# Generalized Weierstrass semigroups at several points on certain maximal
curves which cannot be covered by the Hermitian curve

M. Montanucci and G. Tizziotti

###### Abstract.

In this paper we determine the generalized Weierstrass semigroup
$\widehat{H}(P_{\infty},P_{1},\ldots,P_{m})$, and consequently the Weierstrass
semigroup $H(P_{\infty},P_{1},\ldots,P_{m})$, at $m+1$ points on the curves
$\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$. These curves has been
introduced in Tafazolian et al. [29] as new examples of curves which cannot be
covered by the Hermitian curve.

###### Key words and phrases:

AG codes, maximal curves, Generalized Weierstrass semigroup

###### 2010 Mathematics Subject Classification:

14Q05, 94B05

##  1\. Introduction

Let $\mathcal{X}$ be a nonsingular, projective, geometrically irreducible
algebraic curve of positive genus $g$ defined over a finite field
$\mathbb{F}_{q}$ with $q$ elements and let $\mathcal{X}(\mathbb{F}_{q})$ be
the set of its $\mathbb{F}_{q}$-rational points. The curve $\mathcal{X}$ is
called $\mathbb{F}_{q}$-maximal if its number of $\mathbb{F}_{q}$-rational
point attains the Hasse-Weil upper bound, namely equals $2g\sqrt{q}+q+1$.
Clearly, maximal curves can only exist over fields whose cardinality is a
perfect square. Apart for being of theoretical interest as extremal objects,
maximal curves over finite fields has attracted a lot of attention in recent
decades due to their applications to coding theory and cryptography. Maximal
curves are indeed special for the structure of the so-called Weiestrass
semigroup at one point, which is the main ingredient used in the literature to
construct AG codes with good parameters. In [10], Delgado proposed an
interesting generalization of Weierstrass semigroups, where instead of one
single point several points of the curve are considered simultaneously. His
original approach was on curves over algebraically closed fields, but later
Beelen and Tutas [4] studied such object, called generalized Weierstrass
semigroup, for curves defined over finite fields. Moyano-Fernández, Tenório
and Torres [27] presented importante properties of generalized Weierstrass
semigroup at several points on a curve, which suggested again the use of
maximal curves to obtain good AG codes. In [10] Delgado introduced also the
notion of maximality to generalized Weierstrass semigroups (see Definition
2.5). Interesting connections between the concepts of maximal elements and
generating sets for Weiestrass semigroups was presented in [27], and in [31],
the authors used the notion of maximality in generalized Weierstrass semigroup
to give a characterization of gaps and pure gaps of Weierstrass semigroups.

The most important and well-studied example of a maximal curve is the so-
called Hermitian curve $\mathcal{H}_{q}$ defined over $\mathbb{F}_{q^{2}}$ by
the affine equation $y^{q}+y=x^{q+1}$. A well-known reason is that for fixed
$q$, the curve $\mathcal{H}_{q}$ has the largest possible genus
$g(\mathcal{H}_{q})=q(q-1)/2$ that an $\mathbb{F}_{q^{2}}$-maximal curve can
have. A result commonly attributed to Serre, see [22, Proposition 6], gives
that any $\mathbb{F}_{q^{2}}$-rational curve which is covered by an
$\mathbb{F}_{q^{2}}$-maximal curve is itself also
$\mathbb{F}_{q^{2}}$-maximal. Therefore many maximal curves can be obtained by
constructing subcovers of already known maximal curves, in particular
subcovers of the Hermitian curve. For a while it was speculated in the
research community that perhaps all maximal curves could be obtained as
subcovers of the Hermitian curve, but it was shown by Giulietti and Korchmáros
that this is not the case, see [18]. Giulietti and Korchmáros constructed
indeed a maximal curve over $\mathbb{F}_{q^{6}}$, nowadays referred to as GK
curve, which cannot be covered by the Hermitian curve whenever $q>2$. Garcia,
Güneri, and Stichtenoth, in [16], presented a new family of maximal curves
over $\mathbb{F}_{q^{2n}}$ ($n$ odd), known as GGS curves, which generalizes
the GK curve and that are not Galois-covered by the Hermitian curve [19, 11].
Many applications of these curves in coding theory have been made in recent
years, see e.g. [2], [3], [9], [13], [20] and [32]. Another generalization of
the GK curve over $\mathbb{F}_{q^{2n}}$ ($n$ odd) has been introduced by
Beelen and Montanucci in [5], which is now known as BM curves. These curves
are not Galois-covered by the Hermitian curve as well. Applications of of the
BM curves to coding theory can be found in [23] and [26].

Tafazolian, Teherán-Herrera, and Torres [29] presented two further examples of
maximal curves, denoted by $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$,
that cannot be covered by the Hermitian curve. These examples are again
closely related to the GK curve. They are not generalizations as the GGS and
BM curves, but instead subcovers of the GK curve. The curves
$\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ can be considered as concrete
models of the quotient curves computed by Fanali and Giulietti in [14, Theorem
4.5] while for $s=1$ the curve $\mathcal{Y}_{n,1}$ is the GGS curve mentioned
above. Recently, Brás-Amorós and Castellanos [6] determined the Weierstrass
semigroup at certain $m+1$ points on the curves $\mathcal{X}_{a,b,n,1}$ and
$\mathcal{Y}_{n,1}$ (case $s=1$) and present some conditions to find certain
pure gaps for certain specific semigroups, yielding to some AG codes with good
parameters. It is so natural to ask: what happens if $s>1$? In this paper we
determine the generalized Weierstrass semigroup
$\widehat{H}(P_{\infty},P_{1},\ldots,P_{m})$, and consequently the Weierstrass
semigroup $H(P_{\infty},P_{1},\ldots,P_{m})$, at $m+1$ points on the curves
$\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ for arbitrary $s\geq 1$,
generalizing the results in [6].

The paper is organized as follows. In Section 2 we fix the notation that will
be used throughout the paper and recall some basic facts about generalized
Weierstrass semigroups and their maximal elements. In the same section the
curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$, together with their
most important properties, are also presented. Section 3 is devoted to
determine the generalized Weierstrass semigroup is at several points on the
maximal curves $\mathcal{X}_{a,b,n,s}$, while in Section 4 we present similar
results for the curve $\mathcal{Y}_{n,s}$.

##  2\. Preliminaries

Let $\mathbb{F}_{q}$ denote a finite field with $q$ elements. Let
$\mathcal{X}$ be a nonsingular, projective, geometrically irreducible
algebraic curve of positive genus $g$ defined over $\mathbb{F}_{q}$ and let
$\mathbf{Q}=(Q_{1},\ldots,Q_{m})$ an $m$-tuple of pairwise distinct rational
points on $\mathcal{X}$. Throughout the paper the following notation will be
used.

$\bullet$ $\mathbb{F}_{q}(\mathcal{X})$: function field associated to
$\mathcal{X}$.

$\bullet$ $\mbox{Div}(\mathcal{X})$: the set of divisors on $\mathcal{X}$.

$\bullet$ $(f)$, $(f)_{\infty}$ and $(f)_{0}$: principal divisor, divisor of
poles and divisor of zeros of $f\in\mathbb{F}_{q}(\mathcal{X})^{\times}$,
respectively.

$\bullet$ $\mathcal{L}(G)$: the Riemann-Roch space associated to
$G\in\mbox{Div}(\mathcal{X})$, that is, the $\mathbb{F}_{q}$-vector space
$\\{h\in\mathbb{F}_{q}(\mathcal{X})^{\times}\mbox{ : }(h)+G\geq
0\\}\cup\\{0\\}$.

$\bullet$ $\ell(G)$: the dimension of $\mathcal{L}(G)$.

$\bullet$ $R_{\mathbf{Q}}$: the ring of functions in
$\mathbb{F}_{q}(\mathcal{X})^{\times}$ having poles only on the set
$\\{Q_{1},\ldots,Q_{m}\\}$, that is, that are regular outside
$\\{Q_{1},\ldots,Q_{m}\\}$.

$\bullet$ $\mathbb{N}_{0}$: the set of nonnegative integers.

$\bullet$ For a fixed
$\text{\boldmath$\alpha$}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{Z}^{m}$,
$D_{\text{\boldmath$\alpha$}}$ denotes the divisor
$\alpha_{1}Q_{1}+\ldots\alpha_{m}Q_{m}$.

$\bullet$ $v_{P}$: the discrete valuation in the function field
$\mathbb{F}_{q}(\mathcal{X})$ at the point $P\in\mathcal{X}$.

###  2.1. The curves $\mathcal{X}_{a,b,n,s}$

Let $q=p^{a}$ a power of a prime number $p$ and let $n,b,s\geq 1$ be integers
satisfying the following:

$\bullet$ $n$ is odd;

$\bullet$ $b$ is a divisor of $a$;

$\bullet$ $s$ is a divisor of $\dfrac{q^{n}+1}{q+1}$.

For $c\in\mathbb{F}_{q^{2}}$ with $c^{q-1}=-1$, we define the curve
$\mathcal{X}_{a,b,n,s}$ over $\mathbb{F}_{q^{2n}}$ by the affine equations

(1) $\displaystyle cy^{q+1}=t(x):=\sum_{i=0}^{a/b-1}x^{p^{ib}}\mbox{ and
}y^{q^{2}}-y=z^{M},$

where $M=\dfrac{q^{n}+1}{s(q+1)}$. The curve $\mathcal{X}_{a,b,n,s}$ is
$\mathbb{F}_{q^{2n}}$-maximal and it has genus

$g(\mathcal{X}_{a,b,n,s})=\dfrac{q^{n+2}-p^{b}q^{n}-sq^{3}+q^{2}+(s-1)p^{b}}{2sp^{b}}.$

Furthermore, $\mathcal{X}_{a,b,n,s}$ is a subcover of the GGS curve, see [29].

We will denote an affine point
$P=(\alpha,\beta,\gamma)\in\mathcal{X}_{a,b,n,s}(\mathbb{F}_{q^{2n}})$ by
$P_{(\alpha,\beta,\gamma)}$. The unique common pole of the functions $x,y$ and
$z$ will be denote by $P_{\infty}$. Thus, we have the following divisors:

(2) $(x-\alpha)=(q+1)MP_{(\alpha,0,0)}-(q+1)MP_{\infty}\mbox{ };$

(3)
$\displaystyle(y-\beta)=\sum_{i=1}^{q/p^{b}}MP_{(\alpha_{i},\beta,0)}-\dfrac{q}{p^{b}}MP_{\infty},\mbox{
with }t(\alpha_{i})=\beta^{q+1}\mbox{ and }\beta\in\mathbb{F}_{q^{2}}\mbox{
};$

(4)
$\displaystyle(z)=\sum_{j=1}^{q^{2}}\sum_{i=1}^{q/p^{b}}P_{(\alpha_{i},\beta_{j},0)}-\dfrac{q^{3}}{p^{b}}P_{\infty},\mbox{
with }\beta_{j}\in\mathbb{F}_{q^{2}}\mbox{ and
}c\beta_{j}^{q+1}=t(\alpha_{i}),\mbox{ for all }i,j.$

From [29, Proposition 5.1] we have that
$H(P_{\infty})=\langle\frac{q}{p^{b}}M,\frac{q^{3}}{p^{b}},(q+1)M\rangle$,
which is a telescopic semigroup. More details about the curve
$\mathcal{X}_{a,b,n,s}$ can be found in [29].

######  Remark 2.1.

Since telescopic semigroups are symmetric (see [21, Lemma 6.5]), the Frobenius
number of $H(P_{\infty})$ is
$\ell_{g}=2g-1=\dfrac{q^{n+2}-p^{b}q^{n}-sq^{3}+q^{2}+(s-1)p^{b}}{sp^{b}}-1=\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-q^{3}]$.

