroup, many results have been produced with several applications, especially in coding theory. Examples of these applications can be found in [3], [5] and [11].

In this work we study the Weierstrass semigroup $H(P_{1},\ldots,P_{m})$ for points on curves $\mathcal{X}$ with plane model of the type $f(y)=g(x)$ over $\mathbb{F}_{q}$, where $f(T),g(T)\in\mathbb{F}_{q}[T]$. Our results were obtained using the concept of discrepancy, for given points $P$ and $Q$ on $\mathcal{X}$, see Definition 2.5. This concept was introduced by Duursma and Park in [4], and it was our main tool for obtain the set $\Gamma(P_{1},\ldots,P_{m})$, called minimal generating of $H(P_{1},\ldots,P_{m})$, see Theorem 3.3. We observe that this approach is different from that used by Matthews in [12] and Matthews and Peachey in [13].

This paper is organized as follows. Section 2 contains general results about Weierstrass semigroup and discrepancy. In Section 3, we determine the minimal generating for the Weierstrass semigroup $H(P_{1},\ldots,P_{m})$ for the curves $\mathcal{X}$ with plane model of the type $f(y)=g(x)$ cited above. Finally, in Section 4 we present examples for certain specific curves.

2. Weierstrass semigroup and Discrepancy

Let $\mathcal{X}$ be a non-singular, projective, irreducible, algebraic curve of genus $g\geq 1$ over a finite field $\mathbb{F}_{q}$.

Fix $m$ distinct rational points $P_{1},\ldots,P_{m}$ on $\mathcal{X}$. Define a partial order $\preceq$ on $\mathds{N}_{0}^{m}$ by $(n_{1},\ldots,n_{m})\preceq(p_{1},\ldots,p_{m})$ if and only if $n_{i}\leq p_{i}$ for all $i$, $1\leq i\leq m$.

For ${\bf u}_{1},\ldots,{\bf u}_{t}\in\mathds{N}_{0}^{m}$, where, for all $k$, ${\bf u}_{k}=(u_{k_{1}},\ldots,u_{k_{m}})$, we define the least upper bound ($lub$) of the vectors ${\bf u}_{1},\ldots,{\bf u}_{t}$ in the following way:

[TABLE]

For $\mathbf{n}=(n_{1},\ldots,n_{m})\in\mathds{N}_{0}^{m}$ and $i\in\{1,\ldots,m\}$, we set

[TABLE]

Proposition 2.1**.**

[[12], Proposition 3] Let $\mathbf{n}=(n_{1},\ldots,n_{m})\in\mathds{N}_{0}^{m}$. Then $\mathbf{n}$ is minimal, with respect to $\preceq$, in $\nabla_{i}(\mathbf{n})$ for some $i$, $1\leq i\leq m$, if and only if $\mathbf{n}$ is minimal in $\nabla_{i}(\mathbf{n})$ for all $i$, $1\leq i\leq m$.

Proposition 2.2**.**

[[12], Proposition 6] Suppose that $1\leq t\leq m\leq q$ and ${\bf u}_{1},\ldots,{\bf u}_{t}\in H(P_{1},\ldots,P_{m})$. Then $lub\{{\bf u}_{1},\ldots,{\bf u}_{t}\}\in H(P_{1},\ldots,P_{m})$.

Definition 2.3**.**

Let $\Gamma(P_{1})=H(P_{1})$ and, for $m\geq 2$, define

[TABLE]

Lemma 2.4**.**

[[12], Lemma 4] For $m\geq 2$, $\Gamma(P_{1},\ldots,P_{m})\subseteq G(P_{1})\times\cdots\times G(P_{m})\;.$

In [12], Theorem 7, it is shown that, if $2\leq m\leq q$, then $H(P_{1},\ldots,P_{m})=$

[TABLE]

Therefore, the Weierstrass semigroup $H(P_{1},\ldots,P_{m})$ is completely determined by $\Gamma(P_{1},\ldots,P_{m})$. In [12], Matthews called the set $\Gamma(P_{1},\ldots,P_{m})$ of minimal generating of $H(P_{1},\ldots,P_{m})$.

In [4, Section 5], Duursma and Park introduced the concept of discrepancy as follows.

Definition 2.5**.**

A divisor $A\in Div(\mathcal{X})$ is called a discrepancy for two rational points $P$ and $Q$ on $\mathcal{X}$ if $\mathcal{L}(A)\neq\mathcal{L}(A-P)=\mathcal{L}(A-P-Q)$ and $\mathcal{L}(A)\neq\mathcal{L}(A-Q)=\mathcal{L}(A-P-Q)$.

The next result relates the concept of discrepancy with the set $\Gamma(P_{1},\ldots,P_{m})$.

