# Primitive normal pairs with prescribed traces over finite fields

Shikhamoni Nath  Department of Mathematical Sciences, Tezpur University,
Tezpur, Assam, 784028, India<EMAIL_ADDRESS>,  Arpan Chandra Mazumder
Department of Mathematical Sciences, Tezpur University, Tezpur, Assam, 784028,
India<EMAIL_ADDRESS>and  Dhiren Kumar Basnet  Department of
Mathematical Sciences, Tezpur University, Tezpur, Assam, 784028, India
<EMAIL_ADDRESS>
###### Abstract.

Let $q$ be a positive integral power of some prime $p$ and
$\mathbb{F}_{q^{m}}$ be a finite field with $q^{m}$ elements for some
$m\in\mathbb{N}$. Here we establish a sufficient condition for the existence
of primitive normal pairs of the type $(\epsilon,f(\epsilon))$ in
$\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$ with two prescribed traces,
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$ and
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(f(\epsilon))=b$, where
$f(x)\in\mathbb{F}_{q^{m}}(x)$ is a rational function with some restrictions
and $a,b\in\mathbb{F}_{q}$. Furthermore, for $q=5^{k}$, $m\geq 9$ and rational
functions with degree sum 4, we explicitly find at most 12 fields in which the
desired pair may not exist.

###### Key words and phrases:

Finite fields; Primitive elements; Normal elements; Additive and
multiplicative characters; Trace

###### 2020 Mathematics Subject Classification:

12E20, 11T23

##  1\. Introduction

Given a prime power $q$ and a positive integer $m$, we denote the finite field
with $q$ elements by $\mathbb{F}_{q}$ and its extension field of degree $m$ by
$\mathbb{F}_{q^{m}}$. The multiplicative group $\mathbb{F}^{*}_{q^{m}}$ is a
cylic group and a generator of this group is called a primitive element. For
an element $\epsilon\in\mathbb{F}_{q^{m}}$, the elements
$\epsilon,\epsilon^{q},\epsilon^{q^{2}},\ldots,\epsilon^{q^{m-1}}$ are called
conjugates of $\epsilon$ with respect to $\mathbb{F}_{q}$. Sum of all these
conjugates is called trace of $\epsilon$ over $\mathbb{F}_{q}$ and is denoted
by $Tr_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\epsilon)$. Further, an element
$\epsilon\in\mathbb{F}_{q^{m}}$ for which the set
$\\{\epsilon,\epsilon^{q},\epsilon^{q^{2}},\ldots,\epsilon^{q^{m-1}}\\}$ forms
a basis of $\mathbb{F}_{q^{m}}(\mathbb{F}_{q})$, when considered as a vector
space, is called a $normal$ element. An element which is simultaneously
primitive and normal is referred as a primitive normal element. Existence of
such elements is firstly given by the following theorem.

######  Theorem 1.1.

(Primitive normal basis theorem). In the finite field $\mathbb{F}_{q^{m}}$,
there always exists some element which is simultaneously primitive and normal.

Lenstra and Schoof proved this result in [14]. Later, Cohen and Huczynska [6]
presented a computer-free proof of this result with the help of sieving
techniques.

In 1985, Cohen [2], proved existence of consecutive pairs of primitive
elements of the type $(\epsilon,\epsilon^{-1})$ in $\mathbb{F}_{q}$. After
that, it was further developed in [3], [1], [9], among others. Similar studies
on primitive normal pairs emerged following the development of the primitive
normal basis theorem; they can be studied from [5], [12], [16], and so on, as
well as the references therein.

In 2001, Chou and Cohen [4] studied existence of primitive pair
$(\epsilon,\epsilon^{-1})$ in $\mathbb{F}_{q}$ such that both have trace zero
over $\mathbb{F}_{p}$. Cao and Wang [10] proved that, for any prime power $q$
and any positive integer $m$, there exists elements
$\epsilon\in\mathbb{F}_{q^{m}}$ such that $\epsilon+\epsilon^{-1}$ is a
primitive element of $\mathbb{F}_{q^{m}}$ and
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$ and
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon^{-1})=b$ for any
prescribed $a,b\in\mathbb{F}^{*}_{q}$. In [8], Choudhary et al. discussed the
existence of primitive pairs $(\epsilon,f(\epsilon))$ with two prescribed
traces, where $f$ is a rational function with some restrictions. Recently
Sharma et al. [15] considered rational functions $f$ with some minor
restrictions over $\mathbb{F}_{q^{m}}$ for some prime power $q$ and
$m\in\mathbb{N}$ and provided a sufficient condition for the existence of
primitive normal pair of the form $(\epsilon,f(\epsilon))$ over
$\mathbb{F}_{q}$.

Inspired by all these results we are presenting a sufficient condition for
existence of primitive normal pair $(\epsilon,f(\epsilon))$ in
$\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$ such that
$Tr_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\epsilon)=a$ and
$Tr_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(f(\epsilon))=b$ where
$f(x)\in\mathbb{F}_{q^{m}}(x)$ is a rational function with some restrictions
and $a,b\in\mathbb{F}_{q}$. The kind of rational functions we will study in
this paper is defined below:

######  Definition 1.1.

The set $\mathcal{R}_{q^{m}}^{n}$ consists of rational functions of the form
$f=\frac{f_{1}}{f_{2}}$ of degree sum $n=\deg(f_{1})+\deg(f_{2})$ such that

(i) $f\neq cx^{j}h^{d}$ for any $c\in\mathbb{F}^{*}_{q^{m}}$, any
$j\in\mathbb{Z}$, any $h\in\mathbb{F}_{q^{m}}(x)$ and any divisor $d$ other
than $1$ of $q^{m}-1$.

(ii) there exists at least one monic irreducible factor $g$ of $f_{2}$ in
$\mathbb{F}_{q^{m}}[x]$ with multiplicity $r$ such that $q^{m}\nmid r$.

We will use $\deg(f_{1})=n_{1}$ and $\deg(f_{2})=n_{2}$ throughout. We also
note that the second condition of the above definition implies that $n_{2}>0$.
This type of rational functions are popularly known as Cohen’s kind of
rational functions [9] that also includes Carvalho’s kind of rational
functions [5].

In Section 3, we derive a sufficient condition for the existence of desired
primitive normal pairs. In Section 4, we use a sieving inequality to weaken
the sufficient condition for more efficient results. In Section 5, we study
the application of our results over fields of chracteristic $5$ and for
rational functions of degree sum $4$ and proved the following:

######  Theorem 1.2.

Let $q=5^{k},m\geq 9$ and $f\in\mathcal{R}_{q^{m}}^{n}$. Then for any
$a,b\in\mathbb{F}_{q}$, there exists an element
$\epsilon\in\mathbb{F}_{q^{m}}$ with
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$,
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(f(\epsilon))=b$ and
$(\epsilon,f(\epsilon))$ is a primitive normal pair in $\mathbb{F}_{q^{m}}$
over $\mathbb{F}_{q}$ unless $(q,m)$ is one of the $12$ pairs : $(5,9)$,
$(5,10)$, $(5,11)$, $(5,12)$, $(5,14)$, $(5,18)$, $(5,20)$, $(5,24)$,
$(5^{2},9)$, $(5^{2},10)$, $(5^{2},12)$, $(5^{2},24)$.

Now, we present some notations that are significant to this article. We
indicate the algebraic closure of $\mathbb{F}_{q^{m}}$ by $\mathbb{F}$. Let
$\mathcal{A}_{p}^{n}$ be the set of pairs $(q,m)$ such that there exists an
element $\epsilon\in\mathbb{F}_{q^{m}}$ with
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$ and
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(f(\epsilon))=b$, where
$f(x)\in\mathcal{R}_{q^{m}}^{n}$ and $a,b\in\mathbb{F}_{q}$. For each
$n\in\mathbb{N}$, we denote the number of prime divisors of $n$ and the number
of square-free divisors of $n$ by $w(n)$ and $W(n)$, respectively.
Additionally, for $r(x)\in\mathbb{F}_{q}[x]$, we indicate the number of
square-free and monic irreducible $\mathbb{F}_{q}$-divisors of $r$ by $W(r)$
and $w(r)$, respectively.

##  2\. Preliminaries

In this section, we review a few definitions, lemmas, and notations that will
be used in proving a sufficient condition for the existence of primitive
normal pairs of our interest.

######  Definition 2.1.

(Characters) Let $G$ be a finite abelian group. Then a character $\chi$ of $G$
is a homomorphism from $G$ to the group $S^{1}:=\\{z\in\mathbb{C}:|z|=1\\}$.
The characters of $G$ form a group under multiplication called
$\mathit{dual\thinspace group}$ or $\mathit{character\thinspace group}$ of $G$
which is denoted by $\widehat{G}$. It is well known that $\widehat{G}$ is
isomorphic to $G$. Also, $\chi_{1}$ denotes the trivial character of $G$
defined as $\chi_{1}(a)=1_{G}$, for all $a\in G$, where $1_{G}$ is the
identity element of the group.

If the order of a character of the group $G$ is $d$, then it is denoted by
$\chi_{d}$. We also denote all the characters of order $d$ by $(d)$.

Corresponding to two different abelian group structures present in a finite
field $\mathbb{F}_{q^{m}}$ ie $(\mathbb{F}_{q^{m}},+)$ and
$(\mathbb{F}^{*}_{q^{m}},\cdot)$, two types of characters can be formed. We
refer to them as additive character denoted by $\psi$ and multiplicative
character denoted by $\chi$ respectively. Multiplicative characters are
extended from $\mathbb{F}^{*}_{q^{m}}$ to $\mathbb{F}_{q^{m}}$ by the rule
$\chi(0)=\begin{cases}0\,\mbox{ if}\,\chi\neq\chi_{1},\\\ 1\,\mbox{
if}\,\chi=\chi_{1}.\end{cases}$

The additive character $\bar{\psi}(\epsilon)=e^{2{\pi}iTr(\epsilon)/p}$, for
all $\epsilon\in\mathbb{F}_{q^{m}}$, where $Tr$ is the absolute trace function
from $\mathbb{F}_{q^{m}}$ to $\mathbb{F}_{p}$, is called the canonical
additive character of $\mathbb{F}_{q^{m}}$ and every additive character
$\psi_{\alpha}$ for $\alpha\in\mathbb{F}_{q^{m}}$ can be expressed in terms of
the cannonical additive character $\bar{\psi}$ as
$\psi_{\alpha}(\beta)=\bar{\psi}(\alpha\beta)$ for all
$\beta\in\mathbb{F}_{q^{m}}$.