######  Proposition 2.2.

[28, Proposition 1.6.2] A divisor $K$ is canonical if and only if
$\deg(K)=2g-2$ and $\ell(K)\geq g$.

######  Lemma 2.3.

The divisor

$K=\left(\dfrac{q^{n+2}-p^{b}q^{n}-sq^{3}+q^{2}+(s-1)p^{b}}{sp^{b}}-2\right)P_{\infty}=\left(\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-q^{3}]-1\right)P_{\infty}.$

is a canonical divisor of $\mathcal{X}_{a,b,n,s}$.

###### Proof.

First, note that $\deg(K)=2g-2$. Now, by Remark 2.1 the Frobenius number of
$H(P_{\infty})$ is $2g-1$ and thus $\ell(K)=g$. Then the result follows from
Proposition 2.2. ∎

###  2.2. The curves $\mathcal{Y}_{n,s}$

Let $q$ be a prime power. Let $n\geq 3$ be an odd integer and $s$ be a divisor
of $\frac{q^{n}+1}{q+1}$. The $\mathcal{Y}_{n,s}$ curve defined over
$\mathbb{F}_{q^{2n}}$ is the projective curve defined by the equations

(5) $\left\\{\begin{array}[]{rcl}x^{q}+x&=&y^{q+1}\\\
y^{q^{2}}-y&=&z^{M}\end{array}\right.,$

where $M=\dfrac{q^{n}+1}{s(q+1)}$. The curve $\mathcal{Y}_{n,s}$ is
$\mathbb{F}_{q^{2n}}$-maximal and it has genus

$g(\mathcal{Y}_{n,s})=\dfrac{q^{n+2}-q^{n}-sq^{3}+q^{2}+s-1}{2s}.$

We note that, for $s=1$, the curve $\mathcal{Y}_{n,1}$ is the GGS curve. So,
we can say that this curve generalizes the GGS curve.

Let $\mathcal{Y}_{n,s}(\mathbb{F}_{q^{2n}})$ be the set of
$\mathbb{F}_{q^{2n}}$-rational points of $\mathcal{Y}_{n,s}$, and we will
denote $P=(\alpha,\beta,\gamma)\in\mathcal{Y}_{n,s}(\mathbb{F}_{q^{2n}})$ by
$P_{(\alpha,\beta,\gamma)}$. The unique common pole of the functions $x,y$ and
$z$ will be denote by $P_{\infty}$. Thus, we have the following divisors:

(6) $(x-\alpha)=(q+1)MP_{(\alpha,0,0)}-(q+1)MP_{\infty}\mbox{ ;}$

(7)
$\displaystyle(y-\beta)=\sum_{i=1}^{q}MP_{(\alpha_{i},\beta,0)}-qMP_{\infty},\mbox{
with }\alpha_{i}^{q}+\alpha_{i}=\beta^{q+1}\mbox{ and
}\beta\in\mathbb{F}_{q^{2}}\mbox{ ;}$

(8)
$\displaystyle(z)=\sum_{j=1}^{q^{2}}\sum_{i=1}^{q}P_{(\alpha_{i},\beta_{j},0)}-q^{3}P_{\infty},\mbox{
with }\beta_{j}\in\mathbb{F}_{q^{2}}\mbox{ and
}\beta_{j}^{q+1}=\alpha_{i}^{q}+\alpha_{i}\mbox{ for all }i=1,\ldots,q.$

From [29, Proposition 5.1] we have that $H(P_{\infty})=\langle
qM,q^{3},(q+1)M\rangle$ and it is a telescopic semigroup. More details about
the curve $\mathcal{Y}_{n,s}$ can be found in [29].

Analogously to Lemma 2.3 one can prove that the divisor

(9)
$K=\left(\dfrac{q^{n+2}-q^{n}-sq^{3}+q^{2}+s-1}{s}-2\right)P_{\infty}=[(q^{2}-1)(q+1)M-q^{3}-1]P_{\infty}$

is a canonical divisor of $\mathcal{Y}_{n,s}$. Since the proof would be very
similar to that of Lemma 2.3, we will omit it.

###  2.3. Weierstrass semigroups and gaps

As before, let $\mathcal{X}$ be a curve over $\mathbb{F}_{q}$ and let
$\mathbf{Q}=(Q_{1},\ldots,Q_{m})$ be an $m$-tuple of pairwise distinct
rational points on $\mathcal{X}$. The Weierstrass semigroup $H(\mathbf{Q})$ of
$\mathcal{X}$ at $\mathbf{Q}$ is the set

$H(\mathbf{Q}):=\left\\{(a_{1},\ldots,a_{m})\in\mathbb{N}_{0}^{m}\mbox{ :
}\exists\ h\in\mathbb{F}_{q}(\mathcal{X})^{\times}\mbox{ with
}(h)_{\infty}=\sum_{i=1}^{m}a_{i}Q_{i}\right\\}.$

In [24], G. Matthews introduced the concept of minimal generating set for
$H(\mathbf{Q})$, denoted by $\Gamma(\mathbf{Q})=\Gamma(Q_{1},\ldots,Q_{m})$.
The word “generating” comes from the fact that $H(\mathbf{Q})$ can be
reconstructed from $\Gamma(\mathbf{Q})$, as we describe in the following.

Let ${\bf u}_{1},\ldots,{\bf u}_{t}\in\mathbb{N}_{0}^{m}$, where ${\bf
u}_{k}=(u_{k_{1}},\ldots,u_{k_{m}})$ for all $k=1,\ldots t$. The _least upper
bound_ ($\operatorname{lub}$) of the vectors ${\bf u}_{1},\ldots,{\bf u}_{t}$
is defined as

$\operatorname{lub}\\{{\bf u}_{1},\ldots,{\bf
u}_{t}\\}=(\max\\{{u_{1_{1}}},\ldots,{u_{t_{1}}}\\},\ldots,\max\\{{u_{1_{m}}},\ldots,{u_{t_{m}}}\\})\in\mathbb{N}_{0}^{m}.$

In [24, Theorem 7], it is shown that, if $2\leq m\leq q$, then

$H(\mathbf{Q})=\\{\operatorname{lub}\\{{\bf u}_{1},\ldots,{\bf
u}_{m}\\}\in\mathbb{N}_{0}^{m}\mbox{ ; }{\bf u}_{1},\ldots,{\bf
u}_{m}\in\Gamma(\mathbf{Q})\\}.$

The elements in the finite complement
${G(\mathbf{Q}):=\mathbb{N}^{m}_{0}\backslash H(\mathbf{Q})}$, are called
_Weierstrass gaps_ , or simply gaps, of $\mathcal{X}$ at $\mathbf{Q}$ (or
simply _gaps_. A _pure Weierstrass gap_ (or simply pure gap) is an $m$-tuple
$\text{\boldmath$\alpha$}\in G(\mathbf{Q})$ such that
$\ell(D_{\text{\boldmath$\alpha$}})=\ell(D_{\text{\boldmath$\alpha$}}-Q_{j})$
for $j=1,\ldots,m$. The set of pure gaps of $\mathcal{X}$ at $\mathbf{Q}$ will
be denoted by $G_{0}(\mathbf{Q})$. As mentioned in the Introduction,
Weierstrass semigroups and their gaps are an important tool in coding theory,
see e.g. [8], [17] and [25]. For more details about Weierstrass semigroups at
$m$ points see [7] and [8].

###  2.4. Generalized Weierstrass semigroups and maximal elements

Given $\mathcal{X}$ and $\mathbf{Q}=(Q_{1},\ldots,Q_{m})$ as before, we define
the _generalized Weierstrass semigroup_ of $\mathcal{X}$ at $\mathbf{Q}$ to be
the set

(10)
$\widehat{H}(\mathbf{Q}):=\\{(-v_{Q_{1}}(h),\ldots,-v_{Q_{m}}(h))\in\mathbb{Z}^{m}\
:\ h\in R_{\mathbf{Q}}\backslash\\{0\\}\\}.$

For further references and recent applications of these structures see [4, 27,
30]. It is worth mentioning that the generalized Weierstrass semigroup
$\widehat{H}(\mathbf{Q})$ is a different set than the (classical) Weierstrass
semigroup $H(\mathbf{Q})$ of $\mathcal{X}$ at $\mathbf{Q}$ presented in the
previous section. However, the classical and generalized Weierstrass
semigroups are related by
$H(\mathbf{Q})=\widehat{H}(\mathbf{Q})\cap\mathbb{N}_{0}^{m}$ provided that
$q\geq m$; see [4, Proposition 6].

For
${\text{\boldmath$\alpha$}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{Z}^{m}}$
and $i\in\\{1,\ldots,m\\}$, we let

$\nabla_{i}^{m}(\text{\boldmath$\alpha$}):=\\{\text{\boldmath$\beta$}=(\beta_{1},\ldots,\beta_{m})\in\widehat{H}(\mathbf{Q})\
:\ \beta_{i}=\alpha_{i}\ \text{and }\beta_{j}\leq\alpha_{j}\ \text{for }j\neq
i\\}.$

Such sets play an important role in the characterization of
$\widehat{H}(\mathbf{Q})$ as the following proposition shows.

######  Proposition 2.4.

[30, Proposition 2.1] Let $\text{\boldmath$\alpha$}\in\mathbb{Z}^{m}$ and
assume that $q\geq m$. One has

  1. (1)

$\text{\boldmath$\alpha$}\in\widehat{H}(\mathbf{Q})$ if and only if
$\ell(D_{\text{\boldmath$\alpha$}})=\ell(D_{\text{\boldmath$\alpha$}}-Q_{i})+1$,
for all $i\in\\{1,\ldots,m\\}$;

  2. (2)

$\nabla_{i}^{m}(\text{\boldmath$\alpha$})=\emptyset$ if and only if
$\ell(D_{\text{\boldmath$\alpha$}})=\ell(D_{\text{\boldmath$\alpha$}}-Q_{i})$.

Let
$\text{\boldmath$\alpha$}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{Z}^{m}$.
For a nonempty subset $J\subsetneq\\{1,\ldots,m\\}$, define

$\nabla_{J}(\text{\boldmath$\alpha$}):=\\{\text{\boldmath$\beta$}=(\beta_{1},\ldots,\beta_{m})\in\widehat{H}(\mathbf{Q})\
:\ \beta_{j}=\alpha_{j}\ \text{for }j\in J,\ \text{and }\beta_{i}<\alpha_{i}\
\text{for }i\not\in J\\}.$

######  Definition 2.5.

An element $\text{\boldmath$\alpha$}\in\widehat{H}(\mathbf{Q})$ is called
_absolute maximal_ if ${\nabla_{J}(\text{\boldmath$\alpha$})=\emptyset}$ for
every $J\subsetneq\\{1,\ldots,m\\}$ with $\\#J\geq 2$. If otherwise
${\nabla_{J}(\text{\boldmath$\alpha$})\neq\emptyset}$ for every
$J\subsetneq\\{1,\ldots,m\\}$ with $\\#J\geq 2$, we say that
$\text{\boldmath$\alpha$}\in\widehat{H}(\mathbf{Q})$ is _relative maximal_.

The sets of absolute and relative maximal elements in
$\widehat{H}(\mathbf{Q})$ will be denoted, respectively, by
$\widehat{\Gamma}(\mathbf{Q})$ and $\widehat{\Lambda}(\mathbf{Q})$.

