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# Cascaded Residual Density Network for Crowd Counting

###### Abstract

Crowd counting is a challenging task due to the issues such as scale variation
and perspective variation in real crowd scenes. In this paper, we propose a
novel Cascaded Residual Density Network (CRDNet) in a coarse-to-fine approach
to generate the high-quality density map for crowd counting more accurately.
(1) We estimate the residual density maps by multi-scale pyramidal features
through cascaded residual density modules. It can improve the quality of
density map layer by layer effectively. (2) A novel additional local count
loss is presented to refine the accuracy of crowd counting, which reduces the
errors of pixel-wise Euclidean loss by restricting the number of people in the
local crowd areas. Experiments on two public benchmark datasets show that the
proposed method achieves effective improvement compared with the state-of-the-
art methods.

Index Terms—  Crowd counting, Scale variation, CRDNet, Local count loss

##  1 Introduction

Fig. 1: The network structure of CDRNet, which consists of the encoder (PNet)
and the decoder (CNet). PNet generates the pyramid of multi-scale features.
And CNet estimates the density map through the cascaded residual density
modules.

With the extensive use of urban monitoring systems, crowd counting is becoming
increasingly important in the field of public security and attracts a lot of
attention in recent years. Estimating the number of crowds accurately can
effectively solve the problems of crowd control, crowd gathering monitoring
and overload warning in practical applications. Furthermore, the method of
crowd density estimation and counting can be extended to other fields like
vehicle counting, urban building density estimation, etc. However, it is still
a challenging work to generate the high-quality crowd density map for crowd
counting. The major obstacle comes from the large scale variation in
surveillance videos.

Recent works based on Convolutional Neural Networks utilize multi-scale
architectures to address the problem of scale variation and have made great
progress in crowd density estimation. For example, the multi-column networks
[1, 2, 3] were proposed by using multiple parallel columns with different
filter sizes (large, medium, small). In such networks, the different column
CNN learned different scale features to adaptive to a large variation in
people or head size due to the camera perspective. Besides, the Multi-scale
Network [4] based on the Inception-like Module was used to extract the scale-
relevant features which could generate the crowd density map with various
scales.

However, the above methods suffer inherent disadvantages. On the one hand,
those algorithms utilize different filter sizes to extract corresponding
features at different scales, but the performance is restricted by the types
of filter. It may result in a lower quality of the estimated density map. On
the other hand, most of the algorithms adopt pixel-wise Euclidean loss to
optimize networks. As we know, the Euclidean loss is sensitive to outliers and
blur the generated image [5], which may affect the accuracy of crowd counting.

In this paper, to deal with these issues, we propose a novel crowd counting
network named Cascade Residual Density Network (CRDNet), which adopts U-Net
architecture [6] that consists of an encoder and a decoder. Unlike previous
approaches, where multiple filters with different kernel sizes are needed to
extract multi-scale features for estimating crowd density map, the proposed
CRDNet approach utilizes pyramidal feature extraction inspired by [7] in the
encoder, which has more scale context information including strong semantic
features and weak semantic features at different levels from a single input
image scale. More importantly, in the proposed decoder, the high-quality crowd
density map is estimated by summing up the residual density map in a coarse-
to-fine framework. At each level of the pyramid, the proposed residual density
module learns the residual density map for correcting the estimated density
map of the previous level. Firstly, the stacked residual modules integrate the
features at all scales naturally, which can be more robust to scale variation
in the task of crowd counting. Secondly, the cascaded structure progressively
improves the resolution and quality of density maps by estimating the residual
density maps without passing more errors to the next level. Therefore the
proposed architecture estimates the crowd density map layer by layer that can
improve the accuracy of crowd counting effectively.

Furthermore, considering that the Euclidean loss only focuses on the errors of
individual pixels, it is usually affected by outlier pixels. In this paper, an
additional local count loss, which is defined as the errors of crowd counting
between the estimated density map and the ground truth in local areas, is
proposed to describe the accuracy of the total number of the local crowd
rather than the single pixel. It can regularize the estimated density map and
reduce the counting error caused by the Euclidean loss. The experiment results
demonstrate that it can greatly improve the performance. In summary, the key
highlights of the work are:

(1) The cascaded density residual network is proposed to generate the high-
quality crowd density map from coarse to fine through stacked residual density
modules in a pyramid of multi-scale features.

(2) An additional local count loss is designed to minimize the errors of local
crowd counting not only the pixel value, which constrains the Euclidean
distance between estimated density map and the ground truth pixels. It
improves the whole quality of estimated crowd density maps effectively.

##  2 Related works

In recent years, various approaches have been proposed in the literature for
crowd counting. Early traditional crowd counting methods that use hand-crafted
features can be classified into the two categories: (1) Detection-based
methods, (2) Regression-based methods.

In detection-based methods, the number of crowds is calculated by detecting
the individual entities in the scene [8, 9, 10, 11]. Although such approaches
have achieved good results in low-density scenes, the problem of occlusion
among people in the high-density crowds adversely affects the performance of
accurate detection. In regression-based methods, researchers avoid detecting
the people in the scene but turn the crowd counting into regression problem by
learning a mapping between holistic or local features extracted from images to
their count [12, 13, 14]. The regression techniques include linear regression,
ridge regression, Gaussian regression etc.

More recently, a variety of CNN-based methods for crowd counting have achieved
significant improvement and outperformed traditional methods with handcrafted
features due to the powerful feature characterization of CNN. Zhang et al.
[15] trained a CNN to regress the crowd density map, and fine-tuned the
trained network to adapt to a new scene by retrieving image samples that were
similar to the new scene. But this method requires perspective maps to solve
the large scale and perspective variations in a crowd scene, which limits many
practical applications. Zhang et al. [1] proposed a multi-column network
(MCNN) to extract features for crowd counting without perspective maps. The
network of different columns corresponding to different filter sizes (large,
medium, small) ensured the robustness to large variation in object scales.
Instead of training a multi-column network on all the input images, Sam et al.
[2] proposed an extra switching CNN that was trained to select an independent
CNN regressor suited for a given input. The independent CNN regressors were
designed with different filter sizes for different scales. Similarly, Onoro et
al. [16] proposed a scale-aware crowd counting network named Hydra CNN that
learned a multi-scale non-linear regression model by using a pyramid of image
patches extracted at multiple scales. In addition, Li et al [17] utilized
dilated kernels to deliver larger reception fields for crowd counting tasks.
Ranjan et al [18] tackled the problem of crowd counting by combing low-
resolution CNN branch and high-resolution CNN branch respectively.

##  3 Cascaded Residual Density Network

The Cascade Residual Density Network consists of pyramidal feature extraction
and Cascade density map estimation, named PNet and CNet respectively as shown
in Fig. 1. The PNet extracts pyramids of multi-scale features from the
different layers of the backbone network to represent the scale diversity of
features. And the CNet consists of cascaded residual density modules that
learn coarse-to-fine density map for crowd counting.

###  3.1 Pyramidal Feature Extraction

As shown in Fig. 1, PNet is a feature extractor, which utilizes the CSRnet
[17] based on VGG16 [19] as the backbone. We extract the multi-scale pyramidal
features $F_{k}$ from the mid layers of PNet ($conv1\\_2$, $conv2\\_2$,
$conv3\\_3$, $conv5\\_6$), where $k$ denotes the level of pyramid. In this
paper, we construct a four-layer feature pyramid to adapt to the scale
variances in crowd scenes. The resolutions of pyramidal features reduce by a
factor $s$ from top ($k=1$) to bottom ($k=4$) due to the operations of pooling
and stride convolutions.

###  3.2 Cascaded Density Map Estimation

CNet is composed of multiple residual density modules as shown in Fig. 1. At
each pyramid level of PNet, the residual density map is estimated by the
corresponding residual density module, and the cascaded structure further
improves the quality of density map from low-resolution to high-resolution.

Fig. 2: The residual density module $G_{k}$

As shown in Fig. 2, at the $k$-th level of pyramid, the residual density
module $G_{k}$ transforms the upsampled density map ${D^{\prime}_{k}}$ from
the previous level $D_{k-1}$ and the $k$-th level pyramidal features $F_{k}$
to the residual density map $R_{k}$. $D_{0}$ represents the crowd density map
initialized to zero. The calculation process is as follows:

$R_{k}=C_{k}({f(u_{s}(D_{k-1}),F_{k})})$ (1)

where ${u_{s}}(\cdot)$ denotes the function of upsampling by a factor $s$
using bilinear interpolation. $f(\cdot)$ represents that features are
concatenated in the channel dimension. $C_{k}$ denotes the convolution
operation with the filter of size $1\times 1$.

Then, the density map $D_{k}$ at the $k$-th pyramid level can be represented
as:

$D_{k}=u_{s}(D_{k-1})+R_{k}$ (2)

The estimated $D_{k}$ is similarly passed on to the next higher resolution
level until we generate the final density map $D_{1}$ with the same size as
the input image. In summary, the final estimated density map $D_{1}$ is
ceaselessly refined by adding residual density maps of various scales in the
cascaded network, which enhances the quality of crowd density maps.

###  3.3 Crowd Counting Loss

Euclidean Loss. The pixel-wise Euclidean loss is used to measure the distance
between the estimated density map and the ground truth in most works for crowd
counting. The Euclidean loss function $L_{E}(\Theta)$ is defined as follows:

$L_{E}(\Theta)={1\over
M}\sum\limits_{j=1}^{M}{{{\left\|{{D_{Q}}({m_{j}};\Theta)-{Q_{j}}}\right\|}^{2}}}$
(3)

where $\Theta$ is the set of parameters of the CRDNet model. $M$ denotes the
number of training patch images. $m_{j}$ is the input image of the network.
$D_{Q}$ denotes the estimated density map, and $Q_{j}$ is the corresponding
ground truth density map.

Fig. 3: The estimated density map $D_{Q}$ (a) and the ground truth $Q$ (b)

Local Count Loss. The proposed local count loss is used to calculate the
difference of crowd counting between the estimated density map and the ground
truth in local areas of the input image. The loss measures the number of
people in the local crowd rather than a single pixel, which can optimize the
network and pay more attention to the details of local areas on the basis of
Euclidean loss. Then it can be more beneficial to the overall crowd counting.
The local count loss ($L_{Y}(\Theta)$ is defined as follows:

${c_{j}}(\textbf{x}_{i}|\Theta)=\sum\limits_{\sigma\in\left[{0,h}\right]\times\left[{0,h}\right]}{d_{j}({{\textbf{x}_{i}+\sigma;\Theta}})-q_{j}({{\textbf{x}_{i}+\sigma}})}$
(4) ${L_{Y}(\Theta)}={1\over
M}\sum\limits_{j=1}^{M}{\sum\limits_{i\in\left[{0,H}\right]\times\left[{0,W}\right],stride=t}\left|c_{j}(\textbf{x}_{i}|\Theta)\right|}$
(5)

where $d$ and $q$ denote the local patch of estimated density map $D_{Q}$ and
the ground truth $Q$ respectively as shown in Fig. 3. For each square patch of
size $h$ whose upper left coordinate at $\textbf{x}_{i}$, we calculate the
absolute count error $\left|c_{j}(\textbf{x}_{i}|\Theta)\right|$ for the
$j$-th training image with no padding. $t$ represents the stride of the patch.
And the total local count loss $L_{Y}(\Theta)$ is the sum of all patch losses.

Total Loss. We combine the Euclidean loss and the local count loss as the
final crowd counting loss in this paper, and it is defined as follows:

$L(\Theta)=L_{E}(\Theta)+\lambda{L_{Y}}(\Theta)$ (6)

where $\lambda$ denotes the weight of $L_{Y}(\Theta)$. In our experiments, we
empirically set $\lambda$ as 0.0001.

