On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola

TL;DR

This paper establishes bounds on the number of rational points on certain algebraic curves over finite fields, improving known bounds for chords of polygons inscribed in a hyperbola when specific linear series are Frobenius classical.

Contribution

It provides new upper bounds for rational points on generalized Fermat curves, linking algebraic geometry with combinatorial properties of polygons inscribed in hyperbolas.

Findings

Derived upper bounds for rational points on the curves.

Improved bounds for the number of chords passing through a point in polygons.

Established conditions under which the bounds are tighter.

Abstract

Let $\mathcal{G}$ be the projective plane curve defined over $\mathbb{F}_q$ given by $$aX^nY^n-X^nZ^n-Y^nZ^n+bZ^{2n}=0,$$ where $ab\notin\{0,1\}$, and for each $s\in\{2,\ldots,n-1\}$, let $\mathcal{D}_s^{P_1,P_2}$ be the base-point-free linear series cut out on $\mathcal{G}$ by the linear system of all curves of degree $s$ passing through the singular points $P_1=(1:0:0)$ and $P_2=(0:1:0)$ of $\mathcal{G}$. The present work determines an upper bound for the number $N_q(\mathcal{G})$ of $\mathbb{F}_q$-rational points on the nonsingular model of $\mathcal{G}$ in cases where $\mathcal{D}_s^{P_1,P_2}$ is $\mathbb{F}_q$-Frobenius classical. As a consequence, when $\mathbb{F}_q$ is a prime field, the bound obtained for $N_q(\mathcal{G})$ improves in several cases the known bounds for the number $n_P$ of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point $P$ distinct from its vertices.

Figures

Equations

$$a{X}^n{Y}^n-X^n{Z}^n-Y^n{Z}^n+b{Z}^{2n}=0,$$

$$N_q(\mathcal{X})⩽q+1+2g{q}^{1/2}.$$

$$g(X,Y)=a{X}^n{Y}^n-X^n-Y^n+b=0.$$

$$(\cdots :x^i{y}^j:\cdots ):\mathcal{Y}\to {\mathbb{P}}^N,$$

$$3\cdot (2^{-1}\cdot k{)}^{2/3},$$

$$n_P⩽\frac{k+1}{3}.$$

$${\mathcal{F}}^{\prime }\cdot \mathcal{F}=\sum _{Q\in \mathcal{X}}I(Q,{\mathcal{F}}^{\prime }\cap \eta )Q$$

$$\varphi =(x_0:\cdots :x_M):\mathcal{X}\to \varphi (\mathcal{X})\subset {\mathbb{P}}^M(\overline{{\mathbb{F}}_q}),$$

$$\bigg{\{}\text{div}\,(a_{0}x_{0}+\cdots+a_{M}x_{M})+E\,:\,(a_{0}:\cdots:a_{M})\in\mathbb{P}^{M}(\overline{\mathbb{F}_{q}})\bigg{\}},$$

$$E=\sum _{Q\in \mathcal{X}}{e}_{Q}Q,$$

$$e_Q=-\min \{v_Q(x_0),\dots ,v_Q(x_M)\},$$

$$\bigg{\{}v_{Q}(a_{0}x_{0}+\cdots+a_{M}x_{M})+e_{Q}\,:\,(a_{0}:\cdots:a_{M})\in\mathbb{P}^{M}(\overline{\mathbb{F}_{q}})\bigg{\}}=\bigg{\{}j_{0}(Q),\ldots,j_{M}(Q)\bigg{\}},$$

$$\{j_0(Q),\dots ,j_M(Q)\}=\{{ϵ}_0,\dots ,{ϵ}_M\},$$

$$\det (D_t^{({ϵ}_i)}{x}_j)\ne 0,$$

$$({\nu }_0,\dots ,{\nu }_{M-1}),$$

$$\displaystyle\displaystyle\det\left[\begin{array}[]{ccc}x_{0}^{q}&\cdots&x_{M}^{q}\\ D_{t}^{(\nu_{0})}x_{0}&\cdots&D_{t}^{(\nu_{0})}x_{M}\\ \vdots&&\vdots\\ D_{t}^{(\nu_{M-1})}x_{0}&\cdots&D_{t}^{(\nu_{M-1})}x_{M}\end{array}\right]\neq 0.$$