######  Proposition 2.6.

[30, Proposition 3.2] Let $\text{\boldmath$\alpha$}\in\widehat{H}(\mathbf{Q})$
and assume that $q\geq m$. The following statements are equivalent:

  1. (i)

$\text{\boldmath$\alpha$}\in\widehat{\Gamma}(\mathbf{Q})$;

  2. (ii)

$\nabla_{i}^{m}(\text{\boldmath$\alpha$})=\\{\text{\boldmath$\alpha$}\\}$ for
all $i\in\\{1,\ldots,m\\}$;

  3. (iii)

$\nabla_{i}^{m}(\text{\boldmath$\alpha$})=\\{\text{\boldmath$\alpha$}\\}$ for
some $i\in\\{1,\ldots,m\\}$;

  4. (iv)

$\ell(D_{\text{\boldmath$\alpha$}})=\ell(D_{\text{\boldmath$\alpha$}}-\sum_{i=1}^{m}Q_{i})+1$.

Given a finite subset $\mathcal{B}\subseteq\mathbb{Z}^{m}$, we define the
_least upper bound_ ($\operatorname{lub}$) of $\mathcal{B}$ by

$\mbox{lub}(\mathcal{B}):=(\max\\{\beta_{1}\mbox{ :
}\text{\boldmath$\beta$}\in\mathcal{B}\\},\ldots,\max\\{\beta_{m}\mbox{ :
}\text{\boldmath$\beta$}\in\mathcal{B}\\})\in\mathbb{Z}^{m}.$

The next theorem shows that the set $\widehat{\Gamma}(\mathbf{Q})$ determines
$\widehat{H}(\mathbf{Q})$ in terms of least upper bounds.

######  Theorem 2.7.

[30, Theorem 3.4] Assume that $q\geq m$. The generalized Weierstrass semigroup
of $\mathcal{X}$ at $\mathbf{Q}$ can be written as

$\widehat{H}(\mathbf{Q})=\\{\operatorname{lub}(\\{\text{\boldmath$\beta$}^{1},\ldots,\text{\boldmath$\beta$}^{m}\\})\
:\
\text{\boldmath$\beta$}^{1},\ldots,\text{\boldmath$\beta$}^{m}\in\widehat{\Gamma}(\mathbf{Q})\\}.$

In this way, $\widehat{\Gamma}(\mathbf{Q})$ can be seen as a generating set of
$\widehat{H}(\mathbf{Q})$ in the sense of [24]. We observe that, unlike the
case of generating set for classical Weierstrass semigroups of [24], the set
$\widehat{\Gamma}(\mathbf{Q})$ is not finite. Nevertheless, it is finitely
determined, in the sense that we describe in the following.

For $i=2,\ldots,m$, let $a_{i}$ be the smallest positive integer $t$ such that
${tQ_{i}-tQ_{i-1}}$ is a principal divisor on $\mathcal{X}$. We can thus
define the region

$\mathcal{C}_{m}=\mathcal{C}_{m}(\mathbf{Q}):=\\{\text{\boldmath$\alpha$}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{Z}^{m}\
:\ 0\leq\alpha_{i}<a_{i}\ \mbox{for }i=2,\ldots,m\\}.$

Let
$\text{\boldmath$\eta$}^{i}=(\eta_{1}^{i},\ldots,\eta_{m}^{i})\in\mathbb{Z}^{m}$
be the $m$-tuple whose $j$-th coordinate is

$\eta^{i}_{j}=\left\\{\begin{array}[]{rcl}-a_{i}&,&\mbox{if }j=i-1\\\
a_{i}&,&\mbox{if }j=i\\\ 0&,&\mbox{otherwise.}\\\ \end{array}\right.$

Defining

(11)
${\Theta_{m}=\Theta_{m}(\mathbf{Q}):=\\{b_{2}\text{\boldmath$\eta$}^{2}+\ldots+b_{m}\text{\boldmath$\eta$}^{m}\in\mathbb{Z}^{m}\
:b_{i}\in\mathbb{Z}\ \mbox{for }i=2,\ldots,m\\}},$

we can state the following result concerning the absolute maximal elements and
relative maximal elements in generalized Weierstrass semigroups at several
points.

######  Theorem 2.8.

[27, Theorem 3.7] Assume that $q\geq m$. The following holds:

$\widehat{\Gamma}(\mathbf{Q})=(\widehat{\Gamma}(\mathbf{Q})\cap\mathcal{C}_{m}(\mathbf{Q}))+\Theta_{m}(\mathbf{Q})$
$\widehat{\Lambda}(\mathbf{Q})=(\widehat{\Lambda}(\mathbf{Q})\cap\mathcal{C}_{m}(\mathbf{Q}))+\Theta_{m}(\mathbf{Q}).$

######  Remark 2.9.

Note that since $\widehat{\Gamma}(\mathbf{Q})\cap\mathcal{C}_{m}$ and
$\widehat{\Lambda}(\mathbf{Q})\cap\mathcal{C}_{m}(\mathbf{Q})$ are finite and
$\Theta_{m}$ is finitely generated, $\widehat{\Gamma}(\mathbf{Q})$ and
$\widehat{\Lambda}(\mathbf{Q})$ are determined by a finite number of elements
in $\widehat{H}(\mathbf{Q})$. Also, note that
$\Gamma(\mathbf{Q})=\widehat{\Gamma}(\mathbf{Q})\cap\mathbb{N}_{0}^{m}$ and so
if we get $\widehat{\Gamma}(\mathbf{Q})$ we have $\Gamma(\mathbf{Q})$.

For
$\text{\boldmath$\alpha$}=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{Z}^{m}$,
let
$\overline{\nabla}_{i}(\text{\boldmath$\alpha$}):=\\{\text{\boldmath$\beta$}\in\mathbb{Z}^{m}\
:\ \beta_{i}=\alpha_{i}\mbox{ and }\beta_{j}<\alpha_{j}\ \mbox{for }j\neq
i\\}$ and
$\overline{\nabla}(\text{\boldmath$\alpha$}):=\bigcup_{i=1}^{m}\overline{\nabla}_{i}(\text{\boldmath$\alpha$})$,
and define
$\Lambda(\mathbf{Q}):=\widehat{\Lambda}(\mathbf{Q})\cap\mathbb{N}_{0}^{m}$. In
[31] the following characterization of gaps and pure gaps of a Weierstrass
semigroup at several in terms of maximals elements in a generalized
Weierstrass semigroup is provided.

######  Theorem 2.10.

[31, Theorems 3.1 and 3.2] Assume that $q\geq m$. Then

1) $\displaystyle
G(\mathbf{Q})=\bigcup_{\text{\boldmath$\beta$}^{*}\in\Lambda(\mathbf{Q})}(\overline{\nabla}(\text{\boldmath$\beta$}^{*})\cap\mathbb{N}_{0}^{m}),$
and

2) $\displaystyle
G_{0}(\mathbf{Q})=\bigcup_{\scriptsize{(\text{\boldmath$\beta$}^{1},\ldots,\text{\boldmath$\beta$}^{m})\in\Lambda(\mathbf{Q})^{m}}}\left(\bigcap_{i=1}^{m}\overline{\nabla}_{i}(\text{\boldmath$\beta$}^{i})\right).$

An equivalent formulation for the absolute and relative maximal property can
be made with respect to the concept of discrepancy, which has been introduced
in [12]. This reformulation will be especially useful for us in calculations,
when determining absolute and relative maximal elements.

######  Definition 2.11.

Let $P$ and $Q$ be distinct rational points $P$ and $Q$ on $\mathcal{X}$. A
divisor $A\in\mbox{Div}(\mathcal{X})$ is called a discrepancy with respect to
$P$ and $Q$ if $\mathcal{L}(A)\neq\mathcal{L}(A-Q)$ and
$\mathcal{L}(A-P)=\mathcal{L}(A-P-Q)$.

######  Proposition 2.12.

[31, Proposition 2.8] Let $\text{\boldmath$\alpha$}\in\mathbb{Z}^{m}$ and
assume that $q\geq m$. The following statements are equivalent:

  1. (1)

$\text{\boldmath$\alpha$}\in\widehat{\Lambda}(\mathbf{Q})$;

  2. (2)

$\nabla(\text{\boldmath$\alpha$})=\emptyset$ and
$\ell(D_{\text{\boldmath$\alpha$}})=\ell(D_{\text{\boldmath$\alpha$}-\textbf{1}})+(m-1)$;

  3. (3)

for $i,j\in I$ with $j\neq i$,
$D_{\text{\boldmath$\alpha$}-\textbf{1}}+Q_{i}+Q_{j}$ is discrepancy w.r.t.
$Q_{i}$ and $Q_{j}$;

  4. (4)

there exists $i\in I$ such that
$D_{\text{\boldmath$\alpha$}-\textbf{1}}+Q_{i}+Q_{j}$ is discrepancy w.r.t.
$Q_{i}$ and $Q_{j}$, for any $j\neq i$;

  5. (5)

there exists $i\in I$ such that
$\nabla_{i}(\text{\boldmath$\alpha$})=\emptyset$ and
$\nabla_{i,j}(\text{\boldmath$\alpha$})\neq\emptyset$ for every $j\neq i$.

######  Proposition 2.13.

[30, Proposition 3] Let $\mathcal{X}$ be a curve over $\mathbb{F}_{q}$, $r$ be
an integer such that $1\leq r\leq q$ and $Q_{1},\ldots,Q_{r}$ be rational
points on $\mathcal{X}$. Let
$\text{\boldmath$\alpha$}=(\alpha_{1},\ldots,\alpha_{r})\in\mathbb{Z}^{r}$ and
$\mathbf{Q}=(Q_{1},\ldots,Q_{r})$. The following statements are equivalent:

  1. (i)

$\text{\boldmath$\alpha$}\in\widehat{\Gamma}(\mathbf{Q})$;

  2. (ii)

$\alpha_{1}Q_{1}+\ldots+\alpha_{r}Q_{r}$ is a discrepancy with respect to any
pair of distinct points in $\\{Q_{1},\ldots,Q_{r}\\}$.

We end this section presenting a technical lemma that will be an important
tool in computations with discrepancies in the next sections.

######  Lemma 2.14.

[15, Noether’s Reduction Lemma] Let $D$ be a divisor, $P\in\mathcal{X}$ and
let $K$ be a canonical divisor. If $\dim(\mathcal{L}(D))>0$ and
$\dim(\mathcal{L}(K-D-P))\neq\dim(\mathcal{L}(K-D))$, then
$\dim(\mathcal{L}(D+P))=\dim(\mathcal{L}(D))$.

##  3\. Generalized Weierstrass Semigroup at certain $m+1$ points on
$\mathcal{X}_{a,b,n,s}$

In this section we present our main results about Weierstrass semigroups and
generalized Weierstrass semigroups at certain $m+1$ points on the curve
$\mathcal{X}_{a,b,n,s}$ over $\mathbb{F}_{q^{2n}}$, where $1\leq m\leq
q/p^{b}$. Let $P_{\infty}$ and $P_{(\alpha,\beta,0)}$ on
$\mathcal{X}_{a,b,n,s}$ as in Subsection 2.1. To simplify the notation, we
will denote $P_{(\alpha_{i},0,0)}$ by $P_{i}$, for each $i=1,\ldots,q/p^{b}$,
and $\mathbf{P}_{m+1}=(P_{\infty},P_{1},\ldots,P_{m})$. Define
$\mathcal{D}:=(z)_{0}=\sum_{j=1}^{q^{2}}\sum_{i=1}^{q/p^{b}}P_{(\alpha_{i},\beta_{j},0)}$
as already introduced in Equation (4).

We observe that with the conditions on $a,b$ and $n$ given in Section 2.1, if
$m\leq q/p^{b}$, then $m\leq q^{2n}$. This allows us to use the results given
in Sections 2.3 and 2.4 where this hypothesis was required.