##  4 Experiments

In the training stage, firstly we train each residual density module $G_{k}$
independently to estimate the corresponding crowd density map at the $k$-th
layer of the pyramid. The target residual density map $\tilde{R}_{k}$ is the
difference target density map $H_{k}$ and the estimated density map in the
previous layer. $\tilde{R}_{k}=H_{k}-u_{s}(D_{k-1})$. Finally, we fine-tune
the whole network in the public datasets.

We evaluate the crowd counting methods with the absolute error (MAE) and mean
square error (MSE), which are defined as follows:

$MAE={1\over{N}}\sum\limits_{j=1}^{N}{\left|{{n_{j}}-{{\hat{n}}_{j}}}\right|},MSE=\sqrt{{1\over{N}}\sum\limits_{j=1}^{N}{{{({n_{j}}-{{\hat{n}}_{j}})}^{2}}}}$
(7)

where $N$ is the number of test images. For the $j$-th sample, $n_{j}$ denotes
the ground truth count and $\hat{n}_{j}$ denotes the estimated crowd count
that is the integral of the estimated density map.

The proposed method is tested on two public datasets including ShanghaiTech
dataset and UCF$\\_$CC$\\_$50 dataset. The results show the effectiveness and
the superiority compared with the state-of-the-art methods.

###  4.1 ShanghaiTech Dataset

The ShanghaiTech dataset [1] is divided into two parts: Part$\\_$A and
Part$\\_$B. In the Part$\\_$A sub-dataset, there are 482 images crawled from
the internet, and in the Part$\\_$B sub-dataset, there are 716 images
collected from the busy street.

We compare our method with the existing state-of-the-art methods using MAE and
MSE metrics. In Table 1, it can be observed that the proposed CRDNet can
achieve superior performance compared with those networks with multiple filter
size [1, 2, 4], which demonstrates that the pyramidal features have strong
semantics at all scales.

Table 1: Comparison of CRDNet with other state-of-the-art methods on the ShanghaiTech dataset. Method | Part$\\_$A | Part$\\_$B
---|---|---
MAE |  MSE |  MAE |  MSE
Zhang et al. [15] |  181.8 |  277.7 |  32.0 |  49.8
MCNN [1] |  110.2 |  173.2 |  26.4 |  41.3
Switch-CNN [2] |  90.4 |  135.0 |  21.6 |  33.4
CP-CNN [3] |  73.6 |  106.4 |  20.1 |  30.1
MSCNN [4] |  83.8 |  127.4 |  26.4 |  41.3
CSRNet [17] |  68.2 |  115.0 |  10.6 |  16.0
ic-CNN [18] |  68.5 |  116.2 |  10.7 |  16.0
ACSCP [20] |  75.7 |  102.7 |  17.2 |  27.4
CRDNet (proposed) |  68.5 |  108.4 |  8.5 |  13.6

###  4.2 UCF$\\_$CC$\\_$50 Dataset

The UCF$\\_$CC$\\_$50 dataset [21] consists of 50 images collected from the
publicly available web. We perform 5-fold cross-validation to evaluate the
effectiveness of our proposed CRDNet method. As shown in Table 2, in the high-
density crowd scenes, the proposed CRDNet generates high-quality density maps
by the cascaded network, which makes great progress compared with the state-
of-the-art methods in the MAE metric. In the MSE metric, it can also obtain a
state-of-the-art score.

Table 2: Comparison of CRDNet with other state-of-the-art methods on the UCF$\\_$CC$\\_$50 dataset. Method |  MAE |  MSE
---|---|---
Zhang et al. [15] |  467.0 |  498.5
MCNN [1] |  377.6 |  509.1
Switch-CNN [2] |  318.1 |  439.2
CP-CNN [3] |  295.8 |  320.9
MSCNN [4] |  363.7 |  468.4
CSRNet [17] |  266.1 |  397.5
ic-CNN [18] |  260.9 |  365.5
ACSCP [20] |  291.0 |  404.6
CRDNet (proposed) |  250.7 |  346.6

###  4.3 Loss Function

To evaluate the effectiveness of the local count loss, we compare the
performance of different loss functions on public datasets as shown in Table
3. The results demonstrate that $L_{Y}(\Theta)$ can increase the accuracy of
crowd counting.

Table 3:  Comparisons of errors with different losses on the public benchmark datasets. Loss Function | Part$\\_$A | Part$\\_$B | UCF$\\_$CC$\\_$50
---|---|---|---
MAE | MSE | MAE | MSE | MAE | MSE
$L_{E}$ | 70.3 | 110.2 | 8.5 | 13.7 | 255.6 | 363.1
$L_{E}$, $L_{Y}$ | 68.5 | 108.4 | 8.5 | 13.6 | 250.7 | 346.6

##  5 conclusion

In this paper, the proposed Cascaded Residual Density Network takes advantage
of the feature pyramid structure to extract multi-layer feature maps at
various scales and refines the density map by cascaded coarse-to-fine
architecture. To better improve the accuracy of crowd counting, an additional
local crowd loss is proposed to regularize the estimated density map.
Extensive experiments performed on the ShanghaiTech and UCF$\\_$CC$\\_$50
datasets demonstrate the effectiveness compared with recent state-of-the-art
methods.

Acknowledgement. This work is supported by the National Natural Science
Foundation of China (Grant No. 61371192), the Key Laboratory Foundation of the
Chinese Academy of Sciences (CXJJ-17S044) and the Fundamental Research Funds
for the Central Universities(WK2100330002, WK3480000005).

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# On pairs of $r$-primitive and $k$-normal elements with prescribed traces
over finite fields

Aakash Choudhary  and  R.K. Sharma∗

###### Abstract.

Given $\mathbb{F}_{q^{n}}$, a field with $q^{n}$ elements, where $q$ is a
prime power and $n$ is positive integer. For
$r_{1},r_{2},m_{1},m_{2}\in\mathbb{N}$, $k_{1},k_{2}\in\mathbb{N}\cup\\{0\\}$,
a rational function $F=\frac{F_{1}}{F_{2}}$ in $\mathbb{F}_{q}[x]$ with
deg($F_{i}$) $\leq m_{i}$; $i=1,2,$ satisfying some conditions, and
$a,b\in\mathbb{F}_{q}$, we construct a sufficient condition on $(q,n)$ which
guarantees the existence of an $r_{1}$-primitive, $k_{1}$-normal element
$\epsilon\in\mathbb{F}_{q^{n}}$ such that $F(\epsilon)$ is $r_{2}$-primitive,
$k_{2}$-normal with
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)=a$ and
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon^{-1})=b$. For
$m_{1}=10,\;m_{2}=11,\;r_{1}=3,\;r_{2}=2,\;k_{1}=2,\;k_{2}=1$, we establish
bounds on $q$, for various $n$, to determine the existence of such elements in
$\mathbb{F}_{q^{n}}$. Furthermore, we identify all such pairs $(q,n)$
excluding 10 possible values of $(q,n)$, in fields of characteristics 13.

###### Key words and phrases:

Character, Finite field, $r$-Primitive element, $k$-Normal element
Indian Institute of Technology Delhi, India
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### 1991 Mathematics Subject Classification:

12E20, 11T23

##  1\. Introduction

Given $q$ is a prime power and $n\in\mathbb{N}$, suppose $\mathbb{F}_{q^{n}}$
represent a field extension of degree $n$ over $\mathbb{F}_{q}$. The
multiplicative group $\mathbb{F}_{q^{n}}^{*}$ is cyclic, and its generator is
referred to as primitive element in $\mathbb{F}_{q^{n}}$. Let $r$ be a divisor
of $q^{n}-1$, an element $\epsilon\in\mathbb{F}_{q^{n}}$ with multiplicative
order $\frac{q^{n}-1}{r}$ is referred to as $r$-primitive in
$\mathbb{F}_{q^{n}}$. In fact, a primitive element refers to a 1-primitive
element. It is evident that, for a primitive element
$\epsilon\in\mathbb{F}_{q^{n}}$ and $r\mid q^{n}-1$, the element
$\epsilon^{r}$ is an $r$-primitive element, and there are precisely
$\phi(\frac{q^{n}-1}{r})$ such elements in $\mathbb{F}_{q^{n}}$, where $\phi$
is the Euler-totient function. An element $\epsilon\in\mathbb{F}_{q^{n}}$ is
said to be normal element in $\mathbb{F}_{q^{n}}$ if the set
$V_{\epsilon}=\\{\epsilon,\epsilon^{q},\epsilon^{q^{2}},\dots,\epsilon^{q^{n-1}}\\}$
acts as a basis for the vector space $\mathbb{F}_{q^{n}}$ over
$\mathbb{F}_{q}$. It is well known that $\epsilon\in\mathbb{F}_{q^{n}}$ is
normal element if the greatest common divisor of
$\sum_{i=0}^{n-1}\epsilon^{q^{i}}x^{n-i-1}$ with $x^{n}-1$ is 1. This
motivated Huczynska et al. [12] to develop the concept of $k$-normal elements.
Let $0\leq k\leq n-1$, an element $\epsilon\in\mathbb{F}_{q^{n}}$ is said to
be $k$-normal element if the greatest common divisor of
$\sum_{i=0}^{n-1}\epsilon^{q^{i}}x^{n-i-1}$ with $x^{n}-1$ has degree $k$ in
$\mathbb{F}_{q^{n}}[x]$. Using this terminology, a $0$-normal element is
equivalent to a normal element. For $\epsilon\in\mathbb{F}_{q^{n}}$, the trace
of $\epsilon$ over $\mathbb{F}_{q}$, denoted by
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)$ is defined
as
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)=\epsilon+\epsilon^{q}+\epsilon^{q^{2}}+\dots+\epsilon^{q^{n-1}}$.

Primitive elements are of utmost importance in various cryptographic
applications, such as pseudorandom number generators and discrete logarithmic
problems, owing to their high multiplicative order. For small values of $r$,
$r$-primitive elements can also serve as a substitute for primitive elements
in several applications. Researchers have studied the effective construction
of such highly ordered elements in papers like [9, 14, 19]. $k$-normal
elements has the potential to decrease the computational complexity of
multiplication operations in finite fields, as explained in [16, 17, 18], and
many researchers have studied their existence, including in papers like [21,
24, 26]. In [20, 1], authors presented a condition that is sufficient for the
existence of $r$-primitive, $k$-normal elements in $\mathbb{F}_{q^{n}}$ over
$\mathbb{F}_{q}$.

Numerous articles (see [5, 3, 22, 23, 7, 10]) have been published in which
authors established sufficient criteria for the existence of primitive
elements and normal elements in finite fields with prescribed traces. Recently
[2], proposed a sufficient condition for the existence of $r_{1}$-primitive,
$k_{1}$-normal element $\epsilon\in\mathbb{F}_{q^{n}}$ such that $F(\epsilon)$
is $r_{2}$-primitive, $k_{2}$-normal in $\mathbb{F}_{q^{n}}$ over
$\mathbb{F}_{q}$, where $F\in\Lambda_{q}(m_{1},m_{2})$ (see Definition 1.1).
Prior to this article, $r$-primitive, $k$-normal elements with prescribed
traces were not examined. In this article we present a condition that ensures
the existence of element $\epsilon\in\mathbb{F}_{q^{n}}$ such that $\epsilon$
is $r_{1}$-primitive, $k_{1}$-normal, $F(\epsilon)$ is $r_{2}$-primitive,
$k_{2}$-normal in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ with
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)=a$ and
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon^{-1})=b$ for
any prescribed $a,b\in\mathbb{F}_{q}$, where
$F\in\Lambda_{q^{n}}(m_{1},m_{2})$, $r_{1}$ and $r_{2}$ are positive divisors
of $q^{n}-1$, and $k_{1}$ and $k_{2}$ are degrees of some polynomials over
$\mathbb{F}_{q}$ that divide $x^{n}-1$.