######  Definition 2.2.

($e$-free element) For any divisor $e$ of $q^{m}-1$, an element
$\epsilon\in\mathbb{F}_{q^{m}}$ is called $e$-free if $\epsilon\neq\beta^{d}$
for any $\beta\in\mathbb{F}_{q^{m}}$ and for any divisor $d$ of $e$ other than
$1$.

An element $\epsilon\in\mathbb{F}^{*}_{q^{m}}$ is primitive if and only if
$(q^{m}-1)$-free. The characteristic function for $e$-free elements is defined
as follows:

(2.1)
$\rho_{e}:\mathbb{F}^{*}_{q^{m}}\to\\{0,1\\};\epsilon\mapsto\theta(e)\sum_{d|e}\frac{\mu(d)}{\phi(d)}\sum_{(d)}\chi_{d}(\epsilon),$

where $\theta(e)=\frac{\phi(e)}{e}$ and $\phi,\mu$ denotes the Euler’s totient
function and Mobius function respectively.

The additive group of $\mathbb{F}_{q^{m}}$ is a $\mathbb{F}_{q}[x]$-module
under the rule
$f\circ\alpha=\overset{n}{\underset{i=0}{\sum}}a_{i}\alpha^{q^{i}}$; for
$\alpha\in\mathbb{F}_{q^{m}}$ and
$f\,=\overset{n}{\underset{i=0}{\sum}}a_{i}x^{i}\thinspace\in\mathbb{F}_{q}[x]$.
For $\alpha\in\mathbb{F}_{q^{m}}$, the $\mathbb{F}_{q}$-order of $\alpha$ is
the monic $\mathbb{F}_{q}$-divisor $g$ of $x^{m}-1$ of minimal degree such
that $g\circ\alpha=0$.

Similarly we can define order of an additive character. Let
$\psi\in\widehat{\mathbb{F}_{q^{m}}}$. Then the $\mathbb{F}_{q}$-order of
$\psi$ is the least monic divisor $g$ of $x^{m}-1$ such that $\psi\circ g$ is
the trivial character where $\psi\circ g(\beta)=\psi(g\circ\beta)$.

######  Definition 2.3.

($g$-free element) Let $g$ divides $x^{m}-1$ and
$\epsilon\in\mathbb{F}_{q^{m}}$. If $\epsilon\neq h\circ\gamma$ for any
$\gamma\in\mathbb{F}_{q^{m}}$ and for any divisor $h$ of $g$ other than $1$,
then $\epsilon$ is called a $g$-free element.

An element $\epsilon\in\mathbb{F}_{q^{m}}$ is normal if and only if it is
$(x^{m}-1)$-free. The characteristic function for $g$-free elements is defined
as follows:

(2.2)
$\eta_{g}:\mathbb{F}_{q^{m}}\to\\{0,1\\};\epsilon\mapsto\Theta(g)\sum_{h|g}\frac{\mu^{{}^{\prime}}(g)}{\Phi(g)}\sum_{(h)}\psi_{h}(\epsilon),$

where $\Theta(g)=\frac{\Phi(g)}{q^{\deg(g)}}$ and
$\Phi(g)=|{(\frac{\mathbb{F}_{q}[x]}{<g>})}^{*}|$ and the function
$\mu^{{}^{\prime}}$ is defined as follows:

$\mu^{{}^{\prime}}(g)=\begin{cases}(-1)^{s},\text{ when $g$ is a product of
$s$ distict monic irreducible polynomials}\\\ 0,\text{ otherwise}.\end{cases}$

We shall also need characteristic function for prescribed values of trace. For
each $a\in\mathbb{F}_{q}$ the characteristic function for elements
$\epsilon\in\mathbb{F}_{q^{m}}$ such that
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$ is defined as
follows:

(2.3)
$\tau_{a}:\mathbb{F}_{q^{m}}\to\\{0,1\\};\epsilon\mapsto\frac{1}{q}\sum_{\psi\in\widehat{\mathbb{F}_{q}}}\psi(Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)-a).$

The following lemmas will be handy while finding the sufficient conditions.

######  Lemma 2.1 ([11], Theorem 5.5).

Let $f(x)\in\mathbb{F}_{q^{m}}(x)$ be a rational function. Write
$f(x)=\overset{k}{\underset{j=1}{\prod}}{f_{j}(x)}^{r_{j}}$, where
$f_{j}(x)\in\mathbb{F}_{q^{m}}[x]$ are irreducible polynomials and $r_{j}$ are
nonzero integers. Let $\chi$ be a nontrivial multiplicative character of
$\mathbb{F}_{q^{m}}$ of squarefree order $d$ which is a divisor of $q^{m}-1$.
Suppose that $f(x)$ is not of the form $cg(x)^{d}$ for any rational function
$g(x)\in\mathbb{F}_{q^{m}}(x)$ and $c\in\mathbb{F}^{*}_{q^{m}}$. Then we have,

$\bigg{|}{\underset{\alpha\in\mathbb{F}_{q^{m}},f(\alpha)\neq
0,\infty}{\sum}\chi(f(\alpha))}\bigg{|}\leq\bigg{(}\overset{k}{\underset{j=1}{\sum}}\deg(f_{j})-1\bigg{)}q^{\frac{m}{2}}.$

######  Lemma 2.2 ([11], Theorem 5.6).

Let $f(x),g(x)\in\mathbb{F}_{q^{m}}(x)$ be rational functions. Write
$f(x)=\overset{k}{\underset{j=1}{\prod}}{f_{j}(x)}^{r_{j}}$, where
$f_{j}(x)\in\mathbb{F}_{q^{m}}[x]$ are irreducible polynomials and $r_{j}$ are
nonzero integers. Let $D_{1}=\overset{k}{\underset{j=1}{\sum}}\deg(f_{j})$ ,
$D_{2}=\max(\deg(g),0)$, $D_{3}=\text{ degree of denominator of }g(x)$ and
$D_{4}=$ sum of degrees of irreducible polynomials dividing the denominator of
$g$ but distinct from $f_{j}(x)$ for all $j=1,2,\dots,k.$ Let $\chi$ be a
multiplicative character of $\mathbb{F}_{q^{m}}$, and let $\psi$ be a
nontrivial additive character of $\mathbb{F}_{q^{m}}$. Suppose $g(x)$ is not
of the form $r(x)^{q^{m}}-r(x)$ in $\mathbb{F}_{q^{m}}(x)$, where
$r(x)\in\mathbb{F}_{q^{m}}(x)$ is a rational function. Then we have the
estimate

$\Bigg{|}{\underset{\begin{subarray}{c}\alpha\in\mathbb{F}_{q^{m}},f(\alpha)\neq
0,\infty,\\\ g(\alpha)\neq
0,\infty\end{subarray}}{\sum}\chi(f(\alpha))}\Bigg{|}\leq(D_{1}+D_{2}+D_{3}+D_{4}-1)q^{\frac{m}{2}}$

Using these, we can now proceed to find all the pairs $(q,m)$ for which there
exists a rational function $f\in\mathcal{R}_{q^{m}}^{n}$ of degree sum $n$,
$(\epsilon,f(\epsilon))$ is a primitive normal pair with
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$ and
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(f(\epsilon))=b$ for some
prescribed values $a,b\in\mathbb{F}_{q}$.

##  3\. A sufficient condition for elements in $\mathcal{A}_{p}^{n}$

Let $e_{1},e_{2}$ be two divisors of $q^{m}-1$ and $g_{1},g_{2}$ be two
divisors of $x^{m}-1$. We denote the number of $\epsilon\in\mathbb{F}_{q^{m}}$
such that $\epsilon$ is both $e_{1}$-free and $g_{1}$-free and $f(\epsilon)$
is both $e_{2}$-free and $g_{2}$-free with
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$ and
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(f(\epsilon)=b$, where
$a,b\in\mathbb{F}_{q}$ and $f(x)\in\mathcal{R}_{q^{m}}^{n}$ by
$N_{f,a,b}(e_{1},e_{2},g_{1},g_{2})$. Let $S$ be the set containing all poles
and zeros of $f(x)$ along with $0$. Now using the characteristic functions
2.1), (2.2), and (2.3), we get

$N_{f,a,b}(e_{1},e_{2},g_{1},g_{2})=\underset{\epsilon\in\mathbb{F}_{q^{m}}\textbackslash
S}{\sum}\rho_{e_{1}}(\epsilon)\rho_{e_{2}}(f(\epsilon))\eta_{g_{1}}(\epsilon)\eta_{g_{2}}(f(\epsilon))$

$\displaystyle N_{f,a,b}(e_{1},e_{2},g_{1},g_{2})=$
$\displaystyle\underset{\epsilon\in\mathbb{F}_{q^{m}}\textbackslash
S}{\sum}\rho_{e_{1}}(\epsilon)\rho_{e_{2}}(f(\epsilon))\eta_{g_{1}}(\epsilon)\eta_{g_{2}}(f(\epsilon))$
$\displaystyle=$
$\displaystyle\frac{\theta(e_{1})\theta(e_{2})\Theta(g_{1})\Theta(g_{2})}{q^{2}}\underset{\begin{subarray}{c}d_{1}|e_{1},d_{2}|e_{2}\\\
h_{1}|g_{1},h_{2}|g_{2}\end{subarray}}{\sum}\frac{\mu(d_{1})\mu(d_{2})\mu^{\prime}(h_{1})\mu^{\prime}(h_{2})}{\phi(d_{1})\phi(d_{2})\Phi(h_{1})\Phi(h_{2})}$
$\displaystyle\underset{\begin{subarray}{c}\chi_{d_{1}},\chi_{d_{2}}\\\
\psi_{h_{1}},\psi_{h_{2}}\end{subarray}}{\sum}\chi_{f,a,b}(d_{1},d_{2},h_{1},h_{2})$

where,

$\displaystyle\chi_{f,a,b}(d_{1},d_{2},h_{1},h_{2})=$
$\displaystyle\underset{\epsilon\in\mathbb{F}_{q^{m}}\textbackslash
S}{\sum}\chi_{d_{1}}(\epsilon)\chi_{d_{2}}(f(\epsilon))\psi_{h_{1}}(\epsilon)\psi_{h_{2}}(f(\epsilon))\underset{\psi\in\widehat{\mathbb{F}_{q}}}{\sum}\psi(Tr(\epsilon)-a)$
$\displaystyle\underset{\psi^{\prime}\in\widehat{\mathbb{F}_{q}}}{\sum}\psi^{\prime}(Tr(f(\epsilon))-b)$
$\displaystyle=$
$\displaystyle\underset{\epsilon\in\mathbb{F}_{q^{m}}\textbackslash
S}{\sum}\chi_{d_{1}}(\epsilon)\chi_{d_{2}}(f(\epsilon))\psi_{h_{1}}(\epsilon)\psi_{h_{2}}(f(\epsilon))$
$\displaystyle\underset{u_{1}\in\mathbb{F}_{q}}{\sum}\bar{\psi}(Tr(u_{1}\epsilon)-u_{1}a)\underset{u_{2}\in\mathbb{F}_{q}}{\sum}\bar{\psi}(Tr(u_{2}f(\epsilon))-u_{2}b)$