######  Proposition 3.1.

Let $(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$ and $1\leq
m\leq q/p^{b}$. Then the divisor $\displaystyle
A=\dfrac{1}{p^{b}}[(q^{2}-mp^{b})(q+1)M-iqM-
jq^{3}]P_{\infty}+\sum_{\ell=1}^{m}(iM+j)P_{\ell}$ is a discrepancy to every
pair of distinct points in $\\{P_{\infty},P_{1},\ldots,P_{m}\\}$.

###### Proof.

First, to see that $\mathcal{L}(A)\neq\mathcal{L}(A-P)$ for all
$P\in\\{P_{\infty},P_{1},\ldots,P_{m}\\}$ we just note that the principal
divisor of the function
$f:=\dfrac{z^{M-j}y^{q-i}}{\prod_{k=1}^{m}(x-\alpha_{k})}$ is

$-\dfrac{1}{p^{b}}[(q^{2}-mp^{b})(q+1)M-iqM-
jq^{3}]P_{\infty}-\sum_{\ell=1}^{m}(iM+j)P_{\ell}+\sum_{\ell=m+1}^{q/p^{b}}[(q+1)M-iM-j]P_{\ell}+(M-j)\widetilde{\mathcal{D}},$

where $\widetilde{\mathcal{D}}=\mathcal{D}-(P_{1}+\cdots+P_{q/p^{b}})$.

Hence $f\in\mathcal{L}(A)\setminus\mathcal{L}(A-P)$ for all
$P\in\\{P_{\infty},P_{1},\ldots,P_{m}\\}$.

Now, we must prove that $\mathcal{L}(A-P)=\mathcal{L}(A-P-Q)$ and
$\mathcal{L}(A-Q)=\mathcal{L}(A-Q-P)$, for all
$P,Q\in\\{P_{\infty},P_{1},\ldots,P_{m}\\}$. By Lemma 2.14, it suffices to
prove that $\mathcal{L}(K-A+P)\neq\mathcal{L}(K-A+P+Q)$, where $K$ is a
canonical divisor. Let

$K=\left(\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-q^{3}]-1\right)P_{\infty}$

be the canonical divisor given in Lemma 2.3. Then, we have that

$\begin{array}[]{ll}K+P+Q-A&\displaystyle=\left(\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-q^{3}]-1\right)P_{\infty}+P+Q\\\
&\\\ &\displaystyle-\dfrac{1}{p^{b}}[(q^{2}-mp^{b})(q+1)M-iqM-
jq^{3}]P_{\infty}-\sum_{\ell=1}^{m}(iM+j)P_{\ell}\\\
&\displaystyle=[(m-1)(q+1)M+i\dfrac{q}{p^{b}}M+(j-1)\dfrac{q^{3}}{p^{b}}-1]P_{\infty}+P+Q-\sum_{\ell=1}^{m}(iM+j)P_{\ell}.\end{array}$

If $P_{\infty}\in\\{P,Q\\}$ then we can assume without loss of generality that
$P=P_{\infty}$ and $Q=P_{1}$. Thus,

$K+P+Q-A\displaystyle=[(m-1)(q+1)M+i\dfrac{q}{p^{b}}M+(j-1)\dfrac{q^{3}}{p^{b}}]P_{\infty}-(iM+j-1)P_{1}-\sum_{\ell=2}^{m}(iM+j)P_{\ell}$

and we have that
$z^{j-1}y^{i}(x-\alpha_{2})\cdots(x-\alpha_{m})\in\mathcal{L}(K+P+Q-A)\setminus\mathcal{L}(K+Q-A)$.

If $P_{\infty}\not\in\\{P,Q\\}$, we can suppose that $P=P_{1}$ and $Q=P_{2}.$
In this case, we have

$K+P+Q-A\displaystyle=[(m-1)(q+1)M+i\dfrac{q}{p^{b}}M+(j-1)\dfrac{q^{3}}{p^{b}}-1]P_{\infty}-(iM+j-1)P_{1}-(iM+j-1)P_{2}-\sum_{\ell=3}^{m}(iM+j)P_{\ell}$

and
$z^{j-1}y^{i}(x-\alpha_{3})\cdots(x-\alpha_{m})\in\mathcal{L}(K+P+Q-A)\setminus\mathcal{L}(K+Q-A)$.

So, $\mathcal{L}(K+P+Q-A)\neq\mathcal{L}(K+Q-A)$ and, by Lemma 2.14, we can
conclude that $\mathcal{L}(A-P)=\mathcal{L}(A-P-Q).$

Therefore, $A$ is a discrepancy with respect to $P$ and $Q$ for any two
distinct points $P,Q\in\\{P_{\infty},P_{1},\ldots,P_{m}\\}$. ∎

Our next aim is to compute explicitly the absolute and relative maximal
elements in generalized Weierstrass semigroups of $\mathcal{X}_{a,b,n,s}$ at
$(m+1)$-tuples of rational points, where $1\leq m\leq q/p^{b}$. To do so, we
observe that from Section 2.1 and Section 2.4 one has

(12) $\mathcal{C}_{m+1}=\\{\text{\boldmath$\beta$}\in\mathbb{Z}^{m+1}\ :\
0\leq\beta_{i}<(q+1)M\ \mbox{for }i=2,\ldots,m+1\\},$

and $\Theta_{m+1}$ is generated by the $(m+1)$-tuples

(13)
$\text{\boldmath$\eta$}^{i}=(0,\ldots,0,-(q+1)M,\underbrace{(q+1)M}_{i\text{-th
entry}},0,\ldots,0)\in\mathbb{Z}^{m+1}\quad\mbox{for}\quad i=2,\ldots,m+1.$

For $(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$ define

(14)
$\text{\boldmath$\alpha$}^{i,j,m}:=\left(\dfrac{1}{p^{b}}[(q^{2}-mp^{b})(q+1)M-iqM-
jq^{3}],iM+j,\ldots,iM+j\right)\in\mathbb{Z}^{m+1}.$

Note that when writing
$\text{\boldmath$\alpha$}^{i,j,m}=(\alpha^{i,j,m}_{0},\alpha^{i,j,m}_{1},\ldots,\alpha^{i,j,m}_{m})$
one has $\alpha^{i,j,m}_{0}=\alpha^{i,j,m-1}_{0}-(q+1)M$, for all $m\geq 2$.

######  Lemma 3.2.

Let
$\text{\boldmath$\alpha$}^{i,j,m}=(\alpha^{i,j,m}_{0},\alpha^{i,j,m}_{1},\ldots,\alpha^{i,j,m}_{m})$
be as above with $m\geq 2$. Then

$\nabla_{1}^{m}(\alpha^{i,j,m}_{0},\alpha^{i,j,m}_{1}-1+(q+1)M,\ldots,\alpha^{i,j,m}_{m-1}-1+(q+1)M,\alpha^{i,j,m}_{m}-1)=\emptyset.$

###### Proof.

Let
$\bar{\text{\boldmath$\alpha$}}=(\alpha^{i,j,m}_{0},\alpha^{i,j,m}_{1}+(q+1)M,\ldots,\alpha^{i,j,m}_{m-1}+(q+1)M,\alpha^{i,j,m}_{m})$
and
$D_{\bar{\text{\boldmath$\alpha$}}}=\alpha^{i,j,m}_{0}P_{\infty}+(\alpha^{i,j,m}_{1}+(q+1)M)P_{1}+\ldots+(\alpha^{i,j,m}_{m-1}+(q+1)M)P_{m-1}+\alpha^{i,j,m}_{m}P_{m}$.
In view of Proposition 2.4(2), it is sufficient to prove that

$\ell\left(D_{\bar{\text{\boldmath$\alpha$}}}-(P_{1}+\ldots+P_{m})\right)=\ell\left(D_{\bar{\text{\boldmath$\alpha$}}}-(P_{\infty}+P_{1}+\ldots+P_{m})\right).$

Equivalently, by Lemma 2.14, let us show that

$\mathcal{L}(K-D_{\bar{\text{\boldmath$\alpha$}}}+P_{1}+\ldots+P_{m})\neq\mathcal{L}(K-D_{\bar{\text{\boldmath$\alpha$}}}+P_{\infty}+P_{1}+\ldots+P_{m})$

where $K$ as before denotes the canonical divisor
$K=\left(\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-q^{3}]-1\right)P_{\infty}$ given
in Lemma 2.3.

Hence

$\begin{array}[]{rl}K-D_{\bar{\text{\boldmath$\alpha$}}}+P_{\infty}+P_{1}+\ldots+P_{m}&=\left((m-1)(q+1)M+i\dfrac{qM}{p^{b}}+(j-1)\dfrac{q^{3}}{p^{b}}\right)P_{\infty}\\\
&\displaystyle-\sum_{k=1}^{m-1}[iM+(j-1)+(q+1)M]P_{k}-(iM+j-1)P_{m}.\end{array}$

Let $f:=z^{j-1}y^{i}(x-\alpha_{1})\ldots(x-\alpha_{m-1})$. Thus

$\begin{array}[]{rl}(f)&=(j-1)P_{1}+\cdots+(j-1)P_{q/p^{b}}-(j-1)\dfrac{q^{3}}{p^{b}}P_{\infty}\\\
&+iMP_{1}+\cdots+iMP_{q/p^{b}}-i\dfrac{q}{p^{b}}MP_{\infty}\\\
&+(q+1)MP_{1}+\cdots+(q+1)MP_{m-1}-(m-1)(q+1)MP_{\infty}\\\
&=\displaystyle\sum_{k=1}^{m-1}[iM+(j-1)+(q+1)M]P_{k}+(iM+j-1)P_{m}\\\
&+\displaystyle\sum_{k=m+1}^{q/p^{b}}(iM+j-1)P_{k}-\left((m-1)(q+1)M+i\dfrac{qM}{p^{b}}+(j-1)\dfrac{q^{3}}{p^{b}}\right)P_{\infty}.\end{array}$

So,
$f\in\mathcal{L}(K-D_{\bar{\text{\boldmath$\alpha$}}}+P_{\infty}+P_{1}+\ldots+P_{m})\setminus\mathcal{L}(K-D_{\bar{\text{\boldmath$\alpha$}}}+P_{1}+\ldots+P_{m})$,
and the proof is complete. ∎

We are now ready to give an explicit description for the absolute and the
relative maximal elements of $\widehat{H}(\mathbf{P}_{m+1})$ in the region
$\mathcal{C}_{m+1}$ given in Equation (12).

######  Theorem 3.3.

Let $1\leq m\leq q/p^{b}$, $\mathbf{P}_{m+1}=(P_{\infty},P_{1},\ldots,P_{m})$,
and $\widehat{S}_{m+1}$ be as above. Then

$\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}=\widehat{S}_{m+1}\cup\\{{\bf
0}\\},$

where $\widehat{S}_{m+1}$ is the set of all $\text{\boldmath$\alpha$}^{i,j,m}$
as in (14).

###### Proof.

It follows from Propositions 2.13 and 3.1 that
$\widehat{S}_{m+1}\subseteq\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}$.
To prove the reverse inclusion we will use induction on $m$.

First, for $m=1$ if
$(\lambda,\mu)\in\widehat{\Gamma}(\mathbf{P}_{2})\cap\mathcal{C}_{2}$, then
$\mu<(q+1)M$ and so $\mu=iM+j$, for some
$(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$. Now, by
Propositions 2.13 and 3.1, we have that
$\left(\dfrac{1}{p^{b}}[(q^{2}-mp^{b})(q+1)M-iqM-
jq^{3}],iM+j\right)\in\widehat{\Gamma}(\mathbf{P}_{2})\cap\mathcal{C}_{2}$ and
the results follows from Proposition 2.6.