######  Definition 1.1.

For positive integers $m_{1},m_{2}$, Define $\Lambda_{q}(m_{1},m_{2})$ as the
set of rational functions $F=\frac{F_{1}}{F_{2}}\in\mathbb{F}_{q}(x)$, such
that $F_{1}$ and $F_{2}$ are relatively prime, deg$(F_{i})\leq m_{i}$;
$i=1,2$, and there exist $m\in\mathbb{N}$ and an irreducible monic polynomial
$g\in\mathbb{F}_{q}[x]\setminus\\{x\\}$ such that gcd$(m,q-1)=1$, $g^{m}\mid
F_{1}F_{2}$ but $g^{m+1}\nmid F_{1}F_{2}$.

Let $A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$ denote the set consisting of pairs
$(q,n)\in\mathbb{N}\times\mathbb{N}$ such that for any
$F\in\Lambda_{q}(m_{1},m_{2})$, $r_{1}$ and $r_{2}$, divisors of $q^{n}-1$,
$k_{1}$ and $k_{2}$, non-negative integers, and $a,b\in\mathbb{F}_{q}$, the
set $\mathbb{F}_{q^{n}}$ contains an element $\epsilon$ such that $\epsilon$
is $r_{1}$-primitive, $k_{1}$-normal, and $F(\epsilon)$ is $r_{2}$-primitive,
$k_{2}$-normal element in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$ with
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)=a$ and
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon^{-1})=b$.

In this article, we begin by considering $F\in\Lambda_{q^{n}}(m_{1},m_{2})$,
$r_{1}$ and $r_{2}$ as divisors of $q^{n}-1$, $k_{1}$ and $k_{2}$ as degrees
of some polynomials $f_{1}$ and $f_{2}$ over $\mathbb{F}_{q}$ that divide
$x^{n}-1$, and $a,b\in\mathbb{F}_{q}$. We then establish a sufficient
condition on $(q,n)$ such that $(q,n)\in A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$.
Furthermore, using some results and seive variation of this sufficient
condition we prove the following result:

######  Theorem 1.2.

Suppose $n\geq 13$, be a positive integer and $q$ be a power of $13$ and
$F\in\Lambda_{q^{n}}(10,11)$ If $x^{n}-1$ has a factor of degree 2 in
$\mathbb{F}_{q}[x]$, and $q$ and $n$ assume none of the following values:

  1. (1)

$q=13$ and $n=13,14,15,16,18,20,24;$

  2. (2)

$q=13^{2}$ and $n=13,14$;

  3. (3)

$q=13^{3}$ and $n=13.$

Then, $(q,n)\in A_{F,a,b}(3,2,2,1)$.

##  2\. Preliminaries

This section serves as an introduction to fundamental concepts, notations, and
findings that will be employed in subsequent sections of this article.
Throughout this article, $n$ represents a positive integer, $q$ stands for an
arbitrary prime power, and $\mathbb{F}_{q}$ denotes a finite field with $q$
elements.

###  2.1. Definitions

  1. (1)

A character of a finite abelian group $G$ is a homomorphism $\chi$ from the
set $G$ into $Z^{1}$, where $Z^{1}$ is the set of all elements of complex
field $\mathbb{C}$ with absolute value 1. The trivial character of $G$ denoted
by $\chi_{0}$, is defined as $\chi_{0}(g)=1$ for all $g\in G$. In addition,
the set of all characters of $G$, denoted by $\widehat{G}$, forms a group
under multiplication, which is isomorphic to $G$. The order of a character
$\chi$ is the least positive integer $d$ such that $\chi^{d}=\chi_{0}$. For a
finite field $\mathbb{F}_{q^{n}}$, a character of the additive group
$\mathbb{F}_{q^{n}}$ is called an additive character and that of the
multiplicative group $\mathbb{F}^{*}_{q^{n}}$ is called a multiplicative
character. For more information on characters, primitive elements and finite
fields, we refer the reader to [13].

  2. (2)

The Euler’s totient function for polynomials $f(x)\in\mathbb{F}_{q}[x]$ is
defined as follows:

$\Phi_{q}(f)=\bigg{|}\bigg{(}\dfrac{\mathbb{F}_{q}[x]}{\langle
f\rangle}\bigg{)}^{*}\bigg{|}=|f|\prod_{\begin{subarray}{c}p|f,\\\
p\;\text{irreducible}\\\
\text{over}\;\mathbb{F}_{q}\end{subarray}}\bigg{(}1-\frac{1}{|p|}\bigg{)},$

where $|f|=q^{deg(f)}$, and $\langle f\rangle$ is the ideal generated by $f$
in $\mathbb{F}_{q}[x].$

  3. (3)

For $a\in\mathbb{F}_{q}$, the characteristic function for the subset of
$\mathbb{F}_{q^{n}}$ whose elements satisfy
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)=a$ is defined
as

$\displaystyle\tau_{a}:\epsilon\mapsto\frac{1}{q}\sum_{\eta\in\widehat{\mathbb{F}}_{q}}\eta(\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)-a).$

According to [13, Theorem 5.7], every additive character $\eta$ of
$\mathbb{F}_{q}$ can be obtained as $\eta(a)=\eta_{0}(u^{\prime}a)$, where
$\eta_{0}$ is the canonical additive character of $\mathbb{F}_{q}$ and
$u^{\prime}$ is an element of $\mathbb{F}_{q}$ corresponding to $\eta$. Thus

$\displaystyle{}\tau_{a}$
$\displaystyle=\frac{1}{q}\sum_{u^{\prime}\in\mathbb{F}_{q}}\eta_{0}(\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(u^{\prime}\epsilon)-u^{\prime}a)$
$\displaystyle=\frac{1}{q}\sum_{u^{\prime}\in\mathbb{F}_{q}}\widehat{\eta_{0}}(u^{\prime}\epsilon)\eta_{0}(-u^{\prime}a),$
(2.1)

where $\widehat{\eta_{0}}$ is the additive character of $\mathbb{F}_{q^{n}}$
defined by
$\widehat{\eta_{0}}(\epsilon)=\eta_{0}(\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon))$.

The additive group of $\mathbb{F}_{q^{n}}$ is an $\mathbb{F}_{q}[x]$-module
under the operation $f\circ\epsilon=\sum_{i=0}^{k}a_{i}\epsilon^{q^{i}}$,
where $\epsilon\in\mathbb{F}_{q^{n}}$ and
$f(x)=\sum_{i=0}^{k}a_{i}x^{i}\in\mathbb{F}_{q}[x]$. The
$\mathbb{F}_{q}$-order of $\epsilon\in\mathbb{F}_{q^{n}}$, denoted by
$\operatorname{Ord(\epsilon)}$, is the unique least degree polynomial $g$ such
that $g\circ\epsilon=0.$ It is an easy observation that $g$ is a factor of
$x^{n}-1$. By defining the action of $\mathbb{F}_{q^{n}}[x]$ over
${\mathbb{\widehat{F}}}_{q^{n}}$, by the operation
$f\circ\eta(\epsilon)=\eta(f\circ\epsilon)$, where
$\eta\in\mathbb{\widehat{F}}_{q^{n}},\epsilon\in\mathbb{F}_{q^{n}}$ and
$f\in\mathbb{F}_{q}[x]$, $\widehat{\mathbb{F}}_{q^{n}}$ becomes an
$\mathbb{F}_{q}$-module. From [11, Theorem 13.4.1],
$\mathbb{\widehat{F}}_{q^{n}}^{*}$ and $\mathbb{F}_{q^{n}}^{*}$ are
$\mathbb{Z}$-isomorphic modules, and $\mathbb{\widehat{F}}_{q^{n}}$ and
$\mathbb{F}_{q^{n}}$ are $\mathbb{F}_{q}[x]$-isomorphic modules. The following
character sum holds true for every $\epsilon\in\mathbb{F}_{q^{n}}$.

$\displaystyle{}\mathcal{I}_{0}(\epsilon)=\dfrac{1}{q^{n}}\sum_{\gamma\in\widehat{\mathbb{F}}_{q^{n}}}\gamma(\epsilon)=\begin{cases}1&\text{if}\;\epsilon=0;\\\
0&\text{otherwise.}\end{cases}$ (2.2)

The unique least degree monic polynomial $g$ such that $\eta\circ g=\chi_{0}$
is called the $\mathbb{F}_{q}$-order of $\eta$, denoted by
$\operatorname{Ord(\eta)}.$ Moreover, there are $\Phi_{q}(g)$ characters of
$\mathbb{F}_{q}$-order $g$.

Let $g\in\mathbb{F}_{q^{n}}$ is a divisor of $x^{n}-1$, an element
$\epsilon\in\mathbb{F}_{q^{n}}$ is $g$-free if $\epsilon=h\circ\sigma$, where
$\sigma\in\mathbb{F}_{q^{n}}$ and $h\mid g$, implies $h=1$. It can easily be
seen that an element in $\mathbb{F}_{q^{n}}$ is $(x^{n}-1)$-free if and only
if it is normal. As in the multiplicative case, from [11, Theorem 13.4.4], for
$g\mid x^{n}-1$, the characteristics function for $g$-free elements is given
by

$\displaystyle{}\Omega_{g}(\epsilon)=\dfrac{\Phi_{q}(g)}{q^{deg(g)}}\int\limits_{h\mid
g}\psi_{h}(\epsilon)=\dfrac{\Phi_{q}(g)}{q^{deg(g)}}\sum_{h\mid
g}\dfrac{\mu_{q}(h)}{\Phi_{q}(h)}\sum_{\psi_{h}}\psi_{h}(\epsilon),$ (2.3)

where $\sum_{h\mid g}$ runs over all the monic divisors
$h\in\mathbb{F}_{q^{n}}[x]$ of $g$, $\psi_{h}$ is an additive character of
$\mathbb{F}_{q^{n}}$, the sum $\sum_{\psi_{h}}\psi_{h}$ runs over all
$\Phi_{q}(h)$ additive characters of $\mathbb{F}_{q}$-order $h$ and $\mu_{q}$
is the polynomial Mobius function defined as

$\mu_{q}(h)=\left\\{\begin{array}[]{ll}(-1)^{r}&\text{if $h$ is product of $r$
distinct monic irreducible polynomial over $\mathbb{F}_{q}$, }\\\
0&\text{otherwise.}\\\ \end{array}\right.$

######  Definition 2.1.

[6, Definition 3.1] Let $r\mid q^{n}-1$. For a divisor $R$ of
$\frac{q^{n}-1}{r}$, let $H_{r}$ be a multiplicative cyclic subgroup of
$\mathbb{F}_{q^{n}}^{*}$ of order $\frac{q^{n}-1}{r}$. An element
$\epsilon\in\mathbb{F}_{q^{n}}$ is referred to as $(R,r)$-free if $\epsilon\in
H_{r}$ and $\epsilon$ is $R$-free in $H_{r}$, i.e, if $\epsilon=\sigma^{e}$
with $\sigma\in H_{r}$ and $e\mid R,$ then $e=1$.