$$\{{\nu }_0,\dots ,{\nu }_{M-1}\}=\{{ϵ}_0,\dots ,{ϵ}_M\}\setminus \{{ϵ}_I\},$$

$$N_q(\mathcal{X})⩽\frac{({\nu }_0+\cdots +{\nu }_{M-1})\cdot (2g-2)+\delta \cdot (q+M)-\sum A(Q)}{M},$$

$$A(Q)=\left\{\begin{array}[]{ll}\displaystyle\sum_{i=1}^{M}(j_{i}(Q)-\nu_{i-1})-M,\mbox{ for {$Q\in\mathcal{X}(\mathbb{F}_{q})$}}\\ \displaystyle\sum_{i=0}^{M-1}(j_{i}(Q)-\nu_{i}),\mbox{ otherwise.}\end{array}\right.$$

$$g(X,Y)=a{X}^n{Y}^n-X^n-Y^n+b=0.$$

$${\phi }_s^{P_1,P_2}=(\cdots :x^i{y}^j:\cdots ):\mathcal{Y}\to {\mathbb{P}}^N(\overline{{\mathbb{F}}_q}),$$

$$\bigg{\{}i+jn\,:\,0\leqslant i,j\leqslant s-1\text{ and }0\leqslant i+j\leqslant s\bigg{\}}.$$

$$\displaystyle\bigg{\{}i+j(n+1)-1\,:\,0\leqslant i,j\text{ and }0\leqslant i+j\leqslant s\bigg{\}}\setminus\bigg{\{}-1,s(n+1)-1\bigg{\}}.$$

$$\tilde{\mathcal{D}_{s}}^{P_{1},P_{2}}=\bigg{\{}\mathcal{F}\cdot\mathcal{G}\,:\,\mathcal{F}\in\Sigma^{P_{1},P_{2}}_{s}\bigg{\}}$$

$${\ell }_{\infty }\cdot \mathcal{G}=Q_1^{(1)}+\cdots +Q_n^{(1)}+Q_1^{(2)}+\cdots +Q_n^{(2)}$$

$$\displaystyle\mathcal{D}_{s}^{P_{1},P_{2}}=\tilde{\mathcal{D}_{s}}^{P_{1},P_{2}}-\ell_{\infty}\cdot\mathcal{G}=\bigg{\{}D-\ell_{\infty}\cdot\mathcal{G}\,:\,D\in\tilde{\mathcal{D}_{s}}^{P_{1},P_{2}}\bigg{\}}$$

$$\bigg{\{}i+jn\,:\,0\leqslant i,j\leqslant s-1\text{ and }0\leqslant i+j\leqslant s\bigg{\}}.$$

$$\bigg{\{}i+j(n+1)-1\,:\,0\leqslant i,j\text{ and }0\leqslant i+j\leqslant s\bigg{\}}\setminus\bigg{\{}-1,s(n+1)-1\bigg{\}},$$

$$N_q(\mathcal{G})$$

$$(N-1)\cdot (n^2-2n)+\frac{\delta \cdot (q+N)}{N}$$

Full text

[TABLE]

where $ab\notin\{0,1\}$, and for each $s\in\{2,\ldots,n-1\}$, let $\mathcal{D}_{s}^{P_{1},P_{2}}$ be the base-point-free linear series cut out on $\mathcal{G}$ by the linear system of all curves of degree $s$ passing through the singular points $P_{1}=(1:0:0)$ and $P_{2}=(0:1:0)$ of $\mathcal{G}$. The present work determines an upper bound for the number $N_{q}(\mathcal{G})$ of $\mathbb{F}_{q}$-rational points on the nonsingular model of $\mathcal{G}$ in cases where $\mathcal{D}_{s}^{P_{1},P_{2}}$ is $\mathbb{F}_{q}$-Frobenius classical. As a consequence, when $\mathbb{F}_{q}$ is a prime field, the bound obtained for $N_{q}(\mathcal{G})$ improves in several cases the known bounds for the number $n_{P}$ of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point $P$ distinct from its vertices.

Mathematics Subject Classifications (2010): 11G20, 14G05, 51E15.