Here we have considered $\psi(\beta)=\bar{\psi}(u_{1}\beta)$ and
$\psi^{\prime}(\beta)=\bar{\psi}(u_{2}\beta)$ for all
$\beta\in\mathbb{F}_{q^{m}}$. Let us denote $\bar{\psi}\circ
Tr=\widehat{\psi}$. Then the above equation takes the form

$\displaystyle\chi_{f,a,b}(d_{1},d_{2},h_{1},h_{2})=$
$\displaystyle\underset{u_{1},u_{2}\in\mathbb{F}_{q}}{\sum}\bar{\psi}(-u_{1}a-u_{2}b)\underset{\epsilon\in\mathbb{F}_{q^{m}}\textbackslash
S}{\sum}\chi_{d_{1}}(\epsilon)\chi_{d_{2}}(f(\epsilon))\psi_{h_{1}}(\epsilon)\psi_{h_{2}}(f(\epsilon))\widehat{\psi}(u_{1}\epsilon+u_{2}\epsilon)$

There exists, $v_{1},v_{2}\in\mathbb{F}_{q^{m}}$ such that
$\psi_{h_{1}}(\epsilon)=\widehat{\psi}(v_{1}\epsilon)$ and
$\psi_{h_{2}}(\epsilon)=\widehat{\psi}(v_{2}f(\epsilon))$. Moreover we
consider $\chi_{d_{1}}=\chi^{m_{1}}_{q^{m}-1}$ and
$\chi_{d_{2}}=\chi^{m_{2}}_{q^{m}-1}$ for some multipicative character
$\chi_{q^{m}-1}$ of order $q^{m}-1$, where $m_{1}\in\\{0,1,\dots,q^{m}-2\\}$
and $m_{2}=\frac{q^{m}-1}{d_{2}}$. Then we get,

$\displaystyle\chi_{f,a,b}(d_{1},d_{2},h_{1},h_{2})=$
$\displaystyle\underset{u_{1},u_{2}\in\mathbb{F}_{q}}{\sum}\bar{\psi}(-u_{1}a-u_{2}b)\underset{\epsilon\in\mathbb{F}_{q^{m}}\textbackslash
S}{\sum}\chi_{q^{m}-1}(\epsilon^{m_{1}}f(\epsilon)^{m_{2}})\widehat{\psi}(v_{1}\epsilon+v_{2}f(\epsilon)+u_{1}\epsilon+u_{2}f(\epsilon))$
$\displaystyle=$
$\displaystyle\underset{u_{1},u_{2}\in\mathbb{F}_{q}}{\sum}\bar{\psi}(-u_{1}a-u_{2}b)\underset{\epsilon\in\mathbb{F}_{q^{m}}\textbackslash
S}{\sum}\chi_{q^{m}-1}(F_{1}(\epsilon))\widehat{\psi}(F_{2}(\epsilon)),$

where,

$\displaystyle F_{1}(x)=$ $\displaystyle x^{m_{1}}f(x)^{m_{2}},$
$\displaystyle F_{2}(x)=$ $\displaystyle(u_{1}+v_{1})x+(u_{2}+v_{2})f(x).$

Case I: When $F_{2}(x)\neq r(x)^{q^{m}}-r(x)$ for any $r(x)$ in
$\mathbb{F}_{q}(x)$, we can use Lemma 2.2 to get the following.

$\big{|}\chi_{f,a,b}(d_{1},d_{2},h_{1},h_{2})\big{|}\leq
q^{2}(2n+1)q^{m/2}+q^{2}(|S|-1).$

Case II: $F_{2}(x)=h(x)^{q^{m}}-h(x)$ for some $h(x)\in\mathbb{F}(x)$ and
$F_{1}(x)=x^{m_{1}}{f(x)}^{m_{2}}\neq{(h(x))}^{q^{m}-1}$ for any
$h(x)\in\mathbb{F}(x)$, where $\mathbb{F}$ is the algebraic closure of
$\mathbb{F}_{q^{m}}$.

For $F_{2}(x)=h(x)^{q^{m}}-h(x)$ for some $h(x)\in\mathbb{F}(x)$, we can show
that $g_{1}=g_{2}=1$. For this let us consider
$f(x)=c_{1}{x^{j}}\frac{f_{1}}{f_{2}}$ and
$h(x)=c_{2}{x^{k}}\frac{h_{1}}{h_{2}}$ where
$c_{1}\in\mathbb{F}^{*}_{q^{m}},c_{2}\in\mathbb{F}^{*},j,k\in\mathbb{Z}$ and
$f_{1},f_{2}\in\mathbb{F}_{q^{m}}[x]$ $h_{1},h_{2}\in\mathbb{F}[x]$ are monic
polynomials such that $x$ divides none of them and
$\gcd(f_{1},f_{2})=1,\gcd(h_{1},h_{2})=1$. Then

$\displaystyle(u_{1}+v_{1})x+(u_{2}+v_{2})f(x)=$ $\displaystyle
h(x)^{q^{m}}-h(x)$

Setting $(u_{1}+v_{1})=k_{1}$ and $(u_{2}+v_{2})=k_{2}$ we get,

(3.1) $\displaystyle(k_{1}xf_{2}+k_{2}c_{1}x^{j}f_{1}){h_{2}}^{q^{m}}=$
$\displaystyle(c_{2}^{q^{m}}x^{kq^{m}}h_{1}^{q^{m}}-c_{2}x^{k}h_{1}h_{2}^{q^{m}-1}){f_{2}}$

Let $k_{2}\neq 0$, then from (3.1) we get $f_{2}|h_{2}^{q^{m}}$ since
$(f_{1},f_{2})=1$. This will imply $h_{2}^{q^{m}}|f_{2}$ since
$\gcd(h_{2}^{q^{m}},c_{2}^{q^{m}}x^{k(q^{m})}h_{1}^{q^{m}}-c_{2}x^{k}h_{1}h_{2}^{q^{m}-1})=1$.
This further gives us $f_{2}=h_{2}^{q^{m}}$ which creates contradiction to the
second condition of Definition 1.1. Therefore we must have $k_{2}=0$ i.e.,
$u_{2}+v_{2}=0.$

So,

(3.2) $\displaystyle k_{1}x{h_{2}}^{q^{m}}$
$\displaystyle=c_{2}^{q^{m}}x^{kq^{m}}h_{1}^{q^{m}}-c_{2}x^{k}h_{1}h_{2}^{q^{m}-1}$

i.e. $h_{2}$ divides
$c_{2}^{q^{m}}x^{kq^{m}}h_{1}^{q^{m}}-c_{2}x^{k}h_{1}h_{2}^{q^{m}-1}$. This
gives a contradiction since
$\gcd(h_{2},c_{2}^{q^{m}}x^{kq^{m}}h_{1}^{q^{m}}-c_{2}x^{k}h_{1}h_{2}^{q^{m}-1})=1$.
Therefore $k_{1}=0$ i.e. $u_{1}+v_{1}=0$.

Let us take $\psi_{f_{1}}$ to be an additive character of
$\mathbb{F}_{q}$-order $f_{1}$. Then we will have,

$\displaystyle\psi_{f_{1}}(f_{1}(\epsilon))=0\implies\widehat{\psi_{1}}(v_{1}f_{1}(\epsilon))=0\implies\widehat{\psi_{1}}(-u_{1}f_{1}(\epsilon))=0$

Since $u_{1}\in\mathbb{F}_{q}$ and $\mathbb{F}_{q}$-order of
$\widehat{\psi_{1}}$ is $x^{m}-1$, therefore we must have
$x^{m}-1|-u_{1}f_{1}$. But we already have, $f_{1}$ is a
$\mathbb{F}_{q}$-divisor of $x^{m}-1$. Therefore, we get $u_{1}=0$. This
further implies $v_{1}=0$ since $k_{1}=0$.

Similarly from $k_{2}=0$ it can be shown that $u_{2}=v_{2}=0$. With this we
arrive at the conclusion that, if $F_{2}(x)=h(x)^{q^{m}}-h(x)$ for some
$h(x)\in\mathbb{F}(x)$, where $\mathbb{F}$ is the algebraic closure of
$\mathbb{F}_{q^{m}}$, then $g_{1}=g_{2}=1$.

So in this case we use Lemma 2.1 to prove that

$\big{|}\chi_{f,a,b}(d_{1},d_{2},h_{1},h_{2})\big{|}\leq
q^{2}(n)q^{m/2}=nq^{\frac{m}{2}+2}$

Case III: $F_{2}(x)=h(x)^{q^{m}}-h(x)$ for some $h(x)\in\mathbb{F}(x)$ and
$F_{1}(x)=x^{m_{1}}{f(x)}^{m_{2}}={(h(x))}^{q^{m}-1}$ for some
$h(x)\in\mathbb{F}(x)$, where $\mathbb{F}$ is the algebraic closure of
$\mathbb{F}_{q^{m}}$.