Lemma 2.6**.**

Let ${\bf n}=(n_{1},\ldots,n_{m})\in H(P_{1},\ldots,P_{m})$. Then ${\bf n}\in\Gamma(P_{1},\ldots,P_{m})$ if and only if the divisor $A=n_{1}P_{1}+\cdots+n_{m}P_{m}$ is a discrepancy with respect to $P$ and $Q$ for any two rational points $P,Q\in\{P_{1},\ldots,P_{m}\}$.

Proof.

Let ${\bf n}=(n_{1},\ldots,n_{m})\in\Gamma(P_{1},\ldots,P_{m})$, then there is a rational function $f\in\mathbb{F}_{q}(\mathcal{X})$ such that $(f)_{\infty}=A=n_{1}P_{1}+\cdots+n_{m}P_{m}$. Let $P_{i},P_{j}\in\{P_{1},\ldots,P_{m}\}$, with $i\neq j$. Suppose that $\mathcal{L}(A)=\mathcal{L}(A-P_{i})$. Then we have that $f\in\mathcal{L}(A-P_{i})$ and so $(f)+A-P_{i}\succeq 0$, contradiction because the pole divisor of $f$ is $A$. Therefore $\mathcal{L}(A)\neq\mathcal{L}(A-P_{i})$. Now, suppose that $\mathcal{L}(A-P_{i})\neq\mathcal{L}(A-P_{i}-P_{j})$. Then there is a rational function $g\in\mathcal{L}(A-P_{i})\setminus\mathcal{L}(A-P_{i}-P_{j})$. So $(g)+A-P_{i}\succeq 0$ and this implies that $(g)_{\infty}=s_{1}P_{1}+\ldots+s_{j-1}P_{j-1}+n_{j}P_{j}+s_{j+1}P_{j+1}+\ldots+s_{m}P_{m}$, with $s_{i}\leq n_{i}-1$ and $s_{k}\leq n_{k}$, for $k\in\{1,\ldots,m\}\setminus\{i,j\}$. It follows that $(s_{1},\ldots,s_{j-1},n_{j},s_{j+1},\ldots,s_{m})\in\triangledown_{j}({\bf n})$, this contradicts the fact that ${\bf n}\in\Gamma(P_{1},\ldots,P_{m})$ is minimal in $\triangledown_{j}({\bf n})$. Similar considerations apply when initializing with $P_{j}$. Therefore, $A=n_{1}P_{1}+\cdots+n_{m}P_{m}$ is a discrepancy with respect to $P$ and $Q$ for any two rational points $P,Q\in\{P_{1},\ldots,P_{m}\}$. Conversely, suppose that ${\bf n}=(n_{1},\ldots,n_{m})\in H(P_{1},\ldots,P_{m})$ and that the divisor $A=n_{1}P_{1}+\cdots+n_{m}P_{m}$ is a discrepancy with respect to any $P_{i},P_{j}\in\{P_{1},\ldots,P_{m}\}$, with $i\neq j$. Since ${\bf n}=(n_{1},\ldots,n_{m})\in H(P_{1},\ldots,P_{m})$, there is a rational function $f\in\mathbb{F}_{q}(\mathcal{X})$ such that $(f)_{\infty}=n_{1}P_{1}+\cdots+n_{m}P_{m}$. Suppose that ${\bf n}=(n_{1},\ldots,n_{m})$ is not minimal in $\triangledown_{i}({\bf n})$ for some $i\in\{1,\ldots,m\}$. Then, there exist a rational function $f_{i}\in\mathbb{F}_{q}(\mathcal{X})$ such that $(f)_{\infty}=s_{1}P_{1}+\cdots+s_{m}P_{m}$ with ${\bf s}=(s_{1},\ldots,s_{m})\in\triangledown_{i}({\bf n})$, ${\bf s}\neq{\bf n}$ and ${\bf s}\preceq{\bf n}$. Therefore, $s_{i}=n_{i}$ and $s_{j}<n_{j}$ for some $j\neq i$. It follows that $f_{i}\in\mathcal{L}(A-P_{j})$ and $f_{i}\not\in\mathcal{L}(A-P_{i})$, this contradicts the fact that $\mathcal{L}(A-P_{j})=\mathcal{L}(A-P_{i})$. ∎

3. Weierstrass semigroup $H(P_{1},\ldots,P_{m})$ for certain types of curves

Consider a curve $\mathcal{X}$ over $\mathbb{F}_{q}$ given by affine equation $f(y)=g(x)$ ,where $f(T),g(T)\in\mathbb{F}_{q}[T]$, $\deg(f(y))=a$ and $\deg(g(x))=b$, with $gdc(a,b)=1$. Suppose that $\mathcal{X}$ is absolutely irreducible with genus $g=(a-1)(b-1)/2$.

Let $P_{1},P_{2},\ldots,P_{a+1}$ be $a+1$ distinct rational points such that

[TABLE]

and

[TABLE]

where “$\sim$” represent the equivalence of divisors. Note that $H(P_{1})=\langle a,b\rangle$.