1 Introduction

Let $\mathbb{F}_{q}$ be the finite field with $q=p^{m}$ elements, where $p$ is a prime number, and let $\mathcal{X}$ be a (projective, nonsingular, geometrically irreducible, algebraic) curve of genus $g$ defined over $\mathbb{F}_{q}$. A fundamental problem in the theory of curves over finite fields is estimating the number $N_{q}(\mathcal{X})$ of $\mathbb{F}_{q}$-rational points on $\mathcal{X}$. Apart from a few classes of curves (see [2], [13]), there is usually no explicit formula for $N_{q}(\mathcal{X})$. Nevertheless, some effective upper bounds for this number can be found in the literature. A famous example, given by the Hasse-Weil Theorem, is

[TABLE]

Another noteworthy approach to bound $N_{q}(\mathcal{X})$ was established by Stöhr and Voloch in $1986$ [14]. Their method, more geometric in nature, provides bounds that are dependent on the choice of an embedding of the curve in some $\mathbb{P}^{M}$, and which improve (1) in several circumstances (see [6], [14]).

For $a,b\in\mathbb{F}_{q}$ satisfying $ab\notin\{0,1\}$, let $\mathcal{G}$ be the projective plane curve with affine equation given by

[TABLE]

This provides an example of a generalized Fermat curve, having recently been studied from the point of view of its automorphism group [3].

The number $N_{q}(\mathcal{G})$ of $\mathbb{F}_{q}$-rational points on the nonsingular model $\mathcal{Y}$ of $\mathcal{G}$ was first investigated in the context of Finite Geometry to study the number of chords of an affinely regular polygon in $\mathbb{A}^{2}(\mathbb{F}_{q})$ passing through a given point (see [1], [7]).

A nondegenerate $k$-gon in the affine plane $\mathbb{A}^{2}(\mathbb{F}_{q})$ is a set of $k$ pairwise distinct points arranged in a cyclic order in such a way that no three vertices are collinear. Here every $k$-gon is considered nondegenerate. If $A_{1}A_{2}\ldots A_{k}$ is a regular $k$-gon in the Euclidean plane, then a $k$-gon $B_{1}B_{2}\ldots B_{k}$ in the affine plane is affinely regular if the bijection $A_{i}\mapsto B_{i}$ preserves all parallelisms between chords (sides and diagonals).

It is well-known that every affinely regular $k$-gon is inscribed in a conic, and for $p>2$, either $k\mid(q+1)$, or $k\mid(q-1)$, or $k=p$, according to whether the circumscribed conic is an ellipse, hyperbola, or parabola (see [4], [8], [11]). Moreover, if $k$ is large enough with respect to $q$, then the chords of any affinely regular $k$-gon cover most points of $\mathbb{A}^{2}(\mathbb{F}_{q})$. The uncovered points are those remaining of the circumscribed conic and, in some cases, the center of the conic when this is either an ellipse or hyperbola (see [9], [10], [15]). This raises a natural question: what is the number $n_{P}$ of chords of an affinely regular $k$-gon of $\mathbb{A}^{2}(\mathbb{F}_{q})$ passing through a given point $P$ distinct from its vertices?

In the particular case of such a $k$-gon being inscribed in an ellipse (resp. hyperbola), it was shown that determining $n_{P}$ is equivalent to determine the number of rational affine points (resp. the number of rational affine points lying in an appropriate subset of the plane) of a curve of the form of curve $\mathcal{G}$ (see [1], [7]). These relations give a connection between the problem of determining $n_{P}$ and that of studying $N_{q}(\mathcal{G})$.

The primary method used in [1] and [7] to give an upper bound for the number $N_{q}(\mathcal{G})$ was the Stöhr-Voloch Theory. More precisely, in [1] Abatangelo and Korchmáros provided an upper bound for $N_{q}(\mathcal{G})$ based on the choice of an embedding of $\mathcal{Y}$ in $\mathbb{P}^{5}$, and then on the study of its $\mathbb{F}_{q}$-Frobenius (non)classicality. Later on, Giulietti remarkably improved Abatangelo and Korchmáros’ bound by considering a suitable embedding of $\mathcal{Y}$ in $\mathbb{P}^{3}$ [7]. Further, more recently, and also using the Stöhr-Voloch Theory, the number $N_{q}(\mathcal{G})$ was investigated in [5] in the context of generalized Fermat curves.

Accordingly, in terms of the Stöhr-Voloch Theory, for $\overline{\mathbb{F}_{q}}(\mathcal{Y})=\overline{\mathbb{F}_{q}}(\mathcal{G})=\overline{\mathbb{F}_{q}}(x,y)$ and $s\in\{2,\ldots,n-1\}$, the present work determines an upper bound for $N_{q}(\mathcal{G})$ with respect to the morphism

[TABLE]

with $i,j$ integers satisfying $0\leqslant i,j\leqslant s-1$, $0\leqslant i+j\leqslant s$, and $N={{s+2}\choose{2}}-3$, in cases where it is $\mathbb{F}_{q}$-Frobenius classical.