When $F_{1}(x)=x^{m_{1}}{f(x)}^{m_{2}}={(h(x))}^{q^{m}-1}$ for some
$h(x)\in\mathbb{F}(x)$, where $\mathbb{F}$ is the algebraic closure of
$\mathbb{F}_{q^{m}}$, we can prove that $d_{1}=d_{2}=1$. We can refer to [15]
for the proof. So we have

$(e_{1},e_{2},g_{1},g_{2})=(1,1,1,1).$

Summarising all the above cases we observe that

$\big{|}\chi_{f,a,b}(d_{1},d_{2},h_{1},h_{2})\big{|}\leq(2n+1)q^{\frac{m}{2}+2}$

where $(d_{1},d_{2},h_{1},h_{2})\neq(1,1,1,1)$. Now,

$\displaystyle N_{f,a,b}(e_{1},e_{2},g_{1},g_{2})\geq$
$\displaystyle\frac{\theta(e_{1})\theta(e_{2})\Theta(g_{1})\Theta(g_{2})}{q^{2}}\frac{}{}[(q^{m}-(2n+1)q^{\frac{m}{2}+2})(W(e_{1})W(e_{2})W(g_{1})W(g_{2})-1)]$
$\displaystyle\geq$
$\displaystyle\theta(e_{1})\theta(e_{2})\Theta(g_{1})\Theta(g_{2})q^{\frac{m}{2}}[q^{\frac{m}{2}-2}-(2n+1)W(e_{1})W(e_{2})W(g_{1})W(g_{2})]$

To show that there exists an element $\epsilon\in\mathbb{F}_{q^{m}}$ such that
for preassigned $a,b\in\mathbb{F}_{q}$ and a rational function
$f(x)\in\mathcal{R}_{q^{m}}^{n}$,$(\epsilon,f(\epsilon))$ is a primitive
normal pair with $Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(\epsilon)=a$ and
$Tr_{{\mathbb{F}_{q^{m}}}/{\mathbb{F}_{q}}}(f(\epsilon)=b$, it is enough to
show $N_{f,a,b}(q^{m}-1,q^{m}-1,x^{m}-1,x^{m}-1)>0$, which is evident if

(3.3) $q^{\frac{m}{2}-2}>(2n+1)W(e_{1})W(e_{2})W(g_{1})W(g_{2}).$

##  4\. The prime sieve

We will use the following two lemmas to deduce a sieving inequality that will
help to improve the sufficient condition (3.3).

######  Lemma 4.1.

Let $d$ be a divisor of $q^{m}-1$ and $p_{1},p_{2},\dots,p_{r}$ are the
remaining distinct primes dividing $q^{m}-1$. Furthermore, let $g$ be a
divisor of $x^{m}-1$ such that $g_{1},g_{2},\dots,g_{s}$ are the remaining
distinct irreducible polynomials dividing $x^{m}-1$. Then

$\displaystyle
N_{f,a,b}(q^{m}-1,q^{m}-1,x^{m}-1,x^{m}-1)\geq\overset{r}{\underset{i=1}{\sum}}N_{f,a,b}(p_{i}d,d,g,g)+\overset{r}{\underset{i=1}{\sum}}N_{f,a,b}(d,p_{i}d,g,g)$
$\displaystyle+\overset{s}{\underset{i=1}{\sum}}N_{f,a,b}(d,d,g_{i}g,g)+\overset{s}{\underset{i=1}{\sum}}N_{f,a,b}(d,d,g,g_{i}g)-(2r+2s-1)N_{f,a,b}(d,d,g,g).$

######  Lemma 4.2.

Let $d,m,q\in\mathbb{N},g\in\mathbb{F}_{q}[x]$ be such that $q$ is a prime
power, $m\geq 5,d|q^{m}-1,$ and $g|x^{m}-1$. Let $e$ be a prime number which
divides $q^{m}-1$ but not $d$ and $h$ be any irreducible polynomial dividing
$x^{m}-1$ but not $g$. Then we have the following bounds:

$\displaystyle\big{|}N_{f,a,b}(ed,d,g,g)-\theta(e)N_{f,a,b}(d,d,g,g)\big{|}\leq$
$\displaystyle(2n+1)\theta(e)\theta(d)^{2}\Theta(g)^{2}W(d)^{2}W(g)^{2}q^{\frac{m}{2}}$
$\displaystyle\big{|}N_{f,a,b}(d,ed,g,g)-\theta(e)N_{f,a,b}(d,d,g,g)\big{|}\leq$
$\displaystyle(2n+1)\theta(e)\theta(d)^{2}\Theta(g)^{2}W(d)^{2}W(g)^{2}q^{\frac{m}{2}}$
$\displaystyle\big{|}N_{f,a,b}(d,d,hg,g)-\Theta(h)N_{f,a,b}(d,d,g,g)\big{|}\leq$
$\displaystyle(2n+1)\Theta(h)\theta(d)^{2}\Theta(g)^{2}W(d)^{2}W(g)^{2}q^{\frac{m}{2}}$
$\displaystyle\big{|}N_{f,a,b}(d,d,g,hg)-\Theta(h)N_{f,a,b}(d,d,g,g)\big{|}\leq$
$\displaystyle(2n+1)\Theta(h)\theta(d)^{2}\Theta(g)^{2}W(d)^{2}W(g)^{2}q^{\frac{m}{2}}$

Using these two lemmas 4.1 and 4.2 we deduce the following:

######  Theorem 4.3.

Let $d,m,q\in\mathbb{N},g\in\mathbb{F}_{q}[x]$ be such that $q$ is a prime
power, $m\geq 5,d|q^{m}-1$ and $g|x^{m}-1$. Let $d$ be a divisor of $q^{m}-1$
and $p_{1},p_{2},\dots,p_{r}$ be the remaining distinct primes dividing
$q^{m}-1$. Furthermore let $g$ be a divisor of $x^{m}-1$ such that
$g_{1},g_{2},\dots,g_{s}$ are the remaining distinct irreducible factors of
$x^{m}-1$. Define:

$l:=1-2\overset{n}{\underset{i=1}{\sum}}\frac{1}{p_{i}}-\overset{k}{\underset{i=1}{\sum}}\frac{1}{q^{{\mathrm{d}eg}(g_{i})}},l>0$

and

$\Lambda:=\frac{2(r+s)-1}{l}+2.$

Then $N_{f,a,b}(q^{m}-1,q^{m}-1,x^{m}-1,x^{m}-1)>0$ if

(4.1) $\displaystyle q^{\frac{m}{2}-2}>(2n+1)W(d)^{2}W(g)^{2}L$

###### Proof.

For our convenience, let us denote
$N_{f,a,b}(q^{m}-1,q^{m}-1,x^{m}-1,x^{m}-1)$ by $N$. Then from Lemma 4.1 and
4.2, we have

$\displaystyle N\geq$
$\displaystyle\overset{r}{\underset{i=1}{\sum}}\\{N_{f,a,b}(p_{i}d,d,g,g)-\theta(p_{i})N_{f,a,b}(d,d,g,g)\\}+\overset{r}{\underset{i=1}{\sum}}\\{N_{f,a,b}(d,p_{i}d,g,g)-\theta(p_{i})N_{f,a,b}(d,d,g,g)\\}$
$\displaystyle+$
$\displaystyle\overset{k}{\underset{i=1}{\sum}}\\{N_{f,a,b}(d,d,g_{i}g,g)-\Theta(g_{i})N_{f,a,b}(d,d,g,g)\\}+\overset{k}{\underset{i=1}{\sum}}\\{N_{f,a,b}(d,d,g,g_{i}g)-\Theta(g_{i})N_{f,a,b}(d,d,g,g)\\}$
$\displaystyle+$
$\displaystyle\\{2\overset{r}{\underset{i=1}{\sum}}\theta(p_{i})+2\overset{s}{\underset{i=1}{\sum}}\Theta(g_{i})\\}N_{f,a,b}(d,d,g,g)-(2r+2s-1)N_{f,a,b}(d,d,g,g).$
$\displaystyle=$
$\displaystyle\overset{r}{\underset{i=1}{\sum}}\\{N_{f,a,b}(p_{i}d,d,g,g)-\theta(p_{i})N_{f,a,b}(d,d,g,g)\\}+\overset{r}{\underset{i=1}{\sum}}\\{N_{f,a,b}(d,p_{i}d,g,g)-\theta(p_{i})N_{f,a,b}(d,d,g,g)\\}$
$\displaystyle+$
$\displaystyle\overset{k}{\underset{i=1}{\sum}}\\{N_{f,a,b}(d,d,g_{i}g,g)-\Theta(g_{i})N_{f,a,b}(d,d,g,g)\\}+\overset{k}{\underset{i=1}{\sum}}\\{N_{f,a,b}(d,d,g,g_{i}g)-\Theta(g_{i})N_{f,a,b}(d,d,g,g)\\}$
$\displaystyle+$ $\displaystyle lN_{f,a,b}(d,d,g,g)$ $\displaystyle\geq$
$\displaystyle\theta(d)^{2}\Theta(g)^{2}l[\\{-(2n+1)W(d)^{2}W(g)^{2}q^{\frac{m}{2}}L\\}+\\{q^{m-2}-(n+1)+(2n+1)q^{\frac{m}{2}}\\}]$

Therefore, when $l>0$, then $N_{f,a,b}(q^{m}-1,q^{m}-1,x^{m}-1,x^{m}-1)>0$ if

(4.2) $\displaystyle q^{\frac{m}{2}-2}>(2n+1)W(d)^{2}W(g)^{2}L.$

∎

##  5\. Numerical computations

In this section, we particularly study the case when finite fields are of
characteristic $5$ and rational functions of degree sum $4$ and try to
determine pairs $(q,m)\in\mathcal{A}_{5}^{4}$. For large calculations we use
sagemath [17].

The following lemma will be used extensively throughout the process.

######  Theorem 5.1.

Let $\nu>0$ be a real number and $t$ be a positive integer.Then
$W(t)<D.(t)^{\frac{1}{\nu}}$, where $D=\frac{2^{r}}{(p_{1}p_{2}\dots
p_{r})^{\frac{1}{\nu}}}$ and $p_{1},p_{2},\dots,p_{r}$ are primes $\leq
2^{\nu}$ that divide $t$.