Let now $m\geq 2$ and suppose that
$\widehat{\Gamma}(\mathbf{P}_{k+1})\cap\mathcal{C}_{k+1}=\widehat{S}_{k+1}$
for $k=1,\ldots,m-1$. Recall that, when writing
$\text{\boldmath$\alpha$}^{i,j,m}=(\alpha^{i,j,m}_{0},\alpha^{i,j,m}_{1},\ldots,\alpha^{i,j,m}_{m})$,
we have $\alpha^{i,j,m}_{0}=\alpha^{i,j,m-1}_{0}-(q+1)M$. Given
${\text{\boldmath$\alpha$}=(\alpha_{0},\alpha_{1},\ldots,\alpha_{m})\in\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}}$,
let
$D_{\text{\boldmath$\alpha$}}=\alpha_{0}P_{\infty}+\alpha_{1}P_{1}\ldots+\alpha_{m}P_{m}$
and let us consider ${\tilde{t}:=\min\\{t\in\mathbb{N}\ :\
\alpha_{0}+t(q+1)M\geq 0\\}}$. Observe that, by Proposition 2.6,
$\ell(D_{\text{\boldmath$\alpha$}})\geq 1$, and thus
$|\text{\boldmath$\alpha$}|=\deg(D_{\text{\boldmath$\alpha$}})\geq 0$. In
addition, $\text{\boldmath$\alpha$}\in\mathcal{C}_{m+1}$ and hence
$\alpha_{0}\geq-\sum_{j=1}^{m}\alpha_{j}\geq-m((q+1)M-1)$. So
$\alpha_{0}+m(q+1)M\geq 0$, which shows that $1\leq\tilde{t}\leq m$. We
distinguish two cases.

$\bullet$ If $1\leq\tilde{t}\leq m-1$, we have $2\leq m+1-\tilde{t}\leq m$.
Without loss of generality, assume that $\alpha_{1}=\min_{1\leq k\leq
m}\\{\alpha_{k}\\}$. Hence Theorem 2.8 and Equation (13) lead to

$\text{\boldmath$\alpha$}^{\prime}=(\alpha_{0}+\tilde{t}(q+1)M,\alpha_{1},\ldots,\alpha_{m-\tilde{t}},\alpha_{m-\tilde{t}+1}-(q+1)M,\ldots,\alpha_{m}-(q+1)M)\in\widehat{H}(\mathbf{P}_{m+1}).$

Since $0\leq\alpha_{k}<(q+1)M$ for $k=1,\ldots,m$, we get

$\text{\boldmath$\beta$}=\mbox{lub}(\text{\boldmath$\alpha$}^{\prime},\textbf{0})=(\alpha_{0}+\tilde{t}(q+1)M,\alpha_{1},\ldots,\alpha_{m-\tilde{t}},0,\ldots,0)\in\widehat{H}(\mathbf{P}_{m+1-\tilde{t}}).$

In particular,
$\nabla_{2}^{m+1-\tilde{t}}(\text{\boldmath$\beta$})\neq\emptyset$. From the
induction hypothesis, there exists
$\text{\boldmath$\alpha$}^{i,j,m+1-\tilde{t}}\in\widehat{S}_{m+1-\tilde{t}}$
such that
$\text{\boldmath$\alpha$}^{i,j,m+1-\tilde{t}}\in\nabla_{2}^{m+1-\tilde{t}}(\text{\boldmath$\beta$})$.
As $\alpha_{0}^{i,j,m}=\alpha_{0}^{i,j,m-\tilde{t}}-\tilde{t}(q+1)M$ and
$\alpha_{1}=iM+j\leq\alpha_{k}$ for $k=2,\ldots,m$, it follows that
$\text{\boldmath$\alpha$}^{i,j,m}\in\nabla_{2}^{m}(\text{\boldmath$\alpha$})$.
Therefore, $\text{\boldmath$\alpha$}=\text{\boldmath$\alpha$}^{i,j,m}$ by
Proposition 2.6, since
$\text{\boldmath$\alpha$}\in\widehat{\Gamma}(\mathbf{P}_{m})$.

$\bullet$ If $\tilde{t}=m$, by using a similar argument, we obtain
$\alpha_{0}+m(q+1)M\in
H(P_{\infty})=\langle\frac{q}{p^{b}}M,\frac{q^{3}}{p^{b}},(q+1)M\rangle$. As
$\alpha_{0}+m(q+1)M<(q+1)M$, we have either $\alpha_{0}+m(q+1)M=0$ or
$\alpha_{0}+m(q+1)M=\lambda\dfrac{qM}{p^{b}}+\eta\dfrac{q^{3}}{p^{b}}$, for
some integers $\lambda,\eta\geq 0$.

If $\alpha_{0}+m(q+1)M=0$ then $\alpha_{0}=-m(q+1)M$, which is a contradiction
since $\alpha_{0}\geq-m((q+1)M-1)>-m(q+1)M$.

In the other case, we can write
$\alpha_{0}+m(q+1)M=(q-i)\dfrac{qM}{p^{b}}+(M-j)\dfrac{q^{3}}{p^{b}}=\dfrac{1}{p^{b}}[q^{2}(q+1)M-iqM-
jq^{3}]$ for some
$(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$. So, we have
that $\alpha_{0}=\dfrac{1}{p^{b}}[(q^{2}-mp^{b})(q+1)M-iqM-
jq^{3}]=\alpha^{i,j,m}_{0}$. Suppose that $\alpha$ and
$\text{\boldmath$\alpha$}^{i,j,m}$ are not comparable in the partial order
$\leq$. Without loss of generality, we may assume that
$\alpha_{m}<\alpha_{m}^{i,j,m}$. In this way, since $\alpha^{i,j,m}_{k}\geq 1$
for all $k=1,\ldots,m-1$, we have that
$\text{\boldmath$\alpha$}\in\nabla_{1}(\alpha^{i,j,m}_{0},\alpha^{i,j,m}_{1}-1+(q+1)M,\ldots,\alpha^{i,j,m}_{m-1}-1+(q+1)M,\alpha^{i,j,m}_{m})$.
However, by Lemma 3.2, we have

$\qquad\nabla_{1}(\alpha^{i,j,m}_{0},\alpha^{i,j,m}_{1}-1+(q+1)M,\ldots,\alpha^{i,j,m}_{m-1}-1+(q+1)M,\alpha^{i,j,m}_{m})=\emptyset.$

Thus $\alpha$ and $\text{\boldmath$\alpha$}^{i,j,m}$ are comparable with
respect to $\leq$, which leads to
$\text{\boldmath$\alpha$}=\text{\boldmath$\alpha$}^{i,j,m}$ by their absolute
maximality.

Therefore,
$\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}\subseteq\widehat{S}_{m+1}$
and the proof is complete. ∎

For $i,j,k_{2},\ldots,k_{m+1}\in\mathbb{Z}$, define

(15)
$\begin{array}[]{ll}\text{\boldmath$\gamma$}_{k_{2},\ldots,k_{m+1}}^{i,j,m}:=&\left(\dfrac{1}{p^{b}}[(q^{2}-mp^{b}-p^{b}(\sum_{\ell=2}^{m+1}k_{\ell}))(q+1)M-iqM-
jq^{3}],\right.\\\
&\left.k_{2}(q+1)M+iM+j,\ldots,k_{m+1}(q+1)M+iM+j\right)\in\mathbb{Z}^{m+1}.\end{array}$

The set of absolute maximal elements in the generalized Weierstrass semigroup
$\widehat{H}(\mathbf{P}_{m+1})$ is presented in the following corollary.

######  Corollary 3.4.

$\widehat{\Gamma}(\mathbf{P}_{m+1})=\\{\text{\boldmath$\gamma$}_{k_{2},\ldots,k_{m+1}}^{i,j,m}\in\mathbb{Z}^{m+1}\mbox{
; }(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\mbox{ and
}k_{2},\ldots,k_{m+1}\in\mathbb{Z}\\}.$

###### Proof.

From Theorem 2.8 we have that

(16)
$\widehat{\Gamma}(\mathbf{P}_{m+1})=(\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1})+\Theta_{m+1},$

where
$\Theta_{m+1}=\Theta_{m+1}(\mathbf{P}_{m+1})=\\{b_{2}\text{\boldmath$\eta$}^{2}+\ldots+b_{m}\text{\boldmath$\eta$}^{m}\in\mathbb{Z}^{m+1}\
:b_{i}\in\mathbb{Z}\mbox{ for }i=2,\ldots,m+1\\}$ and
$\text{\boldmath$\eta$}^{i}=(0,\ldots,0,-(q+1)M,\underbrace{(q+1)M}_{i\text{-th
entry}},0,\ldots,0)\in\mathbb{Z}^{m+1}\quad\mbox{for}\quad i=2,\ldots,m+1,$ as
in (13). Hence

$\begin{array}[]{rl}\Theta_{m+1}=&\\{(-b_{2}(q+1)M,(b_{2}-b_{3})(q+1)M,(b_{3}-b_{4})(q+1)M,\ldots,\\\
&(b_{m}-b_{m+1})(q+1)M,b_{m+1}(q+1)M)\in\mathbb{Z}^{m+1}\mbox{ ;
}b_{i}\in\mathbb{Z}\mbox{ for }i=2,\ldots,m+1\\}\\\
=&\\{(-\sum_{\ell=2}^{m+1}k_{\ell}(q+1)M,k_{2}(q+1)M,k_{3}(q+1)M,\ldots,k_{m+1}(q+1)M)\mbox{
; }\\\ &k_{\ell}\in\mathbb{Z}\mbox{ for }\ell=2,\ldots,m+1\\}.\end{array}$

By the previous theorem we have that
$\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}=\widehat{S}_{m+1}\cup\\{{\bf
0}\\}$, where $\widehat{S}_{m+1}$ is the set of all
$\text{\boldmath$\alpha$}^{i,j,m}$ as in (14). So, the result follows from
Equality (16) above. ∎

In the following corollary we determine the minimal generating set of the
Weierstrass semigroup $H(\mathbf{P}_{m+1})$.

######  Corollary 3.5.

Let $i,j,k_{2},\ldots,k_{m+1}$ and
$\text{\boldmath$\gamma$}_{k_{2},\ldots,k_{m+1}}^{i,j,m}$ be as above. Then,
$\Gamma(\mathbf{P}_{m+1})$ is the set of all
$\text{\boldmath$\gamma$}_{k_{2},\ldots,k_{m+1}}^{i,j,m}\in\mathbb{N}_{0}^{m+1}$.

###### Proof.

The result follows directly from Remark 2.9 and Corollary 3.4. ∎

######  Theorem 3.6.

Let $1\leq m\leq q/p^{b}$. Let

$\text{\boldmath$\beta$}^{0,0,m}=\left((m-1)(q+1)M,0,\ldots,0\right)\in\mathbb{Z}^{m+1}$

and for $(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$, let

$\text{\boldmath$\beta$}^{i,j,m}:=\left(\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-iqM-
jq^{3}],iM+j,\ldots,iM+j\right)\in\mathbb{Z}^{m+1}.$

Then

$\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}=\\{\text{\boldmath$\beta$}^{i,j,m}\
:(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\\}\cup\\{\text{\boldmath$\beta$}^{0,0,m}\\}.$

###### Proof.

If $m=1$, then the result follows from the previous theorem. Therefore, we can
suppose $m\geq 2$. Let $R_{m+1}=\\{\text{\boldmath$\beta$}^{i,j,m}\
:(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\\}\cup\\{\text{\boldmath$\beta$}^{0,0,m}\\}$.
Let us prove that
$R_{m+1}\subseteq\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}$.
Note that $R_{m+1}\subseteq C_{m+1}$, since $(i,j)\in([0,q]\times[1,M]$. To
prove that $R_{m+1}\subseteq\widehat{\Lambda}(\mathbf{P}_{m+1})$, by
Proposition 2.12 (3), it is sufficient to prove that the divisor
$D=D_{\text{\boldmath$\beta$}^{i,j,m}-\textbf{1}+\textbf{e}_{1}+\textbf{e}_{k}}$,
with
$(i,j)\in(([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M))\cup\\{(0,0)\\}$,
is a discrepancy with respect to $P_{\infty}$ and $P_{k}$ for any
$k=1,\ldots,m$.