Further, based on techniques developed by Garcia and Voloch in [6, Section 3], and improved by Mattarei in [12], we focus our attention on the case where $q=p$ is prime. In particular, if $n=(p-1)/k\geqslant 3$ is a proper divisor of $p-1$, and $p\leqslant n^{4}/4$, for an affinely regular $k$-gon inscribed in a hyperbola, we obtain that $n_{P}$ is bounded roughly by

[TABLE]

which improves in several cases the upper bound for $n_{P}$ given in [7, Theorem 4.1]:

[TABLE]

This paper is organized as follows. In Section 2, based on results of [14], elements of the Stöhr-Voloch Theory are recalled. In Section 3, the number $N_{q}(\mathcal{G})$ is studied. For each $s\in\{2,\ldots,n-1\}$, an upper bound for $N_{q}(\mathcal{G})$ is given in cases where the morphism (3) is $\mathbb{F}_{q}$-Frobenius classical. Further, the case where $\mathbb{F}_{q}$ is the prime field $\mathbb{F}_{p}$ is addressed in Section 3.1. Finally, the bound given in (4) is presented in Section 4.

Notation

The following notation is used throughout this text.

• •

$\mathbb{F}_{q}$ is the finite field with $q=p^{m}$ elements, with $p$ a prime number.

• •

$\overline{\mathbb{F}_{q}}$ is the algebraic closure of $\mathbb{F}_{q}$.

• •

$P_{1}$ and $P_{2}$ are the points $(1:0:0)$ and $(0:1:0)$ of $\mathbb{P}^{2}(\overline{\mathbb{F}_{q}})$, respectively.

• •

Unless otherwise stated, a curve denotes a projective, geometrically irreducible, algebraic curve.

• •

For plane curves $\mathcal{F}$ and $\mathcal{F}^{\prime}$, where $\mathcal{F}^{\prime}$ is not necessarily irreducible and does not contain $\mathcal{F}$ as a component,

[TABLE]

is the intersection divisor cut out on $\mathcal{F}$ by $\mathcal{F}^{\prime}$, where $\mathcal{X}$ is the nonsingular model of $\mathcal{F}$, and for each $Q\in\mathcal{X}$, $\eta$ is the corresponding branch of $\mathcal{F}$. Further, if $\mathcal{F}$ is defined over $\mathbb{F}_{q}$, then $N_{q}(\mathcal{F})$ is the number of $\mathbb{F}_{q}$-rational points on $\mathcal{X}$.

• •

For a nonsingular curve $\mathcal{X}$ defined over $\mathbb{F}_{q}$, $\overline{\mathbb{F}_{q}}(\mathcal{X})$ is its function field, $\mathbb{F}_{q}(\mathcal{X})$ is its $\mathbb{F}_{q}$-rational function field, $\mathcal{X}(\mathbb{F}_{q})$ is the set of its $\mathbb{F}_{q}$-rational points, and $N_{q}(\mathcal{X})$ is its number of $\mathbb{F}_{q}$-rational points.

2 Preliminaries

In this section, some elements of the Stöhr-Voloch Theory based on [14] are recalled.

Let $\mathcal{X}$ be a nonsingular curve of genus $g$ defined over $\mathbb{F}_{q}$. For a nondegenerate morphism

[TABLE]

where $x_{0},\ldots,x_{M}$ are functions in $\overline{\mathbb{F}_{q}}(\mathcal{X})$, let $\mathcal{D}$ be the corresponding base-point-free linear series of degree $\delta$ and dimension $M$

[TABLE]

where

[TABLE]

[TABLE]

and $v_{Q}$ is the discrete valuation associated to the point $Q\in\mathcal{X}$.