Remark: For large values of $q$ and $m$, calculation of $W(q^{m}-1)$ is very
time consuming. So using the above lemma we find an upper bound to the term
$W(q^{m}-1)$ as follows:

$W(q^{m}-1)\leq D.(q^{m}-1)^{\frac{1}{\nu}}<D.(q^{m})^{\frac{1}{\nu}}.$

Also since $x^{m}-1$ can have atmost $m$ linear factors in
$\mathbb{F}_{q}[x]$. So,

$W(x^{m}-1)\leq 2^{m}.$

Using these two observations we rewrite our sufficient condition (3.3) as:

(5.1) $q^{\frac{m}{2}-2}>(2n+1)D^{2}q^{\frac{2m}{\nu}}2^{2m}.$

In our case, $n=4,m\geq 5,q=5^{k}$ for some positive integer $k$. We choose
$\nu=21.57$. Then we get $D\leq 1.52\times 10^{4906}$. For these values (5.1)
holds for $k\geq 385897$ and $m\geq 5$.

Now for $3\leq k\leq 385897$, we present a table that shows suitable values of
$\nu$ and an integer $m_{k}$ corresponding to each value of $k$ such that for
all $m\geq m_{k}$ and that particular $\nu$, (5.1) holds.

$\nu$ | $k$ | $m_{k}$
---|---|---
13.7 | 1379,1380, …,385896 | 6
11.3 | 212, 213, …, 1378 | 7
10.1 | 84, 85, …, 211 | 8
9.52 | 48,49, …, 83 | 9
9.2 | 33,34, …,47 | 10
8.9 | 26,27,…, 32 | 11
8.7 | 21,22, …, 25 | 12
8.6 | 18,19,20 | 13
8.5 | 16,17 | 14
8.5 | 14,15 | 15
8.4 | 13 | 16
8.4 | 12 | 17
8.5 | 11 | 18
8.4 | 10 | 19
8.4 | 9 | 21
8.5 | 8 | 24
8.4 | 7 | 28
8.5 | 6 | 36
8.8 | 5 | 52
9.4 | 4 | 99
11.3 | 3 | 520
Table 1. Values of $\nu$ and $m_{k}$ corresponding to $k$.

Analysing the values of $m_{k}$ and corresponding range of $k$ from Table 1,
we observe that (5.1) holds for all $k\geq 48$ and $m\geq 9$. Based on this we
claim the following:

######  Lemma 5.2.

For $k\geq 3$ and $m\geq 9$, $(q,m)\in\mathcal{A}_{5}^{4}$.

###### Proof.

Since for $3\leq k\leq 47$, (5.1) holds for all $m\geq m_{k}$, so for $9\leq
k<m_{k}$, we first check the following inequality,

(5.2) $q^{\frac{m}{2}-2}>(2n+1)D^{2}q^{\frac{2m}{\nu}}(W(x^{m}-1))^{2}.$

This inequality reduces the number of possible exceptional pairs to $220$. For
all these possibilities, we can find suitable values of $d$ and $g$ ( as
defined in Theorem 4.3), so that (4.1) holds. ∎

We study the pairs for $k=1,2$ separately. Let us consider $m=m^{\prime}5^{j}$
for $j\geq 0$ such that $\gcd(5,m^{\prime})=1$. Then we can divide our
discussion into the following two cases:
(i) $m^{\prime}|q^{2}-1$,
(ii) $m^{\prime}\nmid q^{2}-1$.

Case I: First let us consider the pairs $(q,m)$, where $m^{\prime}|q^{2}-1$
and $k=1$.

Then possible values of $m^{\prime}$ are $1$, $2$, $3$, $4$, $6$, $8$, $12$,
$24$. Since $W(x^{m}-1)=W(x^{m^{\prime}}-1)$, So we can rewrite (5.1) as

(5.3)
$q^{\frac{m^{\prime}5^{j}}{2}-2}>(2n+1)D^{2}q^{\frac{2m^{\prime}5^{j}}{\nu}}2^{2m^{\prime}}.$

Then (5.3) holds when,
(i) $m^{\prime}=1$ and $j\geq 4$,
(ii) $m^{\prime}=2$ ,$3$, $4$, $6$ and $j\geq 3$,
(iii) $m^{\prime}=8$ ,$12$, $24$ and $j\geq 2$.
Therefore $(5,m)\in\mathcal{A}_{5}^{4}$ unless $m$ $=$ $10$, $12$, $15$, $20$,
$24$, $25$, $30$, $40$, $50$, $60$, $75$, $100$, $120$, $125$, $150$. For
these pairs we directly check sufficient condition (3.3) and reduce the list
of possible exceptional values of $m$ to $10$, $12$, $15$, $20$, $24$, $30$.
Again for these values of $m$, we try to find suitable $d$ and $g$ ( as
defined in Theorem 4.3) for which (4.1) is satisfied ( see Table 2). But for
the pairs $(5,10)$, $(5,12)$, $(5,20)$, $(5,24)$; we could not find such $d$
and $g$. So in this case, these are the possible exceptional pairs.

Now let $k=2$ and $m^{\prime}\nmid q^{2}-1$.
Then possible values of $m^{\prime}$ are $1$, $2$, $3$, $4$, $6$, $8$, $12$,
$13$, $16$, $24$, $26$, $39$, $48$, $52$, $76$, $104$, $156$, $208$, $312$,
$624$. We find out that (5.3) holds when, (i) $m^{\prime}=1$ ,$2$, $3$ and
$j\geq 3$,
(i) $m^{\prime}=4$, $6$, $8$, $12$, $13$, $16$ and $j\geq 2$,
(i) $m^{\prime}=24$, $26$, $39$, $48$, $52$, $76$, $104$, $156$, $208$, $312$,
$624$ and $j\geq 1$.
Therefore $(25,m)\in\mathcal{A}_{5}^{4}$ unless $m=10$, $12$, $13$, $15$,
$16$, $20$, $24$, $25$, $26$, $30$, $39$, $40$, $48$, $50$, $52$, $60$, $65$,
$75$, $76$, $80$, $104$, $156$, $208$, $312$, $624$. We reduce theses
exceptions by removing the pairs $(25,m)$ that satisfies (3.3). This leaves us
with possible exceptional values of $m=10,12,13,15,16,24,48.$ For these pairs
we try to find $d$ and $g$(as defined in Theorem (4.3) such that (4.1) holds(
see Table 2). Since we fail to find such $d$ and $g$ for $m=10,12,15$, so
$(25,10),(25,12),(24,25)$ adds up to the list of exceptional pairs.

SL No. | $(q,m)$ | $d$ | $r$ | $g$ | $s$ | $l$ | $L$
---|---|---|---|---|---|---|---
1 | (5,13) | 2 | 1 | $(x+4)$ | 3 | 0.990 | 9.07
2 | (5,15) | 2 | 5 | $(x+4)$ | 1 | 0.633 | 19.37
3 | (5,21) | 2 | 4 | $(x+4)$ | 4 | 0.850 | 19.65
4 | (5,22) | 6 | 4 | $(x+1)$ | 5 | 0.480 | 37.04
5 | (5,30) | 462 | 7 | $(x+1)$ | 3 | 0.298 | 65.71
6 | (5,80) | 66 | 12 | $(x+1)(x+2)(x+3)(x+4)$ | 4 | 0.466 | 68.50
7 | $(5^{2},11)$ | 6 | 4 | $(x+4)$ | 2 | 0.883 | 14.46
8 | $(5^{2},13)$ | 6 | 3 | $(x+4)$ | 6 | 0.980 | 19.34
9 | $(5^{2},14)$ | 6 | 5 | $(x+1)$ | 5 | 0.692 | 29.45
10 | $(5^{2},15)$ | 6 | 9 | $(x+4)$ | 2 | 0.231 | 93.03
11 | $(5^{2},18)$ | 42 | 8 | $(x+1)$ | 9 | 0.216 | 155.00
12 | $(5^{2},21)$ | 6 | 10 | $(x+4)$ | 8 | 0.348 | 102.67
13 | $(5^{2},36)$ | 546 | 12 | $(x+1)(x+2)(x+3)(x+4)$ | 16 | 0.095 | 579.62
14 | $(5^{2},48)$ | 9282 | 10 | $g^{\prime}$ | 22 | 0.066 | 716.32

Table 2. Pairs $(q,m)$ when $k=1,2$, for which Theorem 4.1 holds for the above
choices of $d$, $r$, $g$ and $s$.

[Here
$g^{\prime}=(x+1)(x+2)(x+3)(x+4)(x+\beta)(x+\beta+1)(x+\beta+2)(x+\beta+3)(x+\beta+4)(x+2\beta)(x+2\beta+1)(x+2\beta+2)(x+2\beta+3)(x+2\beta+4),$
with $\beta$ as an generating element of the field.]

Before moving to the second case let us recall the following lemmas.

######  Lemma 5.3 ([7]).

Suppose $q=5^{k},m^{\prime}>4$ and $\bar{m}=\gcd(q-1,m^{\prime})$. If
$M^{\prime}$ denotes the number of distinct irreducible factors of
$x^{m^{\prime}}-1$ over $\mathbb{F}_{q}$, $e$ is the order of
$q(\mod{m^{\prime}})$ and $\delta(q,m)=\frac{M^{\prime}}{e}$; Then the
following holds.

  * •

$\delta(q,m)\leq\frac{1}{2}$, for $m=2\bar{m}$,

  * •

$\delta(q,m)\leq\frac{3}{8}$, for $m=4\bar{m}$,

  * •

$\delta(q,m)\leq\frac{13}{36}$, for $m=6\bar{m}$,

  * •

$\delta(q,m)\leq\frac{1}{3}$, otherwise.

######  Lemma 5.4 ([15],Lemma 5.3).

Assume that $q=p^{k},k\in\mathbb{N}$ and $m=m^{\prime}p^{j},j\geq 0$ is a
positive integer such that $m^{\prime}\nmid q-1$ and $\gcd(m^{\prime},p)=1$.
Let $e(>2)$ denote the order of $q\mod m^{\prime}$. Then $d=q^{m}-1$ and $g$
as the product of the irreducible factors of $x^{m^{\prime}}-1$ of degree less
than $e$. Then, in the notation of Lemma 4.1, we have $L\leq 2m^{\prime}$.

Since $m^{\prime}\nmid q^{2}-1$ implies $m^{\prime}\nmid q-1$ and $e>2$. We
can transform (5.1) using the above two lemmas as follows:

$q^{\frac{m^{\prime}5^{j}}{2}-2}>9D^{2}q^{\frac{2m^{\prime}5^{j}}{\nu}}2^{2m^{\prime}\delta(q,m^{\prime})}2m^{\prime},$

which is also true if

(5.4)
$q^{\frac{m}{2}-2}>18D^{2}q^{\frac{2m}{\nu}}2^{2m\delta(q,m^{\prime})}m.$

Now we can proceed to the second case.