Examples of such curves can be found in [1], [10], [12] and [13].

Let $1\leq m\leq a+1\leq q$. For

[TABLE]

we have that the equivalences (3.1) and (3.2) yield

[TABLE]

Observe that all coefficients are positive for

[TABLE]

Note that $0<t\leq a$ and thus $0<i<b$.

From the basic equivalence (3.1) and (3.2) we have that

[TABLE]

By (3.2) the divisor on the left can be replaced with an efficient equivalent divisor of the follow form

[TABLE]

Thus, by redistributing $a+1-m=s+t$ over $P_{1}$ and $P_{2}$, the divisors in (3.4), with $s=\sum_{j=2}^{m}s_{j}$, contain the special representative

[TABLE]

The other divisors with same $t,s$ and $i$ are easy obtained from (3.8) by distributing $s$ in all possible ways over $P_{2},\ldots,P_{m+1}$ such that $s=\sum_{j=2}^{m}s_{j}$.

Proposition 3.1**.**

Let $b$ and $i$ be as above. Then, the divisor $(b-i)(aP_{1}-P_{2}-\cdots-P_{m})$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{1},\ldots,P_{m}\}$.

Proof.

It suffices to prove the claim for the equivalent but effective divisor $A^{\prime}=((a+1-m)b-ia)P_{1}+i(P_{2}+\cdots+P_{m}).$ The divisors $A$ and $A^{\prime}$ appear on the left side of the divisor equivalences (3.6) and (3.7), respectively. The equivalence of effective divisors in (3.7) gives a rational function $f\in L(A^{\prime})$ with pole divisor equal to $A^{\prime}$. Thus $L(A^{\prime})\neq L(A^{\prime}-P)$ for any point $P$. To prove that $L(A^{\prime}-P)=L(A^{\prime}-P-Q)$ we consider the equivalent statement $L(K+P+Q-A^{\prime})\neq L(K+P-A^{\prime})$. For the choice of canonical divisor $K=(ab-a-b-1)P_{1}$,

[TABLE]

Consider first the case $P_{1}\in\{P,Q\}$. Without loss of generality we may assume that $P=P_{1},Q=P_{2}$. Let $f_{2},\ldots,f_{m}$ be the functions with divisors

[TABLE]

Then $f_{2}^{i-1}f_{3}\cdots f_{m}\in L(K+P+Q-A^{\prime})\backslash L(K+Q-A^{\prime}).$ Thus $L(A^{\prime}-Q)=L(A^{\prime}-P-Q)$. From $L(A^{\prime})\neq L(A^{\prime}-Q)=L(A^{\prime}-P-Q)$ and $L(A^{\prime})\neq L(A^{\prime}-P)$ it follows that also $L(A^{\prime}-P)=L(A^{\prime}-P-Q).$ Consider next the case $P_{1}\not\in\{P,Q\}$, say $P=P_{2},Q=P_{3}.$ Thus, we have that $f_{2}^{i-1}f_{4}\cdots f_{m}\in L(K+P+Q-A^{\prime})\backslash L(K+Q-A^{\prime}).$ As before, it follows that $L(A^{\prime}-Q)=L(A^{\prime}-P-Q)$ and that $L(A^{\prime}-P)=L(A^{\prime}-P-Q).$ ∎

Corollary 3.2**.**

Let $a$, $b$, $t$, $i$, $s_{2},\ldots,s_{m}$ be as above. Then, the divisor $(tb-ia)P_{1}+\sum_{j=2}^{m}(s_{j}b+i)P_{j}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{1},\ldots,P_{m}\}$.

Proof.

Follows directly from the previous proposition and equations (3.4) and (3.8). ∎

Theorem 3.3**.**

Let $\mathcal{X}$ and $P_{1},P_{2},\ldots,P_{a+1}$ be as above. For $2\leq m\leq a+1$, let

[TABLE]

Then, $\Gamma(P_{1},\ldots,P_{m})=S_{m}$.

Proof.

By Corollary 3.2, the divisor $(tb-ia)P_{1}+\sum_{j=2}^{m}(s_{j}b+i)P_{j}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{1},\ldots,P_{m}\}$. So, by Lemma 2.6, follows that $S_{m}\subseteq\Gamma(P_{1},\ldots,P_{m})$.

Next, we show that $\Gamma(P_{1},\ldots,P_{m})\subseteq S_{m}$. Let $\mathbf{n}=(n_{1},\ldots,n_{m})\in\Gamma(P_{1},\ldots,P_{m})$. By Lemma 2.4, $\mathbf{n}=(n_{1},\ldots,n_{m})\in G(P_{1})\times G(P_{2})\times\cdots\times G(P_{m})$.