If $Q\in\mathcal{X}$, then

[TABLE]

where $0=j_{0}(Q)<\cdots<j_{M}(Q)$ and $v_{Q}(a_{0}x_{0}+\cdots+a_{M}x_{M})+e_{Q}=j_{M}(Q)$ for exactly one $(a_{0}:\cdots:a_{M})\in\mathbb{P}^{M}(\overline{\mathbb{F}_{q}})$. Further, for almost all points $Q\in\mathcal{X}$

[TABLE]

where $0=\epsilon_{0}<\cdots<\epsilon_{M}$. The sequences $(j_{0}(Q),\ldots,j_{M}(Q))$ and $(\epsilon_{0},\ldots,\epsilon_{M})$ are called the $(\mathcal{D},Q)$-order sequence and the $\mathcal{D}$-order sequence, respectively, with $(\epsilon_{0},\ldots,\epsilon_{M})$ being also defined as the minimal sequence in the lexicographic order for which

[TABLE]

where $t\in\overline{\mathbb{F}_{q}}(\mathcal{X})$ is a separable variable and $D_{t}^{(i)}$ is the $i$-th Hasse derivative with respect to $t$. Additionally, $\mathcal{D}$ (or $\phi$) is called classical if $(\epsilon_{0},\ldots,\epsilon_{M})=(0,\ldots,M)$, and nonclassical otherwise.

If $\phi$ is defined over $\mathbb{F}_{q}$, and $t\in\mathbb{F}_{q}(\mathcal{X})$, another important sequence related to $\mathcal{D}$ is the $\mathbb{F}_{q}$-Frobenius order sequence

[TABLE]

which is the minimal sequence in the lexicographic order such that

[TABLE]

This sequence satisfies

[TABLE]

for some $I>0$. In this context, $\mathcal{D}$ (or $\phi$) is called $\mathbb{F}_{q}$-Frobenius classical if $(\nu_{0},\ldots,\nu_{M-1})=(0,\ldots,M-1)$. Otherwise, it is called $\mathbb{F}_{q}$-Frobenius nonclassical.

The following result establishes an useful condition for the classicality and $\mathbb{F}_{q}$-Frobenius classicality of $\mathcal{D}$.

Corollary 2.1** ([14], Corollaries 1.8 and 2.7).**

If $\delta<p$, then $\mathcal{D}$ is classical and $\mathbb{F}_{q}$-Frobenius classical.

In light of the previous considerations, this section ends with the following upper bound for the number $N_{q}(\mathcal{X})$, which is a refinement of [14, Theorem 2.13] obtained from remarks at the beginning of [14, Section 3].

Theorem 2.2** (Stöhr-Voloch).**

[TABLE]

where

[TABLE]

3 The curve $\mathcal{G}$

For $a,b\in\mathbb{F}_{q}$ satisfying $ab\notin\{0,1\}$, let $\mathcal{G}$ be the plane curve defined over $\mathbb{F}_{q}$, with affine equation given by

[TABLE]

The following result, which provides basic information about curve $\mathcal{G}$, can be found in [1, Proposition 3.3].

Proposition 3.1**.**

Let $n$ be a divisor of $q-1$. Then, the following holds:

$\mathcal{G}$* is geometrically irreducible.* 2. 2.

The genus of $\mathcal{G}$ is $(n-1)^{2}$. 3. 3.

The only singular points of $\mathcal{G}$ are $P_{1}$ and $P_{2}$, and each singularity is ordinary with multiplicity $n$. Also, the tangent lines to $\mathcal{G}$ at $P_{1}$ and $P_{2}$ are given by the affine equations $Y=c$ and $X=c$, respectively, where $c^{n}=a^{-1}$, and those tangent lines intersect $\mathcal{G}$ at the corresponding points with multiplicity $2n$. 4. 4.

The intersection multiplicity of a branch centered at $P_{1}$ or $P_{2}$ with its tangent line is $n+1$. 5. 5.

The points $(\xi:0:1)$ and $(0:\xi:1)$, with $\xi^{n}=b$, are inflection points of $\mathcal{G}$. Further, the tangent lines to $\mathcal{G}$ at $(\xi:0:1)$ and $(0:\xi:1)$ are given by the affine equations $X=\xi$ and $Y=\xi$, respectively, and those tangent lines intersect $\mathcal{G}$ at the corresponding points with multiplicity $n$.

Let $\mathcal{Y}$ be the nonsingular model of $\mathcal{G}$ and $\overline{\mathbb{F}_{q}}(\mathcal{Y})=\overline{\mathbb{F}_{q}}(\mathcal{G})=\overline{\mathbb{F}_{q}}(x,y)$. For each $s\in\{2,\ldots,n-1\}$, consider the nondegenerate morphism

[TABLE]

with $i,j$ integers satisfying $0\leqslant i,j\leqslant s-1$ and $0\leqslant i+j\leqslant s$, which corresponds to the base-point-free linear series $\mathcal{D}_{s}^{P_{1},P_{2}}\subset\text{Div}\,(\mathcal{Y})$ of dimension $N={{s+2}\choose{2}}-3$ cut out on $\mathcal{G}$ by all curves (not necessarily irreducible) of degree $s$ passing through the singularities $P_{1}$ and $P_{2}$ of $\mathcal{G}$.