Case II: $m^{\prime}\nmid q^{2}-1$ Let $k=1$. Then possible values of
$m^{\prime}$ are $5$, $7$, $9$, $10$, $11$, $13$, $14$, $15$, $16$, $17$,
$18$, $19$, $20$, $21$, $22$, $23$. Out of these only $16$ is there such that
$16=4.\bar{m}$. So for others we can rewrite (5.5) as

(5.5) $5^{\frac{m}{2}-2}>18D^{2}m5^{\frac{2m}{\nu}}2^{\frac{2m}{3}}.$

For $\nu=11.3$, (5.5) holds for$m\geq 1566$. So we list out the possible
exceptional values of $m$ in this case to be $m=9$, $10$, $11$, $13$, $14$,
$15$, $17$, $18$, $20$, $21$, $22$, $23$, $25$, $35$, $45$, $50$, $55$, $65$,
$70$, $75$, $85$, $90$, $95$, $100$, $105$, $110$, $115$, $125$, $175$, $225$,
$250$, $275$, $325$, $350$, $375$, $425$, $450$, $475$, $500$, $525$, $550$,
$575$, $625$, $875$, $1125$, $1250$, $1375$. Out of them $m=9$, $10$, $11$,
$13$, $14$, $15$, $18$, $20$, $21$, $22$ don’t satisfy (3.3). For these values
we try to calculate $d$ and $g$ so that condition (4.1) is satisfied ( see
Table 2). For $m=9$, $10$, $11$, $14$, $18$ and $20$, we fail to do so.
Therefore they also belong to the list of exceptional values of $m.$

The case $m^{\prime}=16$ is handled separately. For this (5.3) takes the form

(5.6) $5^{\frac{m}{2}-2}>18D^{2}m5^{\frac{2m}{\nu}}2^{\frac{3m}{4}}.$

For $\nu=11.3$, (5.6) holds for $m\geq 342$. This leaves us with $m=16$, $80$.
But for $m=16$, neither (3.3) holds nor we could find any $d$ and $g$ to
satisfy (4.1). So $(5,16)$ is also a possible exceptional pair. With this we
exhaust the case for $k=1.$

For $k=2$, there is no $m^{\prime}$ such that $m^{\prime}=2\bar{m},4\bar{m}$
or $6\bar{m}$. So here we can rewrite (5.3) as

(5.7) $(25)^{\frac{m}{2}-2}>18D^{2}m{(25)}^{\frac{2m}{\nu}}2^{\frac{2m}{3}}.$

With $\nu=11.3$, (5.7) holds for all $m\geq 159$.
Therefore we need to check for
$m\in\\{9,10,..,158\\}\backslash\\{12,13,16,24,26,39,48,52,78,104,156\\}.$ Out
of these, for $m=9,$ $10$, $11$, $14$, $15$, $18$, $21$ and $36$, (3.3) do not
hold. So we search for $d$ and $g$ so that (4.1) holds for them. But for
$m=9$, $10$, we couldnot find such $d$ and $g$, so $(25,9)$, $(25,10)$ are
added up to the list of possible exceptional pairs.

Summarizing the discussions in this section we obtain the proof of Theorem
1.2.

##  6\. Declarations

Conflict of interest The authors declare no competing interests.

Ethical Approval Not applicable.

Funding The first author is supported by NFOBC fellowship (NBCFDC Ref. No.
231610154828).

Data availability Not applicable.

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# Cold traps of hypervolatiles in the protosolar nebula at the origin of comet
C/2016 R2 (PanSTARRS)’s peculiar composition

Olivier Mousis11affiliation: Aix Marseille Univ, CNRS, CNES, LAM, Marseille,
France<EMAIL_ADDRESS>, Artyom Aguichine11affiliation: Aix Marseille
Univ, CNRS, CNES, LAM, Marseille, France<EMAIL_ADDRESS>, Alexis
Bouquet22affiliation: Aix Marseille Univ, CNRS, PIIM, F-13013 Marseille,
France , Jonathan I. Lunine33affiliation: Department of Astronomy, Cornell
University, Ithaca, NY 14853, USA , Grégoire Danger22affiliation: Aix
Marseille Univ, CNRS, PIIM, F-13013 Marseille, France , Kathleen E.
Mandt44affiliation: Applied Physics Laboratory, Johns Hopkins University,
11100 Johns Hopkins Rd., Laurel, MD 20723, USA , and Adrienn Luspay-
Kuti44affiliation: Applied Physics Laboratory, Johns Hopkins University, 11100
Johns Hopkins Rd., Laurel, MD 20723, USA

###### Abstract

Recent observations of the long period comet C/2016 R2 (PanSTARRS) indicate an
unusually high N2/CO abundance ratio, typically larger than $\sim$0.05, and at
least 2–3 times higher than the one measured in 67P/Churyumov-Gerasimenko.
Another striking compositional feature of this comet is its heavy depletion in
H2O (H2O/CO $\sim$0.32%), compared to other comets. Here, we investigate the
formation circumstances of a generic comet whose composition reproduces these
two key features. We first envisage the possibility that this comet
agglomerated from clathrates, but we find that such a scenario does not
explain the observed low water abundance. We then alternatively investigate
the possibility that the building blocks of the comet C/2016 R2 (PanSTARRS)
agglomerated from grains and pebbles made of pure condensates via the use of a
disk model describing the radial transport of volatiles. We show that N2/CO
ratios reproducing the value estimated in this comet can be found in grains
condensed in the vicinity of the CO and N2 icelines. Moreover, high CO/H2O
ratios (> 100 times the initial gas phase value) can be found in grains
condensed in the vicinity of the CO iceline. If the building blocks of a comet
assembled from such grains, they should present N2/CO and CO/H2O ratios
consistent with the measurements made in comet C/2016 R2 (PanSTARRS)’s coma.
Our scenario indicates that comet C/2016 R2 (PanSTARRS) formed in a colder
environment than the other comets that share more usual compositions. Our
model also explains the unusual composition of the interstellar comet
2l/Borisov.

###### Subject headings:

astrobiology – comets: general – comets: individual (C/2016 R2 (PanSTARRS)) –
methods: numerical – solid state: volatile

##  1\. Introduction

Comets are supposed to be water-rich bodies that are relics of the formation
of the solar system. Because they have undergone little alteration during the
4.6 billion years of the solar system evolution, investigating their
composition provides indications of the physico-chemical conditions that were
at play during their formation in the protosolar nebula (PSN). An outstanding
question remaining to be solved is the origin of the apparent N2 deficiency
observed in comets while both Pluto and Triton, also formed in the outer solar
system, harbor N2–rich surfaces and atmospheres (Lellouch et al., 2011;
Cruikshank et al., 2015; Mandt et al., 2017). Decades of remote-sensing
observations of comets suggest they are depleted in N2 (Cochran et al., 2000).
Only upper limits of $\sim$10-4 for N${}_{2}^{+}$/CO+ were derived in the coma
of comets 122P/1995 S1 (deVico) and C/1995 O1 (Hale-Bopp) (Cochran et al.,
2000) while very few detections of N${}_{2}^{+}$ emission lines have been
reported in other comets from ground-based facilities. This apparent N2
depletion was interpreted as the result of the selective trapping of CO at the
expense of N2 in the building blocks of comets presumably agglomerated from
clathrates (Iro et al., 2003), or as the result of their partial
devolatilization due to radiogenic heating (Mousis et al., 2012). To a lesser
extent, these important depletions are in agreement with the measurements of
the N2/CO ratio acquired in comet 67P/Churyumov-Gerasimenko (hereafter
67P/C-G) by the ROSINA mass spectrometer aboard the Rosetta spacecraft very
early in the mission in October 2014, which suggest a value of (5.70 $\pm$
0.66) $\times$ 10-3 that is depleted by a factor of $\sim$25 as compared to
the one derived from protosolar N and C abundances (Rubin et al., 2015).
However, this N2/CO ratio was measured at a heliocentric distance beyond 3 au,
far beyond perihelion. Rubin et al. (2020) later derived a N2/CO ratio of
$\sim$2.87 $\times$ 10-2 in 67P/C-G, a value obtained in May 2015 a few months
before perihelion passage, and roughly 5 times larger than the previous one.

Contrasting with most of these previous measurements, recent optical spectra
of the long period comet C/2016 R2 (PanSTARRS) (hereafter R2) performed at a
heliocentric distance of $\sim$3 AU, showed that its spectrum was largely
dominated by the emission band of CO+, but also surprisingly by the presence
of the emission band of N${}_{2}^{+}$ (Cochran & McKay, 2018). Subsequent
observations confirmed an unusually high N2/CO ratio in R2, a value estimated
to range between 0.06 $\pm$ 0.01 (Opitom et al., 2019) and 0.08 (Biver et al.,
2018), and at least 2–3 times higher than the one measured closer to
perihelion in 67P/C-G. Another striking compositional feature of this comet is
its heavy depletion in H2O (H2O/CO $\sim$0.32%; McKay et al. (2019)). These
trends are at odds with our understanding of the thermodynamic evolution of
cometary nuclei which should be, in principle, depleted in the most volatile
species instead of being enhanced, as R2 shows, after multiple solar passages.
Since sublimation of supervolatile ices can occur possibly beyond 40 AU for CO
(Womack et al., 2017), one should expect CO outgassing, and then cometary
activity over a significant fraction of R2’s orbit. Figure 1 represents a
comparison of the volatile composition of R2 to the average comet made by
McKay et al. (2019). The figure shows that all oxygen appears to be locked in
CO and CO2, and not H2O, as is the case for typical comets.

Figure 1.— Comparison of the volatile composition of R2 to the average comet
as derived by McKay et al. (2019) [Reprinted from The Astronomical Journal,
158:128 (24pp), McKay, A., et al., The Peculiar Volatile Composition of CO-
dominated Comet C/2016 R2 (PanSTARRS), with permission from IOP].

In this paper, we aim at investigating the formation circumstances of a
generic comet whose composition reproduces the two key features of R2’s coma,
namely its high N2/CO ratio and the observed significant water depletion,
assuming they reflect its primordial composition (McKay et al., 2019). We
first investigate the possibility that this comet could have agglomerated from
clathrates as it has been suggested in the case of 67P/C-G (Luspay-Kuti et
al., 2016; Mousis et al., 2016, 2018), but we find that such a scenario cannot
explain the observed low water abundance. We then alternatively investigate
the possibility that the building blocks of this comet agglomerated from
grains and pebbles made of pure condensates crystallized in the vicinity of
the CO iceline in the PSN. This scenario matches both observed features and
suggests that this comet formed in a colder environment than the other comets
sharing usual compositions.