Based on the definition above, it is clear that an element
$\epsilon\in\mathbb{F}_{q^{n}}^{*}$ is $r$-primitive if and only if it is
$(\frac{q^{n}-1}{r},r)$-free. From [6, Theorem 3.8], the characteristic
function for the set of $(R,r)$-free elements is given by

$\displaystyle{}\mathbb{I}_{R,r}(\epsilon)=\dfrac{\theta(R)}{r}\int\limits_{d\mid
Rr}\chi_{d}(\epsilon)=\dfrac{\theta(R)}{r}\sum_{d\mid
Rr}\dfrac{\mu(d_{(r)})}{\phi(d_{(r)})}\sum_{\chi_{d}}\chi_{d}(\epsilon),$
(2.4)

where $\theta(R)=\frac{\phi(R)}{R}$, $\mu$ is the mobius function,
$d_{(r)}=\frac{d}{gcd(d,r)}$, and the sum $\sum_{\chi_{d}}\chi_{d}$ runs over
all the multiplicative characters of $\mathbb{F}_{q^{n}}^{*}$ of order $d$.

For $\kappa$, a positive integer (or a monic polynomial over
$\mathbb{F}_{q}$), we use $\omega(\kappa)$ to represent the number of distinct
prime divisors (irreducible factors) of $\kappa$, and $W(\kappa)$ to represent
the number of square-free divisors (square-free factors) of $\kappa$. Clearly,
$W(\kappa)=2^{\omega(\kappa)}$. We have the following result to bound the sum
(2.4).

######  Lemma 2.2.

[6, Lemma 2.5] For any positive integer $R,r$, we have that

$\displaystyle\sum_{d\mid
Rr}\dfrac{\mu(d_{(r)})}{\phi(d_{(r)})}\phi(d)=gcd(R,r)W(gcd(R,R_{(r)})).$

The results provided by Wang and Fu [8] will play a crucial role in the proof
of Theorem 3.1

######  Lemma 2.3.

[8, Theorem 5.5] Let $f(x)\in\mathbb{F}_{q^{n}}(x)$ be a rational function.
Write $f(x)=\prod_{j=1}^{k}f_{j}(x)^{r_{j}}$, where
$f_{j}(x)\in\mathbb{F}_{q^{n}}[x]$ are irreducible polynomials and
$r_{j}^{\prime}s$ are non zero integers. Let $\chi$ be a multiplicative
character of $\mathbb{F}_{q^{n}}$. Suppose that $f(x)$ is not of the form
$r(x)^{\operatorname{Ord}(\chi)}$ for any, rational function
$r(x)\in\mathbb{F}(x)$, where $\mathbb{F}$ is algebraic closure of
$\mathbb{F}_{q^{n}}$. Then we have

$\bigg{|}\sum_{\epsilon\in\mathbb{F}_{q^{n}},f(\epsilon)\neq
0,\infty}\chi(f(\epsilon))\bigg{|}\leq\bigg{(}\sum_{j=1}^{k}deg(f_{j})-1\bigg{)}q^{\frac{n}{2}}.$

######  Lemma 2.4.

[8, Theorem 5.6] Let $f(x),g(x)$ $\in\mathbb{F}_{q^{n}}(x)$ be rational
functions. Write f(x) = $\prod_{j=1}^{k}f_{j}(x)^{r_{j}}$, where
$f_{j}(x)\in\mathbb{F}_{q^{n}}[x]$ are irreducible polynomials and $r_{j}$ are
non-zero integers. Let $D_{1}=\sum_{j=1}^{k}deg(f_{j})$,
$D_{2}=max\\{deg(g),0\\}$, $D_{3}$ is the degree of denominator of g(x) and
$D_{4}$ is the sum of degrees of those irreducible polynomials dividing
denominator of $g$ but distinct from $f_{j}(x)$ $(j=1,2,...,k)$. Let $\chi$ be
a multiplicative character of $\mathbb{F}_{q^{n}}$, and let $\psi$ be a
nontrivial additive character of $\mathbb{F}_{q^{n}}$. Suppose g(x) is not of
the form $v(x)^{q^{n}}-v(x)$ in $\mathbb{F}(x)$, where $\mathbb{F}$ is
algebraic closure of $\mathbb{F}_{q^{n}}$. Then we have

$\bigg{|}\sum_{\epsilon\in\mathbb{F}_{q^{n}},f(\epsilon)\neq
0,\infty,g(\epsilon)\neq\infty}\chi(f(\epsilon))\psi(g(\epsilon))\bigg{|}\leq(D_{1}+D_{2}+D_{3}+D_{4}-1)q^{\frac{n}{2}}.$

######  Remark 2.5.

In [21, Lemma 3.1], L. Ries determined a technique for constructing $k$-normal
elements. Given a normal element $\sigma$ in $\mathbb{F}_{q^{n}}$ and a
divisor $f$ of $x^{n}-1$ with degree $k$ in $\mathbb{F}_{q}[x]$, the
composition $\epsilon=f\circ\sigma$ is a k-normal element.

##  3\. Sufficient Condition

Suppose $r_{1}$ and $r_{2}$ be positive divisors of $q^{n}-1$, and $f_{1}$ and
$f_{2}\in\mathbb{F}_{q}[x]$ be monic factors of $x^{n}-1$ of degrees $k_{1}$
and $k_{2}$, respectively. Let $m_{1}$ and $m_{2}$ be non-negative integers
such that $1\leq m_{1}+m_{2}<q^{n/2}$, and
$F=\frac{F_{1}}{F_{2}}\in\Lambda_{q^{n}}(m_{1},m_{2})$. Also, let $R_{1}$ and
$R_{2}$ be divisors of $\frac{q^{n}-1}{r_{1}}$ and $\frac{q^{n}-1}{r_{2}}$,
respectively, and $g_{1}$ and $g_{2}\in\mathbb{F}_{q}[x]$ be monic factors of
$x^{n}-1$. Let $a,b\in\mathbb{F}_{q}$. Suppose
$C_{F,a,b}(R_{1},R_{2},g_{1},g_{2})$ (denote $C_{F,a,b}(R_{1},R_{2},g)$ when
$g_{1}=g_{2}=g$) denote the cardinality of the set containing elements
$(\epsilon,\sigma_{1},\sigma_{2})\in\mathbb{F}_{q^{n}}^{*}\times\mathbb{F}_{q^{n}}\times\mathbb{F}_{q^{n}}$
that satisfy the following conditions:

  * •

$\epsilon$ is $(R_{1},r_{1})$-free, $F(\epsilon)$ is $(R_{2},r_{2})$-free,

  * •

$\sigma_{1}$ is $g_{1}$-free, $\sigma_{2}$ is $g_{2}$-free,

  * •

$\epsilon=f_{1}\circ\sigma_{1}$, $F(\epsilon)=f_{2}\circ\sigma_{2}$,

  * •

$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)=a$ and
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon^{-1})=b$.

In particular,
$C_{F,a,b}(\frac{q^{n}-1}{r_{1}},\frac{q^{n}-1}{r_{2}},x^{n}-1)$ denotes the
number of elements $(\epsilon,\sigma_{1},\sigma_{2})$
$\in\mathbb{F}_{q^{n}}^{*}\times\mathbb{F}_{q^{n}}\times\mathbb{F}_{q^{n}}$
such that $\epsilon=f_{1}\circ\sigma_{1}$ is $r_{1}$-primitive,
$k_{1}$-normal, $F(\epsilon)=f_{2}\circ\sigma_{2}$ is $r_{2}$-primitive,
$k_{2}$-normal,
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)=a$ and
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon^{-1})=b$.

We now present a sufficient condition as follows:

######  Theorem 3.1.

Let
$\Omega=r_{1}r_{2}W(R_{1})W(R_{2})W(gcd(g_{1},\dfrac{x^{n}-1}{f_{1}}))W(gcd(g_{2},\dfrac{x^{n}-1}{f_{2}}))$,
if

$\displaystyle
q^{\frac{n}{2}-k_{1}-k_{2}-2}>\begin{cases}\Omega\cdot(2m_{1}+2m_{2}+1)&\text{if}\;\;m_{1}\geq
m_{2},\\\ \Omega\cdot(m_{1}+3m_{2}+1)&\text{if}\;\;m_{1}<m_{2},\end{cases}$

then $C_{F,a,b}(R_{1},R_{2},g_{1},g_{2})>0.$

###### Proof.

Suppose $Z_{F}$ is the set that constitutes all zeroes and poles of
$F\in\Lambda_{q^{n}}(m_{1},m_{2})$. By definition,
$C_{F,a,b}(R_{1},R_{2},g_{1},g_{2})$ is given by

$\sum_{\begin{subarray}{c}\epsilon\in\mathbb{F}_{q^{n}}^{*}\setminus Z_{F}\\\
\sigma_{1},\sigma_{2}\in\mathbb{F}_{q^{n}}\end{subarray}}\mathbb{I}_{R_{1},r_{1}}(\epsilon)\mathbb{I}_{R_{2},r_{2}}(F(\epsilon))\Omega_{g_{1}}(\sigma_{1})\Omega_{g_{2}}(\sigma_{2})\mathcal{I}_{0}(\epsilon-
f_{1}\circ\sigma_{1})\mathcal{I}_{0}(F(\epsilon)-f_{2}\circ\sigma_{2})\tau_{a}(\epsilon)\tau_{b}(\epsilon^{-1}).$

Using (3),(2.2), (2.3) and (2.4), $C_{F,a,b}(R_{1},R_{2},g_{1},g_{2})$ is
equal to

$\displaystyle{}\dfrac{\theta(R_{1})\theta(R_{2})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}\int\limits_{\begin{subarray}{c}d_{1}\mid
R_{1}r_{1}\\\ d_{2}\mid
R_{2}r_{2}\end{subarray}}\int\limits_{\begin{subarray}{c}h_{1}\mid g_{1}\\\
h_{2}\mid
g_{2}\end{subarray}}\sum_{\gamma_{1},\gamma_{2}\in\widehat{\mathbb{F}}_{q^{n}}}\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\psi_{h_{1}},\psi_{h_{2}},\gamma_{1},\gamma_{2},\eta),$
(3.1)

where

$\displaystyle\chi_{F,a,b}$
$\displaystyle(\chi_{d_{1}},\chi_{d_{2}},\psi_{h_{1}},\psi_{h_{2}},\gamma_{1},\gamma_{2},\eta)=\dfrac{1}{q^{2n}}\sum_{u^{\prime},v^{\prime}\in\mathbb{F}_{q}}\eta_{0}(-au^{\prime}-bv^{\prime})\sum_{\epsilon\in\mathbb{F}_{q^{n}}^{*}\setminus
Z_{F}}\chi_{d_{1}}(\epsilon)\chi_{d_{2}}(F(\epsilon))\times$
$\displaystyle\times\gamma_{1}(\epsilon)\gamma_{2}(F(\epsilon))\widehat{\eta_{0}}(u^{\prime}\epsilon+v^{\prime}\epsilon^{-1})\sum_{\sigma_{1}\in\mathbb{F}_{q^{n}}}\psi_{h_{1}}(\sigma_{1})\gamma_{1}^{-1}(f_{1}\circ\sigma_{1})\sum_{\sigma_{2}\in\mathbb{F}_{q^{n}}}\psi_{h_{2}}(\sigma_{2})\gamma_{2}^{-1}(f_{2}\circ\sigma_{2}).$

It follows from [1, Lemma 2.5], if
$\gamma_{i}\in\widehat{f_{i}}^{-1}(\psi_{h_{i}})$; $i\in\\{1,2\\}$, then

$\displaystyle{}\sum_{\sigma_{i}\in\mathbb{F}_{q^{n}}}\psi_{h_{i}}(\sigma_{i})\gamma_{i}^{-1}(f_{i}\circ\sigma_{i})=q^{n}.$
(3.2)