The following proposition presents some important facts related to the linear series $\mathcal{D}_{s}^{P_{1},P_{2}}$.

Proposition 3.2**.**

For each $s\in\{2,\ldots,n-1\}$, the following occurs:

$\mathcal{D}_{s}^{P_{1},P_{2}}$* has degree $\delta=2n\cdot(s-1)$.* 2. 2.

For $Q\in\mathcal{Y}$ corresponding to a point of $\mathcal{G}$ of the form $(\xi:0:1)$ or $(0:\xi:1)$, with $\xi^{n}=b$, the $(\mathcal{D}_{s}^{P_{1},P_{2}},Q)$-order sequence is given by the elements of

[TABLE] 3. 3.

For $Q_{\eta}\in\mathcal{Y}$ corresponding to a branch $\eta$ of $\mathcal{G}$ centered at $P_{1}$ or $P_{2}$, the $(\mathcal{D}_{s}^{P_{1},P_{2}},Q_{\eta})$-order sequence is given by the elements of

[TABLE]

Proof.

For each $s\in\{2,\ldots,n-1\}$, let $\Sigma^{P_{1},P_{2}}_{s}$ be the linear system of all curves (not necessarily irreducible) of degree $s$ passing through the singularities $P_{1}$ and $P_{2}$ of $\mathcal{G}$, and let

[TABLE]

be the linear series cut out on $\mathcal{G}$ by the linear system $\Sigma^{P_{1},P_{2}}_{s}$. By Bézout’s Theorem, $\tilde{\mathcal{D}_{s}}^{P_{1},P_{2}}$ has degree $2ns$.

The base locus of $\tilde{\mathcal{D}_{s}}^{P_{1},P_{2}}$ is the divisor

[TABLE]

of degree $2n$, where the $Q_{j}^{(i)}$’s are all the distinct points on $\mathcal{Y}$ for which the corresponding branches are centered at $P_{i}$, for $i=1,2$ and $j=1,\ldots,n$, and $\ell_{\infty}$ is the line given by the equation $Z=0$.

Therefore,

[TABLE]

has degree $2ns-2n=2n\cdot(s-1)$, which proves statement 1.

To prove statement 2, let $Q\in\mathcal{Y}$ be a point corresponding to $Q^{\prime}\in\mathcal{G}$, where $Q^{\prime}$ is equal to $(\xi:0:1)$ or $(0:\xi:1)$, with $\xi^{n}=b$, and let $\ell_{i}=\overline{Q^{\prime}P_{i}}$, for $i=1,2$. One can verify that none of the lines $\ell_{1}$ and $\ell_{2}$ is equal to $\ell_{\infty}$, and exactly one of them is the tangent line to $\mathcal{G}$ at $Q^{\prime}$. Further, from Proposition 3.1, the $(\mathcal{D}_{1},Q)$-order sequence is $(0,1,n)$, where $\mathcal{D}_{1}$ is the linear series cut out on $\mathcal{G}$ by the linear system of lines. Therefore, considering the reducible curves $\mathcal{C}$ given by the union of $s$ lines chosen (with multiplicity) in the set $\{\ell_{1},\ell_{2},\ell_{\infty}\}$, the $(\mathcal{D}_{s}^{P_{1},P_{2}},Q)$-order sequence is given by the elements of

[TABLE]

Finally, if $Q_{\eta}\in\mathcal{Y}$ corresponds to a branch $\eta$ of $\mathcal{G}$ centered at $P_{1}$ or $P_{2}$, from Proposition 3.1, the $(\mathcal{D}_{1},Q_{\eta})$-order sequence is $(0,1,n+1)$. Thus, considering the reducible curves $\mathcal{C}$ passing through $P_{1}$ and $P_{2}$, and given by the union of $s$ lines (possibly chosen with multiplicity), equation (7) shows that the $(\mathcal{D}_{s}^{P_{1},P_{2}},Q_{\eta})$-order sequence is given by the elements of

[TABLE]

which completes the proof. ∎

Based on Theorem 2.2 and Proposition 3.2, the following result gives an upper bound for the number $N_{q}(\mathcal{G})$ in cases where $\mathcal{D}_{s}^{P_{1},P_{2}}$ is $\mathbb{F}_{q}$-Frobenius classical.