##  2\. Agglomeration from clathrates?

For decades, the agglomeration of comets from clathrates has been regularly
invoked to explain their compositional and thermodynamic properties (Lunine &
Stevenson, 1985; Klinger et al., 1986; Iro et al., 2003; Marboeuf et al.,
2010, 2011, 2012). Recent measurements of 67P/C-G’s composition by the Rosina
mass spectrometer aboard the Rosetta mission have also been interpreted in
favor of the presence of clathrates in its interior (Luspay-Kuti et al., 2016;
Mousis et al., 2016, 2018). This possibility is now investigated in the case
of R2, which presents unusually high CO and N2 abundances in its coma,
compared to typical comets. To do so, we aim at comparing the plausible
composition of multiple guest (MG) clathrates formed in the PSN to that of R2,
assuming this latter would have agglomerated from these icy structures. These
MG clathrates are assumed to be formed from a gaseous mixture composed of N2
and CO, assuming these species are the two main C– and N–bearing volatiles in
the gas phase of the disk, a hypothesis consistent with the thermochemical
models (Lewis & Prinn, 1980; Prinn & Fegley, 1989; Mousis et al., 2002).
Because CO has a much higher propensity for clathration than N2, the
calculations utilized in this work are those performed in the framework of
structure I clathrates, namely the structure predicted for a CO–dominated
clathrate (Mohammadi et al., 2005). Given the fact that half of protosolar
carbon is expected to be used in organic compounds in the PSN (Pollack et al.,
1994), only the remaining half is assumed to be in the form of CO. Assuming
that 90% of protosolar nitrogen is in N2 form (Lewis & Prinn, 1980), we derive
a N2/CO ratio of 2.66 $\times$ 10-1 in the initial gas phase of the PSN, based
on the protosolar abundances given by Lodders et al. (2009). In our model,
clathrates form in the PSN as long as crystalline water is available. Here we
assume that once all the water budget has been used for clathration, the
remaining volatiles are not incorporated into solids.

Our model assumes the formation of a multiple guest (MG) clathrate with an
equilibrium pressure expressed as (Lunine & Stevenson, 1985):

$P_{eq,MG}=\left[\sum_{i}\frac{y_{i}}{P_{eq,i}}\right]^{-1},$ (1)

where $y_{i}$ is the mole fraction of the component $i$ in the fluid phase.
The equilibrium pressure curves of each species are determined by fitting the
available theoretical and laboratory data (Lunine & Stevenson, 1985) with
equations of the form $\log P_{eq,i}=A/T+B$, where $P_{eq,i}$ and $T$ are the
partial equilibrium pressure and temperature of the considered species $i$,
respectively. The relative abundances of guest species incorporated in MG
clathrate formed at a given temperature and pressure from the PSN gas phase
are calculated following the method described in Lunine & Stevenson (1985) and
Mousis et al. (2010). This formalism has been used to interpret 67P/C-G’s ice
structure and composition from Rosetta/ROSINA observations of its coma (Mousis
et al., 2016, 2018).

Figure 2.— N2/CO ratio in cometary grains calculated as a function of the
water abundance (normalized to the oxygen protosolar abundance (Lodders et
al., 2009)) in the PSN for different formation temperatures. The blue and
orange horizontal bars represent the N2/CO ratios measured in comet R2 (Opitom
et al., 2019) and 67P/C-G (Rubin et al., 2015, 2020), respectively. The two
vertical dashed lines with arrows down correspond to upper limits of the N2/CO
ratio measured in 122P/1995 S1 (deVico) and C/1995 O1 (Hale-Bopp) (Cochran et
al., 2000).

Figure 2 shows the influence of the abundance of crystalline water ice in the
outer PSN on the N2/CO ratio in cometary grains at clathration temperatures of
25, 30, 40, and 50 K. These temperatures are all above the formation
temperature of N2 and CO pure condensates. For water abundances below 1.3
$\times$ (O/H2)⊙, the N2/CO ratios in icy grains correspond to those
calculated in clathrates at specified formation temperatures. For example, the
N2/CO ratios measured in Hale-Bopp, deVico and 67P/C-G are closely matched at
25, 30, and 50 K, respectively. At higher water abundances, CO is fully
entrapped and significant amounts of N2 become progressively trapped in
clathrates, implying an increase of the global N2/CO in icy grains. A water
abundance of $\sim$1.4 $\times$ (O/H2)⊙ in the PSN allows one to retrieve a
N2/CO ratio in clathrates corresponding to the value measured in R2. Water
abundances greater than $\sim$1.7 $\times$ (O/H2)⊙ allows the full trapping of
the available N2 in clathrates crystallized in the PSN, leading to a N2/CO
ratio of $\sim$2.66 $\times$ 10-1 in cometary grains, corresponding to the
value calculated from protosolar N and C (with half C in CO and 90% N in N2).
These calculations illustrate the fact that, in principle, a slight increase
of the water abundance in the PSN can lead to the full trapping of CO and
enough N2 to reproduce the N2/CO ratio in R2.

However, the amount of water required by clathrates formed in the PSN to match
the N2/CO ratio measured in R2’s coma is inconsistent with its estimated low
abundance. To match the N2/CO measured in R2, our clathrate model requires a
H2O/CO ratio of 6.1. In contrast, the H2O/CO mixing ratio measured at 2.8 AU
in R2 is $\sim$0.32%, a value several orders of magnitude lower than those
measured in 67P/C-G ($\sim$200–1900%) at a similar distance range (Gasc et
al., 2017) or in many other comets (McKay et al., 2019). As a result, this
mechanism fails at challenging to explain the H2O depletion in R2-like comets
via this mechanism.

##  3\. Agglomeration from particles formed in the vicinity of CO and N2 ice
lines

Disk models including radial transport of volatiles can display significant
enrichments of both volatile and refractory matter at its condensation
location, implying potentially drastic changes of the local metallicity in the
disk (Stevenson & Lunine, 1988; Cyr et al., 1999; Ali-Dib et al., 2014; Mousis
et al., 2019, 2020; Aguichine et al., 2020). Here we investigate this effect
by considering the possibility that R2 formed in the vicinity of the
condensation locations of CO and N2 crystalline ices in the PSN.

###  3.1. Model

The volatile transport and distribution model used in our work is the one
described in Aguichine et al. (2020) and Mousis et al. (2020), to which the
reader is referred for details. Three energy sources are considered in our
model, namely viscous heating, irradiation from the current Sun, and ambient
constant irradiation giving a background temperature of 10 K (see Eq. 6 of
Aguichine et al. (2020)). Here, the accretion rate onto the star declines over
time, and the disk develops a transition radius between inward and outward gas
flows. The location of this transition radius is moving outward with time. In
a few words, our time-dependent PSN model is governed by the following
differential equation (Lynden-Bell & Pringle, 1974):

$\displaystyle\frac{\partial\Sigma_{\mathrm{g}}}{\partial
t}=\frac{3}{r}\frac{\partial}{\partial r}\left[r^{1/2}\frac{\partial}{\partial
r}\left(r^{1/2}\Sigma_{\mathrm{g}}\nu\right)\right],$ (2)

This equation describes the time evolution of a viscous accretion disk of
surface density $\Sigma_{\mathrm{g}}$ of viscosity $\nu$, assuming hydrostatic
equilibrium in the $z$ direction. The viscosity is calculated in the framework
of the $\alpha$-formalism (Shakura & Sunyaev, 1973) using the following
method. For each distance $r$ to the Sun, the disk’s properties are calculated
by solving the equation of energy balance between viscous heating and
radiative transfer at the midplane level. This gives us $\nu$, as well as the
pressure and temperature profiles of the disk as a function of $r$. The
evolution of the disk starts with an initial profile given by
$\Sigma_{\mathrm{g}}\nu\propto\exp\left(-r^{2-p}\right)$, with $p=\frac{3}{2}$
for an early disk (Lynden-Bell & Pringle, 1974). In our computations, the
initial disk’s mass is fixed to 0.1 M⊙. The computational box is set equal to
500 AU, allowing 99% of the disk mass to be encapsulated within $\sim$100 AU.
Mass can only be lost by accretion onto the Sun or via outward diffusion, the
boundary between these two regimes being defined by location of $R_{c}$, the
centrifugal radius. $R_{c}$ is initially located at $\sim$6 AU and expands up
to $\sim$20 and 45 AU after 0.1 and 1 Myr of evolution, respectively. The
disk’s initial mass accretion rate onto the Sun is set to $10^{-7.6}$ M⊙ yr-1
(Hartmann et al., 1998).

The size of dust particles used in our model is determined by a two-
populations algorithm derived from Birnstiel et al. (2012). This algorithm
computes the representative size of particles through the estimate of the
limiting Stokes number in various dynamical regimes. In our model, dust is
initially present in the form of particles of sizes $a_{0}=10^{-7}$ m, and
grow through mutual collisions. This growth is limited by the maximum sizes
imposed by fragmentation or by the drift velocity of the grains (see Aguichine
et al. (2020) for details). The dust surface density is the sum over all
surface densities of available solids at given time and location, assuming a
protosolar ice-to-rock ratio ($\sim$1.58) (Lodders et al., 2009), and a bulk
density of 1000 kg m-3.

We follow the approaches of Desch et al. (2017) and Drążkowska & Alibert
(2017) for the dynamics of trace species in terms of motion and
thermodynamics, respectively. We assume that the disk is uniformly filled with
H2O, CO, and N2. The abundances of CO, and N2 are those derived in Sec. 2 and
the abundance of H2O is that of the leftover oxygen. No chemistry is assumed
to happen between the trace species. In our simulations, grains are considered
as homogeneous mixtures of solid species in proportions determined from the
aforementioned assumptions (relative abundance ratios, condensation,
sublimation). Sublimation of grains occurs during their inward drift when
partial pressures of trace species become lower than the corresponding vapor
pressures. Once released, vapors diffuse both inward and outward. Because of
the outward diffusion, vapors can recondense back in solid form following the
rates defined by Drążkowska & Alibert (2017), and condensation occurs either
until thermodynamic equilibrium is reached or until no more gas is available
to condense. The position of the iceline of a given species is defined as the
location where its gaseous abundance equals that of its coexisting solid
(Lodders, 2003; Öberg & Wordsworth, 2019; Aguichine et al., 2020). This
approach results in icelines closer to the Sun than those computed via a
simple comparison between the partial and saturation pressures.