As $H(P_{1})=\langle a,b\rangle$, from Lemma 1 in [16] we have that $n_{1}=ab-i_{1}a-j_{1}b=(a-j_{1})b-i_{1}a$, for some $i_{1},j_{1}\in\mathbb{N}$. Let $\lambda=a-j_{1}$. Note that $0<j_{1}<a$ and $0<ai_{1}<b\lambda$.

By Equation (3.2), follows that $b\in H(P_{\ell})$, for $2\leq\ell\leq m$. So, we have that $n_{\ell}=s_{\ell}b+i_{\ell}$, where $0<i_{\ell}<b$ and $s_{\ell}\geq 0$.

Let $i=\min\{i_{\ell}:2\leq\ell\leq m\}$. By Equation (3.1), for each $\ell=2,\ldots,m$, there is a rational function $h_{\ell}$ such that $(h_{\ell})=bP_{1}-bP_{\ell}$. By Equation (3.1), there is a rational function $g$ such that $(g)=P_{2}+\cdots+P_{a+1}-aP_{1}$.

Let $\omega=g^{b-i}.\prod_{\ell=2}^{m}h_{\ell}^{s_{\ell}+1}$. Then, $(\omega)_{\infty}=((a-\sum_{\ell=2}^{m}(s_{\ell}+1))b-ia)P_{1}+(s_{2}b+i)P_{2}+\cdots+(s_{m}b+i)P_{m}$. Taking $t=a-\sum_{\ell=2}^{m}(s_{\ell}+1)$, by Corollary 3.2, $(\omega)_{\infty}$ is a discrepancy with respect to $P$ and $Q$ for any two distinct points $P,Q\in\{P_{1},\ldots,P_{m}\}$. So, by Lemma 2.6, $\mathbf{w}=(tb-ia,s_{2}b+i,\ldots,s_{m}b+i)\in\Gamma(P_{1},\ldots,P_{m})$. Now, we know that $i=i_{k}$, for some $k\in\{2,\ldots,m\}$. Then, $\mathbf{w}\in\nabla_{k}({\bf n})$ and by minimality of $\mathbf{w}$ and $\mathbf{n}$ follows that $\mathbf{w}=\mathbf{n}$ and $\Gamma(P_{1},\ldots,P_{m})\subseteq S_{m}$.

∎

4. Examples

Example 4.1**.**

Let $\mathcal{X}_{q^{2r}}$ be the curve defined over $\mathbb{F}_{q^{2r}}$ by the affine equation

[TABLE]

where $r$ is an odd positive integer and $q$ is a prime power. Note that $\mathcal{X}_{q^{2}}$ is just the Hermitian curve. The curve $\mathcal{X}_{q^{2r}}$ has genus $g=q^{r}(q-1)/2$, a single point at infinity, namely $P_{1}=(0:1:0)$, and others $q^{2r+1}$ rational points. It is important to observe that $\mathcal{X}_{q^{2r}}$ is a quotient of the Hermitian curve and thus is a maximal curve over $\mathbb{F}_{q^{2r}}$. The Weierstrass semigroup $H(P_{1},P_{2})$ was studied in [15] and more details about this curve can be found in [10].

Let $y_{1},\ldots,y_{q}$ be the solutions in $\mathbb{F}_{q^{2r}}$ to $y^{q}+y=0$. Let $P_{2}=(0,y_{1}),P_{3}=(0,y_{2}),\ldots,P_{q+1}=(0,y_{q})$. Since $(x)=P_{2}+\ldots+P_{q+1}-qP_{1}$ and $(y-y_{j})=(q^{r}+1)P_{j+1}-(q^{r}+1)P_{1}$, for all $j=1,\ldots,q$, we have that

[TABLE]

and

[TABLE]

For this curve, take $m=4$, $q=5$ and $r=3$ we have that $a=5$ and $b=126$. Then by Theorem (3.3) follows that $\Gamma(P_{1},P_{2},P_{3},P_{4})$ consists of the following 125 elements

[TABLE]

Example 4.2**.**

Let a Kummer extensions given by $y^{b}=g(x)=\prod\limits_{i=1}^{a}(x-\alpha_{i})$ where $g(x)$ is a separable polynomial over $\mathbb{F}_{q}$ of degree $a$ and $\gcd(a,b)=1$. We know that these curves has a single point $P_{1}$ at infinity and has genus $(b-1)(a-1)/2$, see [1]. Then we have the following divisors in $F(\mathcal{X})$:

1. (1)

$(x-\alpha_{i})=bP_{i}-bP_{1}$ for every $i$, $2\leq i\leq a+1$, 2. (2)

$(y)=P_{2}+\cdots+P_{a+1}-aP_{1}$,

For $a=5$ and $b=7$ we have that

[TABLE]

5. Acknowledgment

The authors would like to thank I. Duursma for very useful suggestions and comments that improved the results and the presentation of this work.