Corollary 3.3**.**

Let $s\in\{2,\ldots,n-1\}$. If $\mathcal{D}_{s}^{P_{1},P_{2}}$ is $\mathbb{F}_{q}$-Frobenius classical, then

[TABLE]

where

• •

$n_{1}$* and $n_{2}$ are the number of roots in $\mathbb{F}_{q}$ of the polynomials $T^{n}-b$ and $T^{n}-a^{-1}$, respectively*

• •

$\alpha=1+(s-1)\cdot n-N$**

• •

$\beta=(s-1)\cdot(n+1)-N$**

• •

$\gamma=2(n+1)-s\cdot(4n+3)-N\cdot(N-1)+(s\cdot(2n+3)-3)\cdot\frac{N+3}{3}$.

Proof.

The part

[TABLE]

of (8) follows directly from Theorem 2.2, since the genus of $\mathcal{G}$ is $(n-1)^{2}$ by Proposition 3.1.

Further, let $Q\in\mathcal{Y}$ be a point corresponding to $(\xi:0:1)$ or $(0:\xi:1)\in\mathcal{G}$, with $\xi^{n}=b$, and let $Q_{\eta}\in\mathcal{Y}$ be a point corresponding to a branch $\eta$ of $\mathcal{G}$ centered at $P_{1}$ or $P_{2}$. Using the notation as in Theorem 2.2, from Proposition 3.2, the numbers $A(Q)$ and $A(Q_{\eta})$ are given by the following expressions:

[TABLE]

and

[TABLE]

since

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

where

$\bullet\,$ $n_{1}$ and $n_{2}$ are the number of roots in $\mathbb{F}_{q}$ of the polynomials $T^{n}-b$ and $T^{n}-a^{-1}$, respectively

$\bullet\,$ $\alpha=A(Q)^{(\ref{AP1})}-A(Q)^{(\ref{AP2})}=1+(s-1)\cdot n-N$

$\bullet\,$ $\beta=A(Q_{\eta})^{(\ref{APgamma1})}-A(Q_{\eta})^{(\ref{APgamma2})}=(s-1)\cdot(n+1)-N$

$\bullet\,$ $\gamma=A(Q)^{(\ref{AP2})}+A(Q_{\eta})^{(\ref{APgamma2})}=2(n+1)-s\cdot(4n+3)-N\cdot(N-1)+(s\cdot(2n+3)-3)\cdot\frac{N+3}{3}$. ∎

3.1 The case $q=p$

Hereafter, let $\mathbb{F}_{q}$ be the prime field $\mathbb{F}_{p}$. For each $u\in[2,+\infty)$ and $t_{0}\in[6,+\infty)$, let us consider

[TABLE]

and

[TABLE]

An important step in the proof of Theorem 3.5 is the following lemma, whose proof is straightforward.

Lemma 3.4**.**

Let $t_{0}\in[6,+\infty)$. For $u\in[2,+\infty)$,

[TABLE]

Theorem 3.5**.**

Let $n\geqslant 3$ be a proper divisor of $p-1$ and $k=(p-1)/n$. If p-1\leqslant n\cdot\bigg{(}\frac{(n+3)\cdot(n+4)\cdot(3n-1)}{12}-3\bigg{)}, then

[TABLE]

Proof.

From Corollary 3.3 and its proof,

[TABLE]

for each $s\in\{2,\ldots,n-1\}$ satisfying $2n\cdot(s-1)\leqslant p-1$, since the latter condition, together with Corollary 2.1 and Proposition 3.2, implies that $\mathcal{D}_{s}^{P_{1},P_{2}}$ is $\mathbb{F}_{p}$-Frobenius classical. Hence, defining $t:=s+4$, it follows that

[TABLE]

for all $t\in\{6,\ldots,n+3\}$ satisfying $t\leqslant\frac{k}{2}+5$, and

[TABLE]

One can check that $k<k_{t_{0}}$, for all integers $t_{0}\geqslant\frac{k}{2}+5$. Thus, an iterated application of Lemma 3.4 yields

[TABLE]

Likewise, from the assumption $k\leqslant k_{n+3}=\frac{(n+3)\cdot(n+4)\cdot(3n-1)}{12}-3$, we further have