First, suppose $(i,j)\neq(0,0)$. Thus

$\displaystyle D=\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-iqM-
jq^{3}]P_{\infty}+(iM+j)P_{k}+\sum_{\ell=1\atop\ell\neq
k}^{m}(iM+j-1)P_{\ell}.$

Let
$\mathcal{D}:=\sum_{j=1}^{q^{2}}\sum_{i=1}^{q/p^{b}}P_{(\alpha_{i},\beta_{j},0)}$
(as above). Since,

$\begin{array}[]{ll}\left(\dfrac{z^{M-j}y^{q-i}}{x-\alpha_{k}}\right)=&-\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-iqM-
jq^{3}]P_{\infty}-(iM+j)P_{k}\\\ &\\\
&\displaystyle+(M-j)(\mathcal{D}-P_{k})+(q-i)M\sum_{\ell=1\atop\ell\neq
k}^{m}P_{\ell},\end{array}$

we have that
$\dfrac{z^{M-j}y^{q-i}}{x-\alpha_{k}}\in\mathcal{L}(D)\setminus\mathcal{L}(D-P_{k})$.

Now, we must prove that
$\mathcal{L}(D-P_{\infty})=\mathcal{L}(D-P_{\infty}-P_{k})$. By Lemma 2.14, it
suffices to prove that
$\mathcal{L}(K-D+P_{\infty}+P_{k})\neq\mathcal{L}(K-D+P_{\infty})$, where $K$
is the canonical divisor
$K=\left(\dfrac{1}{p^{b}}[(q^{2}-p^{b})(q+1)M-q^{3}]-1\right)P_{\infty}$. Note
that

$K-D+P_{\infty}+P_{k}=\dfrac{1}{p^{b}}(iqM+(j-1)q^{3})P_{\infty}-\sum_{\ell=1}^{m}(iM+j-1)P_{\ell}.$

Thus,
$z^{j-1}y^{i}\in\mathcal{L}(K-D+P_{\infty}+P_{k})\setminus\mathcal{L}(K-D+P_{\infty})$,
and it follows that $D$ is discrepancy with respect to $P_{\infty}$ and
$P_{k}$ for any $k=1,\ldots,m$.

Therefore, to conclude that
$R_{m+1}\subseteq\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}$, it
remains to verify that
$\text{\boldmath$\beta$}^{0,0,m}\in\widehat{\Lambda}(\mathbf{P}_{m+1})$.

For $(i,j)=(0,0)$, we have $\displaystyle
D=(m-1)(q+1)MP_{\infty}-\sum_{\ell=1\atop\ell\neq k}^{m}P_{\ell}$. Again, we
will prove that $\mathcal{L}(D)\neq\mathcal{L}(D-P_{k})$ and
$\mathcal{L}(K-D+P_{\infty})\neq\mathcal{L}(K-D+P_{\infty}+P_{k})$.

Note that $\displaystyle\prod_{\ell=1\atop\ell\neq
k}^{m}(x-\alpha_{\ell})\in\mathcal{L}(D)\setminus\mathcal{L}(D-P_{k})$.
Moreover, since

$\displaystyle
K-D+P_{k}+P_{\infty}=\dfrac{1}{p^{b}}[(q^{2}-mp^{b})(q+1)M-q^{3}]P_{\infty}+\sum_{\ell=1}^{m}P_{\ell},$

we get
$\displaystyle\dfrac{z^{M-1}y^{q}}{(x-\alpha_{1})\ldots(x-\alpha_{m})}\in\mathcal{L}(K-D+P_{k}+P_{\infty})\setminus\mathcal{L}(K-D+P_{\infty})$.

Therefore, we conclude that
$R_{m+1}\subseteq\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}$.

Now, let
$\text{\boldmath$\beta$}\in\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}$.
Since $\text{\boldmath$\beta$}\in\widehat{\Lambda}(\mathbf{P}_{m+1})$, from
definition we have that
$\nabla_{k}^{m+1}(\text{\boldmath$\beta$})\neq\emptyset$ for any $k\in I$. So,
there exists an absolute maximal element
$\text{\boldmath$\alpha$}^{i,j,m}\in\nabla_{2}^{m+1}(\text{\boldmath$\beta$})$,
where $\text{\boldmath$\alpha$}^{i,j,m}$ is given in (14). Thus
$\beta_{2}=iM+j$ and $\beta_{3}\geq iM+j$. Similarly, there exists an absolute
maximal element
$\text{\boldmath$\alpha$}^{i^{\prime},j^{\prime},m}\in\nabla_{3}^{m+1}(\text{\boldmath$\beta$})$,
and thus $\beta_{3}=i^{\prime}M+j^{\prime}$ and $\beta_{2}\geq
i^{\prime}M+j^{\prime}$. Hence $iM+j=i^{\prime}M+j^{\prime}$, and therefore
$(i,j)=(i^{\prime},j^{\prime})$. Proceeding in the same way with pairs of the
remaining indexes, we conclude that there exists an absolute maximal element
$\text{\boldmath$\alpha$}^{i,j,m}\in\bigcap_{k=2}^{m+1}\nabla_{k}^{m+1}(\text{\boldmath$\beta$})$
and, in particular, we can conclude that $\beta_{k}=iM+j$ for
$k=2,\ldots,m+1$. As
$\text{\boldmath$\beta$}\in\widehat{\Lambda}(\mathbf{P}_{m+1})$, it follows
that $\text{\boldmath$\beta$}\neq\text{\boldmath$\alpha$}^{i,j,m}$ and thus
$\beta_{1}>\alpha^{i,j,m}_{1}$. Hence, for each
$\text{\boldmath$\beta$}\in\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}(\mathbf{P}_{m+1})$,
there exists a unique
$\text{\boldmath$\alpha$}^{i,j,m}\in\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}$
such that
$\text{\boldmath$\alpha$}^{i,j,m}\in\nabla_{I\backslash\\{1\\}}(\text{\boldmath$\beta$})$.
Therefore,
$\\#(\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1})\geq\\#(\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1})$.
As $\\#R_{m+1}=\\#(\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1})$
and
$R_{m+1}\subseteq\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}$, we
have $\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}=R_{m+1}$, which
proves the result. ∎

Now, for $i,j,k_{2},\ldots,k_{m+1}\in\mathbb{Z}$, define

$\begin{array}[]{ll}\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}:=&\left(\dfrac{1}{p^{b}}[(q^{2}-p^{b}(1+\displaystyle\sum_{\ell=2}^{m+1}k_{\ell}))(q+1)M-iqM-
jq^{3}],\right.\\\
&\left.k_{2}(q+1)M+iM+j,\ldots,k_{m+1}(q+1)M+iM+j\right)\in\mathbb{Z}^{m+1},\end{array}$

and

$\text{\boldmath$\lambda$}_{k_{2},\ldots,k_{m+1}}:=\left((m-1-\sum_{\ell=2}^{m+1}k_{\ell})(q+1)M,k_{2}(q+1)M,\ldots,k_{m+1}(q+1)M\right)\in\mathbb{Z}^{m+1}.$

Note that, if $(i,j)\in([0,q]\times[1,M]$, then $k(q+1)M+iM+j\geq 0$ if and
only if $k\geq 0$.

From the previous result and Theorem 2.8 we get the set of relative maximal
elements in the generalized Weierstrass semigroup
$\widehat{H}(\mathbf{P}_{m+1})$.

######  Corollary 3.7.

Let $\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}$ and
$\text{\boldmath$\lambda$}_{k_{2},\ldots,k_{m+1}}$ be as above. Then

$\begin{array}[]{ll}\widehat{\Lambda}(\mathbf{P}_{m+1})=&\\{\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\in\mathbb{Z}^{m+1}\mbox{
; }(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\mbox{ ,
}k_{2},\ldots,k_{m+1}\in\mathbb{Z}\\}\\\
&\bigcup\\{\text{\boldmath$\lambda$}_{k_{2},\ldots,k_{m+1}}\in\mathbb{Z}^{m+1}\mbox{
; }k_{2},\ldots,k_{m+1}\in\mathbb{Z}\\}.\end{array}$

By Theorem 2.10 we have that the gaps and pure gaps of $H(\mathbf{P}_{m+1})$
can be obtained from elements in the set $\Lambda(\mathbf{P}_{m+1})$. Since
$\Lambda(\mathbf{P}_{m+1})=\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathbb{N}_{0}^{m}$,
using the previous result we have the following.

######  Corollary 3.8.

Let $\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}$ and
$\text{\boldmath$\lambda$}_{k_{2},\ldots,k_{m+1}}$ be as above and let
$\tau_{(i,j)}:=\lfloor(q^{3}(M-j)+qM(q-i)-p^{b}(q+1)M)/p^{b}(q+1)M\rfloor$.
Then

$\begin{array}[]{ll}\Lambda(\mathbf{P}_{m+1})=&\\{\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\in\widehat{\Lambda}(\mathbf{P}_{m+1})\mbox{
; }k_{2},\ldots,k_{m+1}\in\mathbb{N}_{0}\mbox{ with
}\sum_{\ell=2}^{m+1}k_{\ell}\leq\tau_{(i,j)}\\}\\\
&\bigcup\\{\text{\boldmath$\lambda$}_{{k_{2}},\ldots,k_{m+1}}\in\widehat{\Lambda}(\mathbf{P}_{m+1})\mbox{
; }k_{2},\ldots,k_{m+1}\in\mathbb{N}_{0}\mbox{ with
}\sum_{\ell=2}^{m+1}k_{\ell}\leq m-1\\}.\end{array}$

###### Proof.

By definition, we have that
$\Lambda(\mathbf{P}_{m+1})=\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathbb{N}_{0}^{m+1}$.
Note that,
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\in\mathbb{N}_{0}^{m+1}$
if and only if $k_{2},\ldots,k_{m+1}\in\mathbb{N}_{0}^{m+1}$ and
$\sum_{\ell=2}^{m+1}k_{\ell}\leq\tau_{(i,j)}$. And
$\text{\boldmath$\lambda$}_{k_{2},\ldots,k_{m+1}}\in\mathbb{N}_{0}^{m+1}$ if
and only if
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\in\mathbb{N}_{0}^{m}$
and $\sum_{\ell=2}^{m+1}k_{\ell}\leq m-1$. So, the result follows from
Corollary 3.7. ∎

######  Lemma 3.9.

Let
$k_{2},\ldots,k_{m+1},k_{2}^{\prime},\ldots,k_{m+1}^{\prime}\in\mathbb{N}_{0}$
and
$(i,j),(i^{\prime},j^{\prime})\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$.
Then, we have that

  1. (1)

$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\neq\text{\boldmath$\lambda$}_{k_{2}^{\prime},\ldots,k_{m+1}^{\prime}}$;

  2. (2)

if $(k_{2},\ldots,k_{m+1})\neq(k_{2}^{\prime},\ldots,k_{m+1}^{\prime})$, then
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\neq\text{\boldmath$\delta$}_{k_{2}^{\prime},\ldots,k_{m+1}^{\prime}}^{i^{\prime},j^{\prime}}$,
and
$\text{\boldmath$\lambda$}_{k_{2},\ldots,k_{m+1}}\neq\text{\boldmath$\lambda$}_{{k_{2}}^{\prime},\ldots,k_{m+1}^{\prime}}$;
and

  3. (3)

if $(i,j)\neq(i^{\prime},j^{\prime})$, then
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\neq\text{\boldmath$\delta$}_{k_{2}^{\prime},\ldots,k_{m+1}^{\prime}}^{i^{\prime},j^{\prime}}$.