The motion of dust and vapor is computed by integrating the 1D radial
advection-diffusion equation derived from Birnstiel et al. (2012) and Desch et
al. (2017), and detailed in Aguichine et al. (2020). The vapor pressures of
trace species are taken from Fray & Schmitt (2009).

###  3.2. Results

Figure 3.— From left to right: radial profiles of the N2/CO ratio relative to
its initial abundance (defined by the enrichment factor $f$) calculated as a
function of time in the PSN for viscosity parameters $\alpha$ = 5 $\times$
10-3, 10-3, and 10-4. Dashed and solid lines correspond to vapor and solid
phases, respectively. The blue bar corresponds to the N2/CO ratio measured in
comet R2.

Figure 3 represents the time evolution of the radial profiles of the N2/CO
ratio relative to its assumed initial ratio ($\sim$2.66 $\times$ 10-1),
defined by the enrichment factor $f$, in both solid and gas phases of the PSN,
and for three values of the viscosity parameter $\alpha$, namely 5 $\times$
10-3, 10-3, and 10-4. The adopted $\alpha$ values are well within the range of
those typically used in models of protoplanetary disks (Hersant et al., 2001;
Nelson et al., 2013; Simon et al., 2015). The figure shows that $f$ is flat
almost everywhere in the PSN, except in the vicinity of the CO and N2
icelines, which are located at very close distances from each other, i.e. less
than a few tenths of AU, in the 10–15 AU region of the PSN, with the CO
iceline situated a bit closer in in the disk.

Depending on the choice of the $\alpha$ parameter, the N2/CO ratio increases
to more than $\sim$10 in the gas phase in the area of the two icelines. This
peak corresponds to the supply of N2 vapor when N2-rich dust drifts inward the
N2 iceline, which is in excess compared to the CO vapor supplied via backward
diffusion beyond the CO iceline. When significant, this peak is preceded by a
decrease of the N2/CO, which corresponds to the the supply of CO vapor when
CO-rich dust drifts inward the CO iceline. The N2/CO ratio in solid phase also
experiences important depletions, which correspond to the location where N2
essentially forms vapor while CO remains in solid phase. The N2/CO ratio in
solid phase also reaches peaks up to $f$ $\sim$10, depending on the $\alpha$
value, at the location of the N2 iceline, corresponding to the formation of
solid N2 from N2 vapor diffusing backward its corresponding iceline. The
figure also shows it is possible to form dust in a narrow region of the PSN,
i.e. within 10–15 AU, with N2/CO ratios matching the value estimated in comet
R2. A comet agglomerated from these grains would present a N2/CO ratio
consistent with the R2 value, both in its interior and its coma, because the
two molecules present similar volatilities, as testified by the proximity of
their icelines in the PSN.

Figure 4.— From left to right: radial profiles of the CO/H2O ratio relative to
its initial abundance (defined by the enrichment factor $f$) calculated as a
function of time in the PSN for viscosity parameters $\alpha$ = 5 $\times$
10-3, 10-3, and 10-4. Dashed and solid lines correspond to vapor and solid
phases, respectively.

Figure 4 represents the time evolution of the radial profiles of the CO/H2O
ratio relative to its assumed initial ratio ($\sim$2.96 $\times$ 10-1),
defined by the enrichment factor $f$, in both solid and gas phases of the PSN,
and for three values of the viscosity parameter $\alpha$, namely 5 $\times$
10-3, 10-3, and 10-4. The CO/H2O ratio in the gas phase increases steeply in
the vicinity of the H2O iceline, which is located at much closer distances
from the Sun than the CO and N2 icelines. This steep increase is due to the
freezing of the H2O vapor when it diffuses outward through its iceline. The
CO/H2O ratio is deeply depleted in the solid phase at distances inward of the
CO iceline because this species remains in gaseous form. On the other hand,
when approaching the CO iceline, the CO/H2O ratio presents values exceeding
its initial value. A peak forms at this location, and its magnitude depends on
the adopted viscosity parameter $\alpha$ of the disk. The CO/H2O enrichment
factor $f$ can typically reach a value of $\sim$10 with $\alpha$ = 10-3, while
it peaks at $\sim$200 with $\alpha$ = 10-4. If the building blocks of a comet
were assembled from grains formed at this location of the PSN, they should
present a CO–rich and H2O–poor composition. Assuming $f$ = 100, the CO/H2O
ratio would be $\sim$30, a value exceeding by far those known in typical
comets (Bockelée-Morvan & Biver, 2017). We note that this value is still about
10 times lower than the one inferred in R2’s coma but this latter can be seen
as an overestimate of the bulk CO abundance in the comet. Indeed, CO is much
more volatile than H2O and its vapor comes from deeper layers than H2O vapor
in the nucleus, potentially inducing an enriched CO/H2O ratio in the coma.
This effect has been investigated in depth by Marboeuf & Schmitt (2014) who
found that the abundance of volatile molecules (relative to H2O) released from
the interiors of nuclei can vary by several orders of magnitude along the
comet’s orbital evolution.

Interestingly, Figures 3 and 4 show that the locations of the CO and N2
icelines do not significantly evolve with time and are only weakly affected by
the value adopted for the viscosity parameter. The main reason for this is
that viscous heating occurs mainly in the dense regions of the PSN. With our
parametrization, the gas density quickly decreases after 10 AU, implying that
the temperature profile beyond this distance is mostly ruled by solar
irradiation, here assumed to be constant.

##  4\. Conclusions

In this work, we first show that a comet agglomerated from clathrates
crystallized in the PSN could, in principle, display N2/CO ratios
corresponding to the value measured in R2’s coma. This finding contrasts with
the suggestion of Wierzchos & Womack (2018) to rule out the presence of
clathrates in R2 only because its N2/CO ratio is higher than those measured in
other comets. This statement is correct in the case of a limited budget of
available crystalline water in the PSN. However, if water is abundant enough,
as shown in this paper, both molecules are efficient clathrate formers. On the
other hand, the amount of water required by clathrates formed in the PSN to
match the N2/CO ratio measured in R2’s coma, namely H2O/CO $\sim$6, is
inconsistent with its estimated extremely low value ($\sim$0.32%).

We have then used a disk model describing the radial transport of volatiles in
the PSN, evolving both in gaseous and solid (pure condensates) phases. We show
that N2/CO ratios reproducing the value estimated in comet R2 can be found in
dust formed in the vicinity of the CO and N2 icelines, i.e. within the 10–15
AU region of the PSN, depending on the adopted viscosity parameter $\alpha$
parameter of the disk. Meanwhile, very high CO/H2O ratios, i.e. up to more
than 200 times the value derived from a protosolar gaseous mixture, can be
found in dust formed in the vicinity of the CO iceline ($\sim$10 AU), the
extent of which also depending on the adopted viscosity parameter. If the
building blocks of a comet assembled from grains formed close to the CO
iceline in the PSN, they should present N2/CO and CO/H2O ratios consistent
with the measurements made in comet R2’s coma.

The likelihood of finding R2-like comets versus H2O-rich comets in the PSN can
be, to first order, assessed via comparing the masses of their respective
reservoirs. To do so, we calculated the mass of H2O and CO ices contained in a
PSN annulus in which CO was found to be more abundant than H2O (R2-like comets
– reservoir 1) , and in another annulus where the CO/H2O ratio was set to 2.96
$\times$ 10-1, i.e. the value derived from protosolar O and C abundances
(average comets – reservoir 2). The time evolution of the reservoir
1/reservoir 2 mass ratio is represented in Fig. 5 for $\alpha$ = 10-3 and
10-4. Higher $\alpha$ values do not allow for the formation of reservoir 1 in
the PSN. This is also shown by Fig. 4 in the case $\alpha$ = 5 $\times$ 10-3,
where the CO/H2O ratio in the solid phase never exceeds twice the initial
ratio. Figure 5 illustrates the fact that the mass ratio between the two
reservoirs strongly depends on the adopted $\alpha$ parameter and the epoch of
the PSN evolution. In the case $\alpha$ = 10-3, the mass of R2-like comets is
about 1% that of the average comets, if the grains from which they assembled
condensed between 0.1 and 1 Myr in the PSN. In the case $\alpha$ = 10-4, the
mass of R2-like comets varies between less than $\sim$1% and 100$\%$ that of
the average comets, depending on the particular grain formation epoch in the
0.1–1 Myr timeframe. We conclude that the likelihood of finding R2-like comets
versus H2O-rich comets in the PSN strongly depends on the adopted disk
parameters, and thus remains difficult to predict a priori.

Figure 5.— Time evolution of the mass ratio between reservoir 1 (R2-like
comets) and reservoir 2 (average comets) for viscosity parameters $\alpha$ =
10-3 and 10-4.

The heliocentric distances indicated for the different icelines must be taken
with caution and are model-dependent. However, the agglomeration of R2 from
dust condensed in the region of the CO iceline indicates that this comet
formed at a greater heliocentric distance than a H2O-rich comet formed from
clathrates because the formation temperatures of N2– and CO–rich clathrates
are always higher than those of the pure condensates at PSN conditions. Our
model also applies to other protoplanetary disks and can explain the unusual
composition of the interstellar comet 2l/Borisov, which seems to contain
substantially more CO than H2O gas in its coma (Bodewits et al., 2020).

Interestingly, physicochemical models of nitrogen chemistry in protostellar
disks show that photodissociation of N2 leads to production of HCN (Hily-Blant
et al., 2017). To overcome this issue, R2 may have formed in a location of the
PSN where significant N2 shielding led to the high N2 and decreased HCN
abundances (Wierzchos & Womack, 2018).

We thank the two Referees for their valuable comments on the content of our
manuscript and their suggestions for improving the document. O.M. and A.B.
acknowledge support from CNES, and J.L. from the JWST project. K.E.M.
acknowledges support from the NASA RDAP grant 80NSSC19K1306, and A.L-K.
acknowledges support from the NASA RDAP grant 80NSSC18K1620.

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