[TABLE]

The objective is to determine a suitable function $W$ such that $W(k)\geqslant V(k)$, for all prime numbers $p$ and for all proper divisors $n\geqslant 3$ of $p-1$, where $k=(p-1)/n$. Then, defining for each $u\in[2,+\infty)$

[TABLE]

Lemma 3.4 implies that

[TABLE]

and that

[TABLE]

Thus one may choose a convenient concave function $W$ such that $W(k_{t_{0}})\geqslant\tilde{V}(k_{t_{0}})$ for all $t_{0}\in\mathbb{Z}\cap[6,+\infty)$, and such that $W(u)\geqslant\tilde{V}(u)$, for all integers $2\leqslant u<k_{6}$. Indeed, fixed $u\in[k_{6},+\infty)$, if $k_{t_{0}}\leqslant u\leqslant k_{t_{0}+1}$, for some $t_{0}\in\mathbb{Z}\cap[6,+\infty)$, let $u=u_{1}k_{t_{0}}+u_{2}k_{t_{0}+1}$, where $0\leqslant u_{1},u_{2}\leqslant 1$ and $u_{1}+u_{2}=1$. Then

[TABLE]

Let us consider the family of concave functions $W_{\lambda}$ defined in $[2,+\infty)$ by

[TABLE]

where $\lambda$ is a constant.

Direct computation shows that the smallest $\lambda$ such that $W_{\lambda}(k_{t_{0}})\geqslant\tilde{V}(k_{t_{0}})$ for all $t_{0}\in\mathbb{Z}\cap[6,+\infty)$, and such that $W_{\lambda}(u)\geqslant\tilde{V}(u)=f_{u}(6)$ for all integers $2\leqslant u<k_{6}$, is

[TABLE]

Therefore, the proof completes considering $W$ in $[2,+\infty)$ defined as $W(u)=3\cdot(2^{1/2}\cdot u)^{2/3}-\frac{103}{19}\cdot(2^{1/2}\cdot u)^{1/3}+\frac{13}{3}.$ ∎

4 Number of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point

Let $A=A_{1}A_{2}\ldots A_{k}$ be an affinely regular polygon with $k=(p-1)/n$ vertices inscribed in a hyperbola $\mathcal{H}$ of $\mathbb{A}^{2}(\mathbb{F}_{p})$. Changing variable, one may consider $\mathcal{H}$ given by the equation $XY=1$.

For $a,b\in\mathbb{F}_{p}$ satisfying $ab\notin\{0,1\}$, let $\mathcal{G}$ be the plane curve defined over $\mathbb{F}_{p}$, with affine equation

[TABLE]

The following result provides a relation between the number of chords of $A$ passing through the point $P=(a,b)\in\mathbb{A}^{2}(\mathbb{F}_{p})$ and the number $N_{p}(\mathcal{G})$.

Proposition 4.1** ([7], Proposition 2.1).**

Let $N_{p}$ be the number of $\mathbb{F}_{p}$-rational affine points of $\mathcal{G}$ not lying in the coordinate axes or the line of equation $X=Y$. Then $n_{P}=\frac{N_{p}}{2n^{2}}$.

Therefore, from Theorem 3.5 and Proposition 4.1, the following upper bound for $n_{P}$ is obtained.

Corollary 4.2**.**

If $n\geqslant 3$ is a proper divisor of $p-1$ such that p-1\leqslant n\cdot\bigg{(}\frac{(n+3)\cdot(n+4)\cdot(3n-1)}{12}-3\bigg{)}, then

[TABLE]

Remark 4.3**.**

One can check that bound (16) improves that given in (5) for $k\geqslant 44$. On the other hand, for each $2\leqslant k<44$, bounds (5) and (16) effectively differ by at most one unit. Moreover, considering the definition of $V(k)=\tilde{V}(k)$ as given in the proof of Theorem 3.5, for $25<k<44$, we obtain that $n_{P}\leqslant\lfloor\frac{1}{2}V(k)\rfloor\leqslant\lfloor\frac{k+1}{3}\rfloor$, with $\frac{k+1}{3}$ as in (5). For an illustration of the region where (16) is better than the bound for $n_{P}$ derived from the Hasse-Weil bound, see also Figure 1.

Aknowlegments

The first author was supported by FAPESP (Brazil), grant 2017/04681-3. The study of the second author was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and CNPq (Brazil), grant 154359/2016-5.