###### Proof.

$(1)$. Suppose that
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}=\text{\boldmath$\lambda$}_{k_{2}^{\prime},\ldots,k_{m+1}^{\prime}}$
for some
$k_{2},\ldots,k_{m+1},k_{2}^{\prime},\ldots,k_{m+1}^{\prime}\in\mathbb{N}_{0}$.
Then, $k_{2}(q+1)M+iM+j=k_{2}^{\prime}(q+1)M$, and we have that
$j=[(k_{2}^{\prime}-k_{2})(q+1)-i]M$, a contradiction, since
$(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$.

$(2)$. Let
$(k_{2},\ldots,k_{m+1})\neq(k_{2}^{\prime},\ldots,k_{m+1}^{\prime})$. So,
there is $t\in\\{2,\ldots,m+1\\}$ such that $k_{t}\neq k_{t}^{\prime}$.
Suppose that
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}=\text{\boldmath$\delta$}_{k_{2}^{\prime},\ldots,k_{m+1}^{\prime}}^{i^{\prime},j^{\prime}}$.
Then, $k_{t}(q+1)M+iM+j=k_{t}^{\prime}(q+1)M+i^{\prime}M+j^{\prime}$. Thus, we
have $j^{\prime}-j=[(k_{t}-k_{t}^{\prime})(q+1)+i-i^{\prime}]M$, a
contradiction, since $k_{t}-k_{t}^{\prime}\neq 0$ and
$(i,j),(i^{\prime},j^{\prime})\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$.

It is clear that
$\text{\boldmath$\lambda$}_{k_{2},\ldots,k_{m+1}}\neq\text{\boldmath$\lambda$}_{{k_{2}}^{\prime},\ldots,k_{m+1}^{\prime}}$,
since $k_{t}(q+1)M\neq k_{t}^{\prime}(q+1)M$.

$(3)$. Suppose that
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}=\text{\boldmath$\delta$}_{k_{2}^{\prime},\ldots,k_{m+1}^{\prime}}^{i^{\prime},j^{\prime}}$.
By previous item, we have that $k_{t}=k_{t}^{\prime}$, for all
$t\in\\{2,\ldots,m+1\\}$. So, we get $j^{\prime}-j=(i-i^{\prime})M$. Now,
since $j,j^{\prime}\in[1,M]$, we have $j=j^{\prime}$ and $i=i^{\prime}$.
Therefore, we conclude that if $(i,j)\neq(i^{\prime},j^{\prime})$, then
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\neq\text{\boldmath$\delta$}_{k_{2}^{\prime},\ldots,k_{m+1}^{\prime}}^{i^{\prime},j^{\prime}}$.
∎

######  Proposition 3.10.

For $(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$, let
$\tau_{(i,j)}:=\lfloor(q^{3}(M-j)+qM(q-i)-p^{b}(q+1)M)/p^{b}(q+1)M\rfloor$ be
as above. Then

$\displaystyle|\Lambda(\mathbf{P}_{m+1})|=\dfrac{(2m-1)!}{(m-1)!m!}+\sum_{i=0}^{q}\sum_{\begin{subarray}{c}j=1\\\
\tau_{(i,j)}\geq
0\end{subarray}}^{M}\dfrac{(\tau_{(i,j)}+m)!}{\tau_{(i,j)}!m!}.$

###### Proof.

Let
$\mathbb{A}:=\\{\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\in\widehat{\Lambda}(\mathbf{P}_{m+1})\mbox{
; }k_{2},\ldots,k_{m+1}\in\mathbb{N}_{0}\mbox{ with
}\sum_{\ell=2}^{m+1}k_{\ell}\leq\tau_{(i,j)}\\}$ and
$\mathbb{B}:=\\{\text{\boldmath$\lambda$}_{{k_{2}},\ldots,k_{m+1}}\in\mathbb{N}_{0}^{m+1}\mbox{
; }k_{2},\ldots,k_{m+1}\in\mathbb{N}_{0}\mbox{ such that
}\sum_{\ell=2}^{m+1}k_{\ell}\leq m-1\\}.$ Thus, by Corollary 3.8,
$\Lambda(\mathbf{P}_{m+1})=\mathbb{A}\cup\mathbb{B}$. For each
$\ell\in\\{2,\ldots,m+1\\}$, $k_{\ell}(q+1)M+iM+j\geq 0$ if and only if
$k_{\ell}\geq 0$. Now, note that if $\tau_{(i,j)}<0$, then
$\text{\boldmath$\delta$}_{k_{2},\ldots,k_{m+1}}^{i,j}\notin\mathbb{A}$, since
there are not $k_{2},\ldots,k_{m+1}\in\mathbb{N}_{0}\mbox{ with
}\sum_{\ell=2}^{m+1}k_{\ell}\leq\tau_{(i,j)}$. So, using Lemma 3.9 and the
number of non-negative integer solutions ${k_{2}},\ldots,k_{m+1}$ to
$\sum_{\ell=2}^{m+1}k_{\ell}\leq\tau_{(i,j)}$ and
$\sum_{\ell=2}^{m+1}k_{\ell}\leq m-1$ we can conclude that
$\displaystyle|\mathbb{A}|=\sum_{i=0}^{q}\sum_{\begin{subarray}{c}j=1\\\
\tau_{(i,j)}\geq
0\end{subarray}}^{M}\dfrac{(\tau_{(i,j)}+m)!}{\tau_{(i,j)}!m!}$ and
$|\mathbb{B}|=\dfrac{(2m-1)!}{(m-1)!m!}$. Now, by Lemma 3.9, we have that
$\mathbb{A}\cap\mathbb{B}=\emptyset$ and the result follows. ∎

By Theorem 2.10, we have that $\displaystyle
G(\mathbf{P}_{m+1})=\bigcup_{\text{\boldmath$\beta$}\in\Lambda(\mathbf{P}_{m+1})}(\overline{\nabla}(\text{\boldmath$\beta$})\cap\mathbb{N}_{0}^{m+1})$.
In the follow we present a bound for the cardinality of $G(\mathbf{P}_{m+1})$.

######  Proposition 3.11.

Let $\Lambda_{m+1}:=|\Lambda(\mathbf{P}_{m+1})|$ and suppose that
$\Lambda(\mathbf{P}_{m+1})=\\{\text{\boldmath$\beta$}_{1},\ldots,\text{\boldmath$\beta$}_{\Lambda}\\}$,
with $\text{\boldmath$\beta$}_{k}=(\beta_{1}^{(k)},\ldots,\beta_{m+1}^{(k)})$
for each $k\in\\{1,\ldots,\Lambda\\}$. Then

$\displaystyle|G(\mathbf{P}_{m+1})|\leq\sum_{k=1}^{\Lambda_{m+1}}\left(\sum_{r=1}^{m+1}\prod_{s\neq
r}\beta_{s}^{(k)}\right).$

###### Proof.

For each $\text{\boldmath$\beta$}_{k}\in\Lambda(\mathbf{P}_{m+1})$, we have
that
$\displaystyle|\overline{\nabla}(\text{\boldmath$\beta$}_{k})\cap\mathbb{N}_{0}^{m+1}|=\sum_{r=1}^{m+1}\prod_{s\neq
r}\beta_{s}^{(k)}$ and the result follows. ∎

For the particular case $m=1$ we have that

$\displaystyle\Lambda_{2}=|\Lambda(\mathbf{P}_{2})|=1+\sum_{i=0}^{q}\sum_{\begin{subarray}{c}j=1\\\
\tau_{(i,j)}\geq 0\end{subarray}}^{M}\tau_{(i,j)}.$

Given $(m_{1},m_{2}),(n_{1},n_{2})\in\mathbb{N}_{0}^{2}$, we write
$(m_{1},m_{2})\prec(n_{1},n_{2})$ if $m_{2}<n_{2}$.

######  Remark 3.12.

Note that, by Lemma 3.9 and since
$(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$, for all
$\text{\boldmath$\alpha$}=(\alpha_{1},\alpha_{2}),\text{\boldmath$\beta$}=(\beta_{1},\beta_{2})\in\Lambda(\mathbf{P}_{2})$,
with $\text{\boldmath$\alpha$}\neq\text{\boldmath$\beta$}$, we have that
$\alpha_{1}\neq\beta_{1}$ and $\alpha_{2}\neq\beta_{2}$. So, we have
$\text{\boldmath$\alpha$}\prec\text{\boldmath$\beta$}$ or
$\text{\boldmath$\beta$}\prec\text{\boldmath$\alpha$}$.

Let $n\geq 1$ be an integer and let
$A=\\{\text{\boldmath$\alpha$}_{1}\prec\text{\boldmath$\alpha$}_{2}\prec\cdots\prec\text{\boldmath$\alpha$}_{n}\\}\subset\mathbb{N}_{0}^{2}$,
where $\text{\boldmath$\alpha$}_{k}=(\alpha_{1}^{(k)},\alpha_{2}^{(k)})$, for
each $k=1,\ldots,n$. Define $\zeta_{1}(A):=0$ and, for $t=2,\ldots,n$,

(17)
$\zeta_{t}(A):=|\\{\text{\boldmath$\alpha$}_{k}=(\alpha_{1}^{(k)},\alpha_{2}^{(k)})\in
A\mbox{ ; }\text{\boldmath$\alpha$}_{k}\prec\text{\boldmath$\alpha$}_{t}\mbox{
and }\alpha_{1}^{(k)}>\alpha_{1}^{(t)}\\}|.$

######  Proposition 3.13.

Let
$\Lambda(\mathbf{P}_{2})=\\{\text{\boldmath$\beta$}_{1}\prec\text{\boldmath$\beta$}_{2}\prec\cdots\prec\text{\boldmath$\beta$}_{\Lambda_{2}}\\}$,
where, for each $t=1,\ldots,\Lambda_{2}$,
$\text{\boldmath$\beta$}_{t}=(\beta_{1}^{(t)},\beta_{2}^{(t)})$, and
$\zeta_{t}(\Lambda(\mathbf{P}_{2}))$ is given (17). Then

$\displaystyle|G(\mathbf{P}_{2})|=\sum_{t=1}^{\Lambda_{2}}[\beta_{1}^{(t)}+\beta_{2}^{(t)}-\zeta_{t}(\Lambda(\mathbf{P}_{2}))].$

###### Proof.

The results follows directly from Theorem 2.10 and the definition of
$\zeta_{t}(\Lambda(\mathbf{P}_{2}))$. ∎

##  4\. Generalized Weierstrass Semigroup at certain $m+1$ points on
$\mathcal{Y}_{n,s}$

Denoting for simplicity the point $P_{(\alpha_{i},0,0)}\in\mathcal{Y}_{n,s}$
as given in Equation (8) by $P_{i}$, where $1\leq i\leq q$, and taking
$p^{b}=1$ in the equations, results and proofs in the previous section, we get
the similar results for the absolute and relative maximals elements in
$\widehat{H}(\mathbf{P}_{m+1})$ and the minimal generating set of the
Weierstrass semigroup $H(\mathbf{P}_{m+1})$, for $1\leq m\leq q$, where
$\mathbf{P}_{m+1}=(P_{\infty},P_{1},\ldots,P_{m})$. We will summarize the main
results below. The proofs will be omitted because they are analogous to those
presented in the previous section.