Left hand side of (3.2) is $0$, if
$\gamma_{i}\notin\widehat{f_{i}}^{-1}(\psi_{h_{i}})$, and the set
$\widehat{f_{i}}^{-1}(\psi_{h_{i}})$ is empty if $\mathbb{F}_{q}$ -order of
$\psi_{h_{i}}=h_{i}$ does not divide $\frac{x^{n}-1}{f_{i}}$; $i\in\\{1,2\\}$.
Define $\tilde{g_{i}}=gcd(g_{i},\frac{x^{n}-1}{f_{i}})$, then
$C_{F,a,b}(R_{1},R_{2},g_{1},g_{2})$ becomes

$\dfrac{\theta(R_{1})\theta(R_{2})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}\int\limits_{\begin{subarray}{c}d_{1}\mid
R_{1}r_{1}\\\ d_{2}\mid
R_{2}r_{2}\end{subarray}}\int\limits_{\begin{subarray}{c}h_{1}\mid g_{1}\\\
h_{2}\mid
g_{2}\end{subarray}}\sum_{\begin{subarray}{c}\gamma_{1}\in\widehat{f_{1}}^{-1}(\psi_{h_{1}})\\\
\gamma_{2}\in\widehat{f_{1}}^{-1}(\psi_{h_{2}})\end{subarray}}\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta),$

where

$\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)=\sum_{u^{\prime},v^{\prime}\in\mathbb{F}_{q}}\eta_{0}(-au^{\prime}-bv^{\prime})\sum_{\epsilon\in\mathbb{F}_{q^{n}}^{*}\setminus
Z_{F}}\chi_{d_{1}}(\epsilon)\chi_{d_{2}}(F(\epsilon))\gamma_{1}(\epsilon)\gamma_{2}(F(\epsilon))\widehat{\eta_{0}}(\epsilon^{\prime}),$

where $\epsilon^{\prime}=u^{\prime}\epsilon+v^{\prime}\epsilon^{-1}$. Since
$\gamma_{1}$ and $\gamma_{2}$ are additive characters of $\mathbb{F}_{q^{n}}$,
there exist $w_{1},w_{2}\in\mathbb{F}_{q^{n}}$ such that
$\gamma_{1}(\epsilon)=\widehat{\eta}_{0}(w_{1}\epsilon)$ and
$\gamma_{2}(F(\epsilon))=\widehat{\eta}_{0}(w_{2}F(\epsilon))$, where
$\widehat{\eta}_{0}$ is the canonical additive character of
$\mathbb{F}_{q^{n}}$. Assuming $u_{0}=w_{1}+u^{\prime}$, we have

$\displaystyle\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)=\sum_{u^{\prime},v^{\prime}\in\mathbb{F}_{q}}\eta_{0}(-au^{\prime}-$
$\displaystyle bv^{\prime})\sum_{\epsilon\in\mathbb{F}_{q^{n}}^{*}\setminus
Z_{F}}\chi_{d_{1}}(\epsilon)\chi_{d_{2}}(F(\epsilon))\times$
$\displaystyle\times\widehat{\eta}_{0}(u_{0}\epsilon+w_{2}F(\epsilon)+v^{\prime}\epsilon^{-1}).$

Let deg$(F_{i})=n_{i};\;i=1,2$. First we consider the case when $d_{2}=1$. we
have

$\chi_{F,a,b}(\chi_{d_{1}},\chi_{1},\gamma_{1},\gamma_{2},\eta)=\sum_{u^{\prime},v^{\prime}\in\mathbb{F}_{q}}\eta_{0}(-au^{\prime}-bv^{\prime})\sum_{\epsilon\in\mathbb{F}_{q^{n}}^{*}\setminus
Z_{F}}\chi_{d_{1}}(\epsilon)\widehat{\eta}_{0}(G(\epsilon)),$

where
$G(x)=\dfrac{u_{0}x^{2}F_{2}(x)+w_{2}xF_{1}(x)+v^{\prime}F_{2}(x)}{xF_{2}(x)}$.
If $G(x)\neq r(x)^{q^{n}}-r(x)$ for any $r(x)\in\mathbb{F}(x)$, then we have
two cases:
Case 1:- If $n_{1}>n_{2}+1$, then in accordance with Lemma 2.4, we have
$D_{2}=n_{1}-n_{2}$, and

$\displaystyle{}|\chi_{F,a,b}(\chi_{d_{1}},\chi_{1},\gamma_{1},\gamma_{2},\eta)|\leq(2(m_{1}+m_{2})+1)q^{\frac{n}{2}+2}.$
(3.3)

Case 2:- If $n_{1}\leq n_{2}+1$, then $D_{2}=1$, and

$\displaystyle{}|\chi_{F,a,b}(\chi_{d_{1}},\chi_{1},\gamma_{1},\gamma_{2},\eta)|\leq(m_{1}+3m_{2}+2)q^{\frac{n}{2}+2}.$
(3.4)

If $G(x)=r(x)^{q^{n}}-r(x)$, for some $r(x)\in\mathbb{F}(x)$, where
$r(x)=\frac{r_{1}(x)}{r_{2}(x)}$ with $(r_{1},r_{2})=1$. We have,
$\dfrac{u_{0}x^{2}F_{2}(x)+w_{2}xF_{1}(x)+v^{\prime}F_{2}(x)}{xF_{2}(x)}=\dfrac{r_{1}(x)^{q^{n}}}{r_{2}(x)^{q^{n}}}-\dfrac{r_{1}(x)}{r_{2}(x)}$,
i.e.,

$(r_{1}(x)^{q^{n}}-r_{1}(x)r_{2}(x)^{q^{n}-1})xF_{2}(x)=r_{2}(x)^{q^{n}}(u_{0}x^{2}F_{2}(x)+w_{2}xF_{1}(x)+v^{\prime}F_{2}(x)).$

Since $(r_{1}(x)^{q^{n}}-r_{1}(x)r_{2}(x)^{q^{n}-1},r_{2}(x)^{q^{n}})=1$,
$r_{2}(x)^{q^{n}}\mid xF_{2}(x)$, which is possible only if $r_{2}$ is
constant. Let $r_{2}(x)=c$, then we have

$\displaystyle{}(r_{1}(x)^{q^{n}}-r_{1}(x)c^{q^{n}-1})xF_{2}(x)=c^{q^{n}}(u_{0}x^{2}F_{2}(x)+w_{2}xF_{1}(x)+vF_{2}(x)).$
(3.5)

From 3.5, $xF_{2}(x)$ divides
$(u_{0}x^{2}F_{2}(x)+w_{2}xF_{1}(x)+v^{\prime}F_{2}(x))$, which happens only
when $w_{2}=0$. Now
$(r_{1}(x)^{q^{n}}-r_{1}(x)c^{q^{n}-1})x=c^{q^{n}}(u_{0}x^{2}+v^{\prime})$,
implies $v^{\prime}=0$, and
$(r_{1}(x)^{q^{n}}-r_{1}(x)c^{q^{n}-1})=c^{q^{n}}u_{0}x$ implies $u_{0}=0$ and
$r_{1}(x)$ is constant. From this discussion we get

$\displaystyle{}|\chi_{F,a,b}(\chi_{d_{1}},\chi_{1},\gamma_{1},\gamma_{2},\eta)|\leq(m_{1}+m_{2})q^{\frac{n}{2}+2}.$
(3.6)

Now assume, $d_{2}>1$. There exist $t_{1},t_{2}$ with $0\leq
t_{1},t_{2}<q^{n}-1$ such that $\chi_{d_{i}}(x)=\chi_{q^{n}-1}(x^{t_{i}})$;
$i\in\\{1,2\\}$. We have

$\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)=\sum_{u^{\prime},v^{\prime}\in\mathbb{F}_{q}}\eta_{0}(-au^{\prime}-bv^{\prime})\sum_{\epsilon\in\mathbb{F}_{q^{n}}^{*}\setminus
Z_{F}}\chi_{q^{n}-1}(G_{1}(\epsilon))\widehat{\eta}_{0}(G_{2}(\epsilon)),$

where $G_{1}(x)=x^{t_{1}}F(x)^{t_{2}}\in\mathbb{F}(x)$ and
$G_{2}(x)=u_{0}x+w_{2}F(x)+v^{\prime}x^{-1}\in\mathbb{F}(x)$ . If
$G_{2}(x)\neq h(x)^{q^{n}}-h(x)$ for any $h(x)\in\mathbb{F}(x)$, again we have
two cases:
Case 1:- If $n_{1}>n_{2}+1$, then from Lemma 2.4, we have

$\displaystyle{}|\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)|\leq(2m_{1}+m_{2}+1)q^{\frac{n}{2}+2}.$
(3.7)

Case 2:- If $n_{1}\leq n_{2}+1$, we have $D_{2}=1$, and

$\displaystyle{}|\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)|\leq(m_{1}+2m_{2}+2)q^{\frac{n}{2}+2}.$
(3.8)

On the other hand $G_{2}(x)=h(x)^{q^{n}}-h(x)$ for some
$h(x)\in\mathbb{F}(x)$, leads to $u_{0}=v^{\prime}=w_{2}=0$, and

$\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)=\sum_{u^{\prime},v^{\prime}\in\mathbb{F}_{q}}\eta_{0}(-au^{\prime}-bv^{\prime})\sum_{\epsilon\in\mathbb{F}_{q^{n}}^{*}\setminus
Z_{F}}\chi_{q^{n}-1}(G_{1}(\epsilon)).$

From [4, Theorem 3.2], $G_{1}(x)$ is not of the type $h(x)^{q^{n}-1}$ for any
$h(x)\in\mathbb{F}(x)$, it follows from Lemma 2.3,

$\displaystyle{}|\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)|\leq(m_{1}+m_{2})q^{\frac{n}{2}+2}.$
(3.9)

Suppose $M=\operatorname{max}\\{2(m_{1}+m_{2})+1,m_{1}+3m_{2}+1\\}$. We
observe that

$\displaystyle{}|\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta)|\leq
Mq^{\frac{n}{2}+2},$ (3.10)

for inequalities (3.3), (3.4), (3.6), (3.7), (3.8) and (3.9). Let $\psi_{1}$
be the trivial additive character, and suppose

$U_{1}=\int\limits_{\begin{subarray}{c}d_{1}\mid R_{1}r_{1}\\\ d_{2}\mid
R_{2}r_{2}\end{subarray}}\sum\limits_{\begin{subarray}{c}\gamma_{1}\in\widehat{f_{1}}^{-1}(\psi_{1})\\\
\gamma_{2}\in\widehat{f_{2}}^{-1}(\psi_{1})\\\
(\gamma_{1},\gamma_{2})\neq(\psi_{1},\psi_{1})\end{subarray}}\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta),$

and

$U_{2}=\int\limits_{\begin{subarray}{c}d_{1}\mid R_{1}r_{1}\\\ d_{2}\mid
R_{2}r_{2}\end{subarray}}\int\limits_{\begin{subarray}{c}h_{1}\mid\tilde{g_{1}}\\\
h_{2}\mid\tilde{g_{2}}\\\ \\\
(h_{1},h_{2})\neq(1,1)\end{subarray}}\sum\limits_{\begin{subarray}{c}\gamma_{1}\in\widehat{f_{1}}^{-1}(\psi_{h_{1}})\\\
\gamma_{2}\in\widehat{f_{2}}^{-1}(\psi_{h_{2}})\end{subarray}}\chi_{F,a,b}(\chi_{d_{1}},\chi_{d_{2}},\gamma_{1},\gamma_{2},\eta).$