######  Theorem 4.1.

Let $1\leq m\leq q$ and $\mathbf{P}_{m+1}=(P_{\infty},P_{1},\ldots,P_{m})$.
Let

$\text{\boldmath$\alpha$}^{i,j,m}:=\left((q^{2}-m)(q+1)M-iqM-
jq^{3},iM+j,\ldots,iM+j\right)\in\mathbb{Z}^{m+1}.$

Then

$\widehat{\Gamma}(\mathbf{P}_{m+1})\cap\mathcal{C}_{m+1}=\\{\text{\boldmath$\alpha$}^{i,j,m}\
:(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\\}\cup\\{{\bf
0}\\}.$

######  Corollary 4.2.

$\begin{array}[]{ll}\widehat{\Gamma}(\mathbf{P}_{m+1})=&\left\\{\left((q^{2}-m-\sum_{\ell=2}^{m+1}k_{\ell})(q+1)M-iqM-
jq^{3},\right.\right.\\\
&k_{2}(q+1)M+iM+j,\ldots,k_{m+1}(q+1)M+iM+j)\in\mathbb{Z}^{m+1}\mbox{ ; }\\\
&(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\mbox{ ,
}k_{\ell}\in\mathbb{Z}\mbox{ for }\ell=2,\ldots,m+1\\}.\end{array}$

For $(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$ and
$k_{2},\ldots,k_{m+1}\in\mathbb{Z}$, let

(18)
$\begin{array}[]{ll}\text{\boldmath$\beta$}_{k_{2},\ldots,k_{m+1}}^{i,j,m}:=&\left((q^{2}-m-\sum_{\ell=2}^{m+1}k_{\ell})(q+1)M-iqM-
jq^{3},\right.\\\
&\left.k_{2}(q+1)M+iM+j,\ldots,k_{m+1}(q+1)M+iM+j\right)\in\mathbb{Z}^{m+1}.\end{array}$

######  Corollary 4.3.

Let $i,j,k_{2},\ldots,k_{m+1}$ and
$\text{\boldmath$\beta$}_{k_{2},\ldots,k_{m+1}}^{i,j,m}$ be as above. Then,
$\Gamma(\mathbf{P}_{m+1})$ is the set of all
$\text{\boldmath$\beta$}_{k_{2},\ldots,k_{m+1}}^{i,j,m}\in\mathbb{N}_{0}^{m+1}$.

######  Theorem 4.4.

Let $1\leq m\leq q$ and $\mathbf{P}_{m+1}=(P_{\infty},P_{1},\ldots,P_{m})$.
Let

$\text{\boldmath$\beta$}^{0,0,m}=\left((m-1)(q+1)M,0,\ldots,0\right)\in\mathbb{Z}^{m+1}$

and for $(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)$, let

$\text{\boldmath$\beta$}^{i,j,m}:=\left((q^{2}-1)(q+1)M-iqM-
jq^{3},iM+j,\ldots,iM+j\right)\in\mathbb{Z}^{m+1}.$

Then

$\widehat{\Lambda}(\mathbf{P}_{m+1})\cap\mathcal{C}(\mathbf{P}_{m+1})=\\{\text{\boldmath$\beta$}^{i,j,m}\
:(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\\}\cup\\{\text{\boldmath$\beta$}^{0,0,m}\\}.$

######  Corollary 4.5.

$\begin{array}[]{ll}\widehat{\Lambda}(\mathbf{P}_{m+1})=&\left\\{\left((q^{2}-1-\sum_{\ell=2}^{m+1}k_{\ell})(q+1)M-iqM-
jq^{3},\right.\right.\\\
&k_{2}(q^{n}+1)M+iM+j,\ldots,k_{m+1}(q^{n}+1)M+iM+j)\in\mathbb{Z}^{m+1}\mbox{
; }\\\ &(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\mbox{ ,
}k_{\ell}\in\mathbb{Z}\mbox{ for }\ell=2,\ldots,m+1\\}\\\
&\bigcup\left\\{(m-1-\sum_{\ell=2}^{m+1}\widetilde{k}_{\ell})(q+1)M,\widetilde{k}_{2}(q^{n}+1)M,\ldots,\widetilde{k}_{m+1}(q^{n}+1)M)\mbox{
; }\widetilde{k}_{\ell}\in\mathbb{Z}\right\\}.\end{array}$

######  Corollary 4.6.

$\begin{array}[]{rl}\Lambda(\mathbf{P}_{m+1})=&\left\\{\left((q^{2}-1-\sum_{\ell=2}^{m+1}k_{\ell})(q+1)M-iqM-
jq^{3},\right.\right.\\\
&k_{2}(q^{n}+1)M+iM+j,\ldots,k_{m+1}(q^{n}+1)M+iM+j)\in\mathbb{Z}^{m+1}\mbox{
; }\\\ &(i,j)\in([0,q]\times[1,M]\cap\mathbb{Z}^{2})\backslash(q,M)\mbox{ ,
}k_{\ell}\in\mathbb{Z}\mbox{ and }k_{\ell}(q^{n}+1)M+iM+j\geq 0\\\ &\mbox{ for
}\ell=2,\ldots,m+1\mbox{ ,and
}(q^{2}-1-\sum_{\ell=2}^{m+1}k_{\ell})(q+1)M-iqM-jq^{3}\geq 0\\}\\\
\bigcup&\\{(m-1-\sum_{\ell=2}^{m+1}\widetilde{k}_{\ell})(q+1)M,\widetilde{k}_{2}(q^{n}+1)M,\ldots,\widetilde{k}_{m+1}(q^{n}+1)M)\mbox{
; }\widetilde{k}_{\ell}\in\mathbb{N}_{0}\mbox{ and }\\\
&\sum_{\ell=2}^{m+1}\widetilde{k}_{\ell}\leq m-1\\}.\end{array}$

## References

  * [1]
  * [2] D. Bartoli, M. Montanucci and G. Zini, Multi point AG codes on the GK maximal curve, Designs, Codes and Cryptography 86 (2018), 161–177.
  * [3] D. Bartoli, M. Montanucci, and G. Zini, _AG codes and AG quantum codes from the GGS curve_ , Designs, Codes and Cryptography 86 (2018), 2315–2344.
  * [4] P. Beelen and N. Tutas, _A generalization of the Weierstrass semigroup_ , J. Pure Appl. Algebra 207 (2006), 243–260.
  * [5] P. Beelen and M. Montanucci, _A new family of maximal curves_ , Journal of the London Math. Soc. 98 (2018), 573–592.
  * [6] M. Brás-Amorós, A.S. Castellanos, Weierstrass semigroup at $m+1$ rational points in maximal curves which cannot be covered by the Hermitian curve, Designs, Codes and Cryptography 88 (2020), 1595–1616.
  * [7] C. Carvalho and T. Kato, On Weierstrass semigroup and sets: a review with new results, Geom. Dedicata 139 (2009), 139–195.
  * [8] C. Carvalho and F. Torres, _On Goppa codes and Weierstrass gaps at several points_ , Designs, Codes and Cryptography 35 (2005), 211–225.
  * [9] A. S. Castellanos, G. Tizziotti, Two-Point AG Codes on the GK Maximal Curves. IEEE Transactions on Information Theory 62 (2016), 681–686.
  * [10] F. Delgado, _The symmetry of the Weierstrass generalized semigroups and affine embeddings_ , Proc. Amer. Math. Soc. 108 (2990), 627–631.
  * [11] I. Duursma, K.H. Mak, On maximal curves which are not Galois subcovers of the Hermitian curve, Bull. Braz. Math. Soc. 43 (2012), 453–465.
  * [12] I. M. Duursma and S. Park, _Delta sets for divisors supported in two points_ , Finite Fields Appl. 18 (2012), 865–885.
  * [13] S. Fanali and M. Giulietti, One-point AG Codes on the GK Maximal Curves, IEEE Trans. on Information Theory 56 (2010), 202–210.
  * [14] S. Fanali and M. Giulietti, Quotient curves of the GK curve, Adv. Geom. 12 (2012), 239– 268.
  * [15] W. Fulton, Algebraic Curves: an introduction to Algebraic Geometry, Addison Wesley, (1969).
  * [16] A. Garcia, C. Güneri, and H. Stichtenoth, _A generalization of the Giulietti-Korchmáros maximal curve_ , Adv. Geom. 10 (2010), 427–434.
  * [17] A. Garcia, S. J. Kim, and R. F. Lax, Consecutive Weierstrass gaps and minimum distance of Goppa codes, J. Pure Appl. Algebra 84 (1993), 199–207.
  * [18] M. Giulietti and G. Korchmáros, _A new family of maximal curves over a finite field_ , Math. Ann. 343 (2009), 229–245.
  * [19] M. Giulietti, M. Montanucci and G. Zini, On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl. 41 (2016), 72–88.
  * [20] C. Hu and S. Yang, _Multi-point Codes from the GGS Curves_ , Advances in Mathematics of Communications 14 (2020), 279–299.
  * [21] C. Kirfel and R. Pellikaan, The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inform. Theory41 (1995), 1720–1732.
  * [22] G. Lachaud, Sommes d’ Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis, C.R. Acad. Sci. Paris 305 (Serie I) (1987), 729–732.
  * [23] L. Landi and L. Vicino, Two-point AG codes from the Beelen-Montanucci maximal curve, preprint, arXiv:2106.14564.
  * [24] G. L. Matthews, The Weierstrass semigroup of an $m$-tuple of collinear points on a Hermitian curve, Lecture note in Comput. Sci., Spinger, Berlin, 2948 (2004), 12–24.
  * [25] G. L. Matthews, Weierstrass semigroups and codes from a quotient of the Hermitian curve, Designs, Codes and Cryptography 37 (2005), 473–492.
  * [26] M. Montanucci and V. Pallozzi Lavorante, AG codes from the second generalization of the GK maximal curve, Discrete Math. 343 (2020), 111810.
  * [27] J. J. Moyano-Fernández, W. Tenório, and F. Torres, Generalized Weierstrass semigroups and their Poincaré series, Finite Fields and Their Applications 58 (2019), 46–69.
  * [28] H. Stichtenoth, Algebraic Function Fields and Codes, Berlin, Germany: Springer, 1993.
  * [29] S. Tafazolian, A. Teherán-Herrera, and F. Torres, Further examples of maximal curves which cannot be covered by the Hermitian curve, J. Pure Appl. Algebra 220 (2016), 1122–1132.
  * [30] W. Tenório and G. Tizziotti, Generalized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables, Finite Fields and Their Applications 57 (2019), 230–248.
  * [31] W. Tenório , G. Tizziotti, _On Weierstrass gaps at several points_ , Bulletin of the Brazilian Math. Society 50 (2019), 543–559.
  * [32] G. Tizziotti and A. S. Castellanos, Weierstrass Semigroup and Pure Gaps at Several Points on the GK Curve. Buletin Brazilian Mathematical Society (ONLINE) 49 (2018), 419–429.


FlexCross: High-Speed and Flexible Packet Processing via a Crosspoint-Queued Crossbar
We acknowledge the financial support from the Bavarian Ministry of Economic Affairs, Regional Development and Energy in the context of the project "6G Future Lab Bavaria".

get arXiv to do 4 passes: Label(s) may have changed. Rerun