It follows from [1, Lemma 2.5], Lemma 2.2 and (3.10),

$|U_{1}|\leq Mq^{\frac{n}{2}+2}(q^{k_{1}+k_{2}}-1)r_{1}r_{2}W(R_{1})W(R_{2}),$

and

$|U_{2}|\leq
Mq^{\frac{n}{2}+2}q^{k_{1}+k_{2}}r_{1}r_{2}W(R_{1})W(R_{2})(W(\tilde{g_{1}})W(\tilde{g_{2}})-1).$

Hence, from the above discussion, along with (3.1), we get that

$\displaystyle C_{F,a,b}(R_{1},R_{2},g_{1},g_{2})\geq$
$\displaystyle\;\dfrac{\theta(R_{1})\theta(R_{2})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}((q^{n}-Z_{F}\cdot
q^{2})-(|U_{1}|+|U_{2}|))$ $\displaystyle\geq$
$\displaystyle\;\dfrac{\theta(R_{1})\theta(R_{2})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}((q^{n}-(m_{1}+m_{2}+1)q^{2})$
$\displaystyle-(|U_{1}|+|U_{2}|))$ $\displaystyle>$
$\displaystyle\dfrac{\theta(R_{1})\theta(R_{2})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}(q^{n}-Mq^{\frac{n}{2}+2}q^{k_{1}+k_{2}}r_{1}r_{2}W(R_{1})\times$
$\displaystyle\times W(R_{2})W(\tilde{g_{1}})W(\tilde{g_{2}}))$

Thus if, $q^{\frac{n}{2}-k_{1}-k_{2}-2}>M\Omega$, then
$C_{F,a,b}(R_{1},R_{2},g_{1},g_{2})>0.$

∎

######  Corollary 3.2.

Suppose $M=max\\{2(m_{1}+m_{2})+1,m_{1}+3m_{2}+1\\}$. If

$q^{\frac{n}{2}-k_{1}-k_{2}-2}>Mr_{1}r_{2}W\bigg{(}\frac{q^{n}-1}{r_{1}}\bigg{)}W\bigg{(}\frac{q^{n}-1}{r_{2}}\bigg{)}W\bigg{(}\frac{x^{n}-1}{f_{1}}\bigg{)}W\bigg{(}\frac{x^{n}-1}{f_{2}}\bigg{)}.$

Then, $(q,n)\in A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$.

###### Proof.

It follows by taking $R_{i}=\dfrac{q^{n}-1}{r_{i}}$ and $g_{i}=x^{n}-1$;
$i=1,2$ in Theorem 3.1. ∎

From now onwards $M=\operatorname{max}\\{2(m_{1}+m_{2})+1,m_{1}+3m_{2}+1\\}$.
Proofs of Lemma 3.3 and Lemma 3.4 are omitted as they follow from the idea of
[10].

######  Lemma 3.3.

Suppose $l_{i}$ is a divisor of $\frac{q^{n}-1}{r_{i}};\;i=1,2$. Let
$\\{p_{1},p_{2},\dots,p_{u}\\}$ be the set of all primes dividing
$\frac{q^{n}-1}{r_{1}}$ but not $l_{1}$, and $\\{q_{1},q_{2},\dots,q_{v}\\}$
be the set of all primes dividing $\frac{q^{n}-1}{r_{2}}$ but not $l_{2}$.
Also, let $\\{P_{1},P_{2},\dots,P_{s}\\}$ be the set of all monic irreducible
polynomials which divide $x^{n}-1$ but not $g_{1}$, and
$\\{Q_{1},Q_{2},\dots,Q_{t}\\}$ be the set of all monic irreducible
polynomials which divide $x^{n}-1$ but not $g_{2}$. Then

$\displaystyle C_{F,a,b}(\frac{q^{n}-1}{r_{1}},\frac{q^{n}-1}{r_{2}},x^{n}-1)$
$\displaystyle\geq\sum_{i=1}^{u}C_{F,a,b}(l_{1}p_{i},l_{2},g_{1},g_{2})+\sum_{i=1}^{v}C_{F,a,b}(l_{1},l_{2}q_{i},g_{1},g_{2})$
$\displaystyle+\sum_{i=1}^{s}C_{F,a,b}(l_{1},l_{2},g_{1}P_{i},g_{2})+\sum_{i=1}^{t}C_{F,a,b}(l_{1},l_{2},g_{1},g_{2}Q_{i})$
$\displaystyle-(u+v+s+t-1)C_{F,a,b}(l_{1},l_{2},g_{1},g_{2}).$

######  Lemma 3.4.

Suppose $l_{i}$ be a divisor of $\frac{q^{n}-1}{r_{i}};\;i=1,2$. Let
$\\{p_{1},p_{2},\dots,p_{u}\\}$ be the set of all primes dividing
$\frac{q^{n}-1}{r_{1}}$ but not $l_{1}$, and $\\{q_{1},q_{2},\dots,q_{v}\\}$
be the set of all primes dividing $\frac{q^{n}-1}{r_{2}}$ but not $l_{2}$.
Also, let $\\{P_{1},P_{2},\dots,P_{s}\\}$ be the set of monic irreducible
polynomials which divide $\frac{x^{n}-1}{f_{1}}$ but not $g_{1}$, and
$\\{Q_{1},Q_{2},\dots,Q_{t}\\}$ be the set of monic irreducible polynomials
which divide $\frac{x^{n}-1}{f_{2}}$ but not $g_{2}$. Suppose
$\tilde{g}_{i}=gcd(g_{i},\frac{x^{n}-1}{f_{i}})$; $i=1,2,$ and
$\Gamma=Mq^{\frac{n}{2}+k_{1}+k_{2}+2}r_{1}r_{2}W(l_{1})W(l_{2})W(\tilde{g}_{1})W(\tilde{g}_{2})$,
then

  1. (1)

$|C_{F,a,b}(l_{1}p_{i},l_{2},g_{1},g_{2})-\theta(p_{i})C_{F,a,b}(l_{1},l_{2},g_{1},g_{2})|\leq\dfrac{\theta(l_{1})\theta(l_{2})\theta(p_{i})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}\Gamma$
for $i=1,2,\dots,u,$

  2. (2)

$|C_{F,a,b}(l_{1},l_{2}q_{i},g_{1},g_{2})-\theta(q_{i})C_{F,a,b}(l_{1},l_{2},g_{1},g_{2})|\leq\dfrac{\theta(l_{1})\theta(l_{2})\theta(q_{i})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}\Gamma$
for $i=1,2,\dots,v,$

  3. (3)

$|C_{F,a,b}(l_{1},l_{2},g_{1}P_{i},g_{2})-\Theta(P_{i})C_{F,a,b}(l_{1},l_{2},g_{1},g_{2})|\leq\dfrac{\theta(l_{1})\theta(l_{2})\Theta(P_{i})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}\Gamma$
for $i=1,2,\dots,s,$ and

  4. (4)

$|C_{F,a,b}(l_{1},l_{2},g_{1},g_{2}Q_{i})-\Theta(Q_{i})C_{F,a,b}(l_{1},l_{2},g_{1},g_{2})|\leq\dfrac{\theta(l_{1})\theta(l_{2})\Theta(Q_{i})\Theta(g_{1})\Theta(g_{2})}{r_{1}r_{2}q^{2}}\Gamma$
for $i=1,2,\dots,t.$

We shall now give Sieve variation of Theorem 3.1 proof of which is omitted as
it follows from Lemma 3.3, Lemma 3.4 and the idea of [10].

######  Theorem 3.5.

Assume the notations and conditions in Lemma 3.4. Let
$\delta=1-\sum_{i=1}^{u}\frac{1}{p_{i}}-\sum_{i=1}^{v}\frac{1}{q_{i}}-\sum_{i=1}^{s}\frac{1}{q^{deg(P_{i})}}-\sum_{i=1}^{t}\frac{1}{q^{deg(Q_{i})}}>0$
and $\Delta=\frac{u+v+s+t-1}{\delta}+2$. If

$q^{\frac{n}{2}-k_{1}-k_{2}-2}>Mr_{1}r_{2}\Delta
W(l_{1})W(l_{2})W(\tilde{g}_{1})W(\tilde{g}_{2}),$

then $(q,n)\in A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$.

Let $\alpha$ and $\beta$ are positive real numbers, we define $A_{\alpha}$ and
$A_{\alpha,\beta}$ as follows:

$A_{\alpha}=\prod\limits_{\begin{subarray}{c}p<2^{\alpha}\\\ p\text{ is
prime}\end{subarray}}\frac{2}{\sqrt[\alpha]{p}}\;\;\;\text{and}\;\;\;A_{\alpha,\beta}=\prod\limits_{\begin{subarray}{c}p<2^{\alpha}\\\
p\text{ is prime}\end{subarray}}\frac{2}{\sqrt[\alpha+\beta]{p}}.$

Following two Lemmas, Lemma 3.6 and 3.7 follow from [1].

######  Lemma 3.6.

Suppose $r_{1}$ and $r_{2}$ are divisors positive of $q^{n}-1$,
$k_{1}+k_{2}+2<n/2$ is such that there exist factors $f_{1}$ and $f_{2}$ of
$x^{n}-1$ of degrees $k_{1}$ and $k_{2}$, respectively in $\mathbb{F}_{q}[x]$.
Let $(2n-k_{1}-k_{2})^{2}<q$. If
$q^{\frac{n}{2}-k_{1}-k_{2}-2}>Mr_{1}r_{2}(2n-k_{1}-k_{2}+2)W(\frac{q^{n}-1}{r_{1}})W(\frac{q^{n}-1}{r_{2}})$,
then $(q,n)\in A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$.

######  Lemma 3.7.

Let $r_{1}$ and $r_{2}$ are positive divisors of $q^{n}-1$,
$k_{1}+k_{2}+2<n/2$ is such that there exist factors $f_{1}$ and $f_{2}$ of
$x^{n}-1$ of degrees $k_{1}$ and $k_{2}$, respectively in $\mathbb{F}_{q}[x]$.
Suppose $\alpha>0$ be such that $\alpha>\frac{4n}{n-2(k_{1}+k_{2}+2)}$, and
let $d=\frac{2\alpha}{\alpha(n-2(k_{1}+k_{2}+2))-4n}$. If

$q\geq\min\\{\mathcal{U},max\\{\mathcal{V},(2n-k_{1}-k_{2})^{2}\\}\\},$

then, $(q,n)\in A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$, where
$\mathcal{U}=(M(r_{1}r_{2})^{1-\frac{1}{\alpha}}2^{2n-k_{1}-k_{2}}A_{\alpha}^{2})^{d}$
and
$\mathcal{V}=(M(r_{1}r_{2})^{1-\frac{1}{\alpha}}(2n-k_{1}-k_{2}+2)A_{\alpha}^{2})^{d}$.

Next Lemma will play an important role in numerical examples.

######  Lemma 3.8.

Let $r_{1}$ and $r_{2}$ are positive divisors of $q^{n}-1$,
$k_{1}+k_{2}+2<n/2$ is such that there exist factors $f_{1}$ and $f_{2}$ of
$x^{n}-1$ of degrees $k_{1}$ and $k_{2}$, respectively in $\mathbb{F}_{q}[x]$
and $(2n-k_{1}-k_{2})^{2}<q$. Suppose $\alpha$ and $\beta$ are positive reals
such that $\alpha+\beta>\frac{4n}{n-2(k_{1}+k_{2}+2)}$,
$\delta_{\alpha,\beta}=1-2S_{\alpha,\beta}-\frac{1}{2n-k_{1}-k_{2}}>0$, where
$S_{\alpha,\beta}=\sum\limits_{\begin{subarray}{c}2^{\alpha}<p<2^{\alpha+\beta}\\\
p\;prime\end{subarray}}\frac{1}{p}$, and
$\Delta_{\alpha,\beta}=2+\frac{2v_{\alpha,\beta}+2n-k_{1}-k_{2}-1}{\delta_{\alpha,\beta}}$,
where $v_{\alpha,\beta}$ is the number of primes between $2^{\alpha}$ and
$2^{\alpha+\beta}$. Let
$d=\frac{2(\alpha+\beta)}{(\alpha+\beta)(n-2(k_{1}+k_{2}+2))-4n}$. If

$q\geq(M(r_{1}r_{2})^{1-\frac{1}{\alpha+\beta}}A_{\alpha,\beta}^{2}\Delta_{\alpha,\beta})^{d},$

then $(q,n)\in A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$.

###### Proof.

Suppose $\alpha$ and $\beta$ are positive reals such that
$\alpha+\beta>\frac{4n}{n-2(k_{1}+k_{2}+2)}$. Let
$\frac{q^{n}-1}{r_{1}}=p_{1}^{a_{1}}\cdot p_{2}^{a_{2}}\cdots
p_{x}^{a_{x}}\cdot q_{1}^{b_{1}}\cdot q_{2}^{b_{2}}\cdots q_{u}^{b_{u}},$
where $2\leq p_{j}\leq 2^{\alpha}$ or $p_{j}\geq 2^{\alpha+\beta}$ for $1\leq
j\leq x$, and $2^{\alpha}<q_{k}<2^{\alpha+\beta}$ for $1\leq k\leq u$ and let
$\frac{q^{n}-1}{r_{2}}=r_{1}^{c_{1}}\cdot r_{2}^{c_{2}}\cdots
r_{y}^{c_{y}}\cdot s_{1}^{d_{1}}\cdot s_{2}^{d_{2}}\cdots s_{v}^{d_{v}},$
where $2\leq r_{j}\leq 2^{\alpha}$ or $r_{j}\geq 2^{\alpha+\beta}$ for $1\leq
j\leq y$, and $2^{\alpha}<s_{k}<2^{\alpha+\beta}$ for $1\leq k\leq v$. Assume
$l_{1}=p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{x}^{a_{x}}$ and
$l_{2}=r_{1}^{c_{1}}\cdot r_{2}^{c_{2}}\cdots r_{y}^{c_{y}}$, $g_{1}$ and
$g_{2}$ are divisors of $x^{n}-1$ such that
$\gcd(g_{i},\frac{x^{n}-1}{f_{i}})=1$, and any irreducible factor of $x^{n}-1$
divides $g_{i}$ or $\frac{x^{n}-1}{f_{i}}$; $i=1,2$.
Let $P_{1},P_{2},\dots,P_{s}$ and $Q_{1},Q_{2},\dots,Q_{t}$ be all irreducible
polynomials such that $rad(\frac{x^{n}-1}{f_{1}})$ $=P_{1}\cdot P_{2}\cdots
P_{s}$ and $rad(\frac{x^{n}-1}{f_{2}})=Q_{1}\cdot Q_{2}\cdots Q_{t}$,
respectively. By definition of $l_{1}$ and $l_{2}$, we have
$\delta=1-\sum_{k=1}^{u}\frac{1}{q_{k}}-\sum_{k=1}^{v}\frac{1}{s_{k}}-\sum_{k=1}^{s}\frac{1}{q^{degP_{k}}}-\sum_{k=1}^{t}\frac{1}{q^{degQ_{k}}}\geq
1-\sum_{k=1}^{u}\frac{1}{q_{k}}-\sum_{k=1}^{v}\frac{1}{s_{k}}-\frac{1}{2n-k_{1}-k_{2}}\geq
1-2S_{\alpha,\beta}-\frac{1}{2n-k_{1}-k_{2}}=\delta_{\alpha,\beta}$, and
$\Delta=\frac{u+v+s+t-1}{\delta}+2\leq\frac{2v_{\alpha,\beta}+2n-k_{1}-k_{2}-1}{\delta_{\alpha,\beta}}+2=\Delta_{\alpha,\beta}$.
From [4, Lemma 3.7], we have $W(m)\leq
A_{\alpha,\beta}m^{\frac{1}{\alpha+\beta}}$, i.e., we have
$W(l_{1})W(l_{2})\leq
A_{\alpha,\beta}^{2}q^{\frac{2n}{\alpha+\beta}}(r_{1}r_{2})^{\frac{-1}{\alpha+\beta}}$.
From Theorem 3.5, we conclude that, if
$q^{\frac{n}{2}-k_{1}-k_{2}-2}>M(r_{1}r_{2})^{1-\frac{1}{\alpha+\beta}}\Delta_{\alpha,\beta}A_{\alpha,\beta}^{2}q^{\frac{2n}{\alpha+\beta}}$
or, if
$q\geq(M(r_{1}r_{2})^{1-\frac{1}{\alpha+\beta}}A_{\alpha,\beta}^{2}\Delta_{\alpha,\beta})^{d}$,
then $(q,n)\in A_{F,a,b}(r_{1},r_{2},k_{1},k_{2})$. ∎

##  4\. Numerical Examples

In this section, we shall present some numerical examples. Specifically, we
aim to identify pairs $(q,n)$ for which $\mathbb{F}_{q^{n}}$ contains a
3-primitive, 2-normal element $\epsilon$, such that $F(\epsilon)$ is a
2-primitive, 1-normal in $\mathbb{F}_{q^{n}}$ satisfying
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon)$ $=a$ and
$\operatorname{Tr}_{\mathbb{F}_{q^{n}}/\mathbb{F}_{q}}(\epsilon^{-1})=b$, for
any prescribed $a,b\in\mathbb{F}_{q}$, where $F\in\Lambda_{q^{n}}(10,11)$.

Table 1.

Value of $n$ |  bound on $q$ corresponding to $\alpha$
---|---
$n=12$ |  $q\geq 1.65\times 10^{1769600}$ for $\alpha=25.5$
$n=13$ |  $q\geq 6.2\times 10^{16610}$ for $\alpha=18.9$
$n=14$ |  $q\geq 1.52\times 10^{1585}$ for $\alpha=15.6$
$n=15$ |  $q\geq 5.51\times 10^{384}$ for $\alpha=13.7$
$n=16$ |  $q\geq 3.14\times 10^{149}$ for $\alpha=12.5$
$n=17$ |  $q\geq 8.65\times 10^{75}$ for $\alpha=11.6$
$n=18$ |  $q\geq 3.59\times 10^{45}$ for $\alpha=10.9$
$n=19$ |  $q\geq 3.16\times 10^{30}$ for $\alpha=10.4$
$n=20$ |  $q\geq 1.07\times 10^{22}$ for $\alpha=10.0$
$n=21$ |  $q\geq 6.53\times 10^{16}$ for $\alpha=9.7$
$n=22$ |  $q\geq 2.37\times 10^{13}$ for $\alpha=9.5$
$n\geq 25$ |  $q\geq 8.42\times 10^{7}$ for $\alpha=8.9$
$n\geq 31$ |  $q\geq 13698$ for $\alpha=8.3$
$n\geq 62$ |  $q\geq 12334$ for $\alpha=9.3$
$n\geq 72$ |  $q\geq 5344$ for $\alpha=9.3$
$n\geq 100$ |  $q\geq 1485$ for $\alpha=9.6$
$n\geq 502$ |  $q\geq 141$ for $\alpha=11.3$

Suppose $q$ is any prime power, $r_{1}=3,r_{2}=2,k_{1}=2,k_{2}=1,m_{1}=10$,
and $m_{2}=11$. Let $F\in\Lambda_{q^{n}}(10,11)$, using Lemma 3.7 and an
appropriate value of $\alpha$, Table 1 consists pairs $(q,n)$ such that
$(q,n)\in A_{F,a,b}(3,2,2,1)$, under the condition that $x^{n}-1$ has a degree
2 factor in $\mathbb{F}_{q}[x]$, and 6 divides $q^{n}-1$. However, due to the
computationally intensive nature of the calculations involved, we were unable
to determine a bound on $q$ when $n=11$, and thus excluded it from
calculations. From Table 1 we see that, for smaller values of $q$, $(q,n)\in
A_{F,a,b}(3,2,2,1)$ when $n$ is large, and for smaller values of $n$, bound on
$q$ is significantly large.

Table 2.

$n$ |  $(\alpha,\beta)$ |  bound on $q$ corresponding to $(\alpha,\beta)$
---|---|---
$n\geq 53$ |  $(5.7,3.6)$ |  $q\geq 13$
$n\geq 36$ |  $(5.7,3.5)$ |  $q\geq 137$
$n\geq 33$ |  $(5.7,3.5)$ |  $q\geq 358$
$n\geq 30$ |  $(5.9,3.6)$ |  $q\geq 1464$
$n\geq 27$ |  $(6.1,3.8)$ |  $q\geq 13177$
$n\geq 24$ |  $(6.4,4.1)$ |  $q\geq 608734$
$n\geq 21$ |  $(6.7,4.3)$ |  $q\geq 1.86\times 10^{9}$
$n\geq 18$ |  $(7.6,4.8)$ |  $q\geq 2.16\times 10^{19}$
$n=16$ |  $(8.5,5.3)$ |  $q\geq 1.5\times 10^{44}$
$n=15$ |  $(9.2,5.7)$ |  $q\geq 4.11\times 10^{83}$
$n=14$ |  $(10.3,6.4)$ |  $q\geq 1.84\times 10^{216}$
$n=13$ |  $(12.2,7.5)$ |  $q\geq 4.75\times 10^{1047}$
$n=12$ |  $(16.4,10)$ |  $q\geq 8.56\times 10^{24648}$

To improve the bound on $q$ for different values of $n$, we employed Lemma 3.8
to those value of $(q,n)$ in Table 1. Accordingly, for suitable values of
$\alpha$ and $\beta$, Table 2 consists of pairs $(q,n)$ such that $(q,n)\in
A_{F,a,b}(3,2,2,1)$ under the condition that $x^{n}-1$ has a degree 2 factor
in $\mathbb{F}_{q}[x]$ and 6 divides $q^{n}-1$.

In next section, we find all pairs $(q,n)$ such that $(q,n)\in
A_{F,a,b}(3,2,2,1)$ in field of characteristic 13. The decision to restrict to
the field of characteristics 13 is due to the fact that calculations within
this field are not time consuming and tedious, whereas similar work in fields
of other characteristics can be conducted using advanced and efficient
computing resources.

##  5\. Proof of theorem 1.2

First we assume, $n\geq 14$. Using the fact, for a divisor $f$ of $x^{n}-1$,
$W(\frac{x^{n}-1}{f})\leq 2^{n-deg(f)}$, in corollary 3.2 we verify the
condition $q^{\frac{n}{2}-5}>44\cdot 6\cdot
W(\frac{q^{n}-1}{3})W(\frac{q^{n}-1}{2})\cdot$ $\cdot 2^{2n-3}$ for the pairs
$(q,n)$ not listed in Table 2 for which $x^{n}-1$ has a factor of degree 2 in
$\mathbb{F}_{q}[x]$. Suppose $f_{i}(x)$ is a degree $i$ factor of $x^{n}-1$ in
$\mathbb{F}_{q}[x]$; $i=1,2$. Using [25], we determine that the condition is
satisfied for all pairs except those that are enumerated in Table 3.

Utilizing precise factorization of $\frac{x^{n}-1}{f_{i}}$; $i=1,2$ in
$\mathbb{F}_{q}[x]$, we evaluated Theorem 3.1 for the values of $(q,n)$ in
Table 3 and we confirm that it is valid for all the pairs except the ones
specified in Table 4.

Table 3.

powers of $q=13$ |  $n$
---|---
13 |  36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52
13, $13^{2}$ |  23, 25, 27, 29, 30, 31, 32, 33, 34, 35
$13,\;13^{2},\;13^{3}$ |  22, 24, 26, 28
$13,\;13^{2},\;13^{3},\;13^{4}$ |  20, 21
$13,\;13^{2},\dots,13^{6}$ |  16
$13,\;13^{2},\dots,13^{8}$ |  15
$13,\;13^{2},\dots,13^{10},\;13^{12}$ |  14

Table 4.

prime powers of $q=13$ |  $n$
---|---
13 |  22 ,27, 28, 30, 32, 36, 40, 42, 48
$13,\;13^{2},\;13^{3}$ |  20, 21, 24
$13,\;13^{2},\;13^{3},\;13^{4}$ |  18
$13,\;13^{2},\dots,13^{5}$ |  16
$13,\;13^{2},\;13^{3},\;13^{4},\;13^{6},\;13^{8}$ |  15
$13,\;13^{2},\dots,13^{6},\;13^{9}$ |  14

For the case $n=13$, first we shall examine prime powers $q\leq 4.75\times
10^{1047}$, and establish the following lemma:

######  Lemma 5.1.

Let $q$ be any prime power such that $2.29\times 10^{7}\leq q\leq 4.75\times
10^{1047}$. If $6\mid(q^{13}-1)$ and $x^{13}-1$ has a degree 2 factor in
$\mathbb{F}_{q}[x]$, then $(q,13)\in A_{F,a,b}(3,2,2,1)$.

###### Proof.

Assume that $q$ is a prime power such that $q\leq 4.75\times 10^{1047}$ and
$6\mid(q^{13}-1)$. Also, let $f_{i}(x)$ is a degree $i$ factor of $x^{13}-1$;
$i=1,2$. We use Theorem 3.5 with $l_{1}=\frac{q-1}{3}$ and
$l_{2}=\frac{q-1}{2}$, and suppose $g_{i}=f_{i}$ if $13\nmid q$ and $g_{i}=1$
if $13\mid q$, that is, $\tilde{g_{i}}=1$ and $s+t\leq 23;i=1,2$. Let $p$ and
$p^{\prime}$ be primes such that $p$ divides $\frac{q^{13}-1}{3}$ but not
$\frac{q-1}{3}$, and $p^{\prime}$ divides $\frac{q^{13}-1}{2}$ but not
$\frac{q-1}{2}$, respectively, i.e., $13\mid 2(p-1)$ and
$13\mid(p^{\prime}-1)$, which means the set $U=\\{p_{1},p_{2},\dots,p_{u}\\}$
constitutes primes of the type $\frac{13i+2}{2}$ and that of
$V=\\{p_{1}^{\prime},p_{2}^{\prime},\dots,p_{v}^{\prime}\\}$ constitutes
primes of the type $13j+1$. Suppose $\mathcal{Q}_{m}$ denote the set of first
$m$ primes of type $\frac{13i+2}{2}$, and $\mathcal{Q}_{m}^{\prime}$ denote
the set of first $m$ primes of type $13j+1$. Let

$\mathcal{S}_{m}=\sum\limits_{r\in\mathcal{Q}_{m}}\frac{1}{r}\;\text{and}\;\mathcal{P}_{m}=\prod\limits_{r\in\mathcal{Q}_{m}}r,\;\text{and}\;\mathcal{S}_{m}^{\prime}=\sum\limits_{r\in\mathcal{Q}_{m}^{\prime}}\frac{1}{r}\;\text{and}\;\mathcal{P}_{m}^{\prime}=\prod\limits_{r\in\mathcal{Q}_{m}^{\prime}}r.$

Since elements of $U$ are primes which divide
$\frac{q^{12}+q^{11}+\dots+q+1}{3}$ and that of $V$ are primes which divides
$\frac{q^{12}+q^{11}+\dots+q+1}{2}$, we have $\mathcal{P}_{u}\leq 4.39\times
10^{12571}$ and $\mathcal{P}_{v}^{{}^{\prime}}\leq 6.597\times 10^{12571}$,
which gives $u\leq 2482$ and $v\leq 2482$, and $\mathcal{S}_{u}\leq 0.111533$
and $\mathcal{S}_{v}^{{}^{\prime}}\leq 0.111533$. Assuming $q\geq 10^{4}$ and
since $s+t\leq 23$, we have
$\delta=1-\sum_{i=1}^{u}\frac{1}{p_{i}}-\sum_{i=1}^{v}\frac{1}{p_{i}^{\prime}}-\sum_{i=1}^{s}\frac{1}{q^{deg(P_{i})}}-\sum_{i=1}^{t}\frac{1}{q^{deg(Q_{i})}}\geq
1-\mathcal{S}_{u}-\mathcal{S}_{v}^{\prime}-\frac{23}{q}>0.774635$, which
implies $\Delta<6438.5799$. For real number $\alpha>\frac{4}{3}$, if
$q\geq(44\cdot 6^{1-\frac{1}{\alpha}}\cdot 6438.5799\cdot
A_{\alpha}^{2})^{\frac{2\alpha}{3\alpha-4}}$, it follows from [4, Lemma 3.7 ],
Theorem 3.5 and $\alpha=4.3$ that, $(q,13)\in A_{F,a,b}(3,2,2,1)$, for $q\geq
2.289\times 10^{7}$. ∎

For $q\leq 2.29\times 10^{7}$, i.e.,
$q=13,13^{2},13^{3},13^{4},13^{5},13^{6}$, Theorem 3.1 does not hold. For
these prime powers along with those in Table 4, we find that seiving variation
holds for majority of them (Table 5) with the exception of those mentioned in
Thereom 1.2. Theorem 1.2 concluded. ∎
We can observe that for most of the values of $n$ in Table 5, $g_{i}=f_{i}$;
$i=1,2.$

## Table 5.

List of pairs $(q,n)$ where criterion of Theorem 3.1 fails but Theorem 3.5
holds true for certain selection of $l_{1}$, $l_{2}$, $f_{1}(x)$, $f_{2}(x)$,
$g_{1}(x)$, and $g_{2}(x)$.

$(q,n)$ |  $(l_{1},l_{2})$ |  $(f_{1}(x),f_{2}(x))$ |  $(g_{1}(x),g_{2}(x))$
---|---|---|---
$(13^{4},13)$ |  (6,10) |  $(x-1,(x-1)^{2})$ |  $(x-1,(x-1)^{2})$
$(13^{5},13)$ |  (6,2) |  $(x-1,(x-1)^{2})$ |  $(x-1,(x-1)^{2})$
$(13^{6},13)$ |  (6,2) |  $(x-1,(x-1)^{2})$ |  $(x-1,(x-1)^{2})$
$(13^{3},14)$ |  (6,6) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{4},14)$ |  (6,10) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{5},14)$ |  (6,14) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{6},14)$ |  (6,6) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{7},14)$ |  (6,14) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{8},14)$ |  (6,10) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{9},14)$ |  (6,6) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{10},14)$ |  (6,10) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{12},14)$ |  (30,30) |  $(x+1,x^{2}+3x+1)$ |  $(x+1,x^{2}+3x+1)$
$(13^{2},15)$ |  (6,6) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$
$(13^{3},15)$ |  (6,6) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$
$(13^{4},15)$ |  (30,30) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$
$(13^{5},15)$ |  (6,6) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$
$(13^{6},15)$ |  (6,6) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$

$(13^{7},15)$ |  (6,6) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$
---|---|---|---
$(13^{8},15)$ |  (30,30) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$
$(13^{2},16)$ |  (6,10) |  $(x+1,x^{2}+5)$ |  $(x+1,x^{2}+5)$
$(13^{3},16)$ |  (6,6) |  $(x+1,x^{2}+5)$ |  $(x+1,x^{2}+5)$
$(13^{4},16)$ |  (6,10) |  $(x+1,x^{2}+5)$ |  $(x+1,x^{2}+5)$
$(13^{5},16)$ |  (6,10) |  $(x+1,x^{2}+5)$ |  $(x+1,x^{2}+5)$
$(13^{6},16)$ |  (6,6) |  $(x+1,x^{2}+5)$ |  $(x+1,x^{2}+5)$
$(13^{2},18)$ |  (6,30) |  $(x+1,x^{2}+4x+3)$ |  $(x+1,x^{2}+4x+3)$
$(13^{3},18)$ |  (6,2) |  $(x+1,x^{2}+4x+3)$ |  $(x+1,x^{2}+4x+3)$
$(13^{4},18)$ |  (210,210) |  $(x+1,x^{2}+4x+3)$ |  $(x+1,x^{2}+4x+3)$
$(13^{5},18)$ |  (2,2) |  $(x+1,x^{2}+4x+3)$ |  $(x+1,x^{2}+4x+3)$
$(13^{6},18)$ |  (30,30) |  $(x+1,x^{2}+4x+3)$ |  $(x+1,x^{2}+4x+3)$
$(13^{2},20)$ |  (6,10) |  $(x+1,x^{2}+6x+5)$ |  $(x+1,x^{2}+6x+5)$
$(13^{3},20)$ |  (30,30) |  $(x+1,x^{2}+6x+5)$ |  $(x+1,x^{2}+6x+5)$
$(13,21)$ |  (6,6) |  $(x+4,x^{2}+x+9)$ |  $(x-3,x^{2}+2x+3)$
$(13^{2},21)$ |  (6,6) |  $(x+4,x^{2}+x+9)$ |  $(x-3,x^{2}+2x+3)$
$(13^{3},21)$ |  (6,6) |  $(x+4,x^{2}+x+9)$ |  $(x-3,x^{2}+2x+3)$
$(13,22)$ |  (6,14) |  $(x+1,x^{2}-1)$ |  $(x+1,x^{2}-1)$
$(13^{2},24)$ |  (6,6) |  $(x+1,x^{2}+2)$ |  $(x+1,x^{2}+2)$
$(13^{3},24)$ |  (6,30) |  $(x+1,x^{2}+2)$ |  $(x+1,x^{2}+2)$
$(13,27)$ |  (6,6) |  $(x+4,x^{2}+x+1)$ |  $(x+4,x^{2}+x+1)$
$(13,28)$ |  (210,1085) |  $(x+1,x^{2}+x-1)$ |  $(x+1,x^{2}+x-1)$
$(13,30)$ |  (42,42) |  $(x+1,x^{2}+4x+3)$ |  $(x+1,x^{2}+4x+3)$
$(13,32)$ |  (6,10) |  $(x+1,x^{2}+6x+5)$ |  $(x+1,x^{2}+6x+5)$
$(13,40)$ |  (30,70) |  $(x+1,x^{2}+6x+5)$ |  $(x+1,x^{2}+6x+5)$
$(13,42)$ |  (1218,52374) |  $(x+1,x^{2}+4x+3)$ |  $(x+1,x^{2}+4x+3)$

For $(q,n)=(13,36)$, we took $(l_{1},l_{2})=(30,210)$,
$(f_{1}(x),f_{2}(x))=(x+1,x^{2}+2x+3)$ and
$g_{1}(x)=g_{2}(x)=(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)(x+9)(x+10)(x+11)(x+12)(x^{3}+2)(x^{3}+3)(x^{3}+4)$.

For $(q,n)=(13,48)$, we took $(l_{1},l_{2})=(30,30)$,
$(f_{1}(x),f_{2}(x))=(x+1,x^{2}+2)$ and
$g_{1}(x)=g_{2}(x)=(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)(x+7)(x+8)(x+9)(x+10)(x+11)(x^{4}+2)(x^{4}+5)(x^{4}+6)(x^{4}+7)(x^{4}+7)(x^{4}+8)(x^{4}+11)$.

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