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\begin{center}
\vskip 1cm{\Large\bf The Arithmetic Jacobian Matrix and Determinant}
\vskip 1cm
\large
Pentti Haukkanen and Jorma K. Merikoski \\
Faculty of Natural Sciences \\
FI-33014 University of Tampere\\ 
Finland\\
\href{mailto:pentti.haukkanen@uta.fi}{\tt pentti.haukkanen@uta.fi} \\
\href{mailto:jorma.merikoski@uta.fi}{\tt jorma.merikoski@uta.fi} \\
\ \\
Mika Mattila\\
Department of Mathematics\\
Tampere University of Technology\\
PO Box 553 \\
FI-33101 Tampere\\
Finland\\
\href{mailto:mika.mattila@tut.fi}{\tt mika.mattila@tut.fi}\\
\ \\
Timo Tossavainen\\
Department of Arts, Communication and Education\\ 
Lulea University of Technology\\
SE-97187 Lulea\\
Sweden\\
\href{mailto:timo.tossavainen@ltu.se}{\tt timo.tossavainen@ltu.se}
\end{center}

\vskip .2 in

\begin{abstract}
Let $a_1,\dots,a_m$ be real numbers that can be expressed as a finite
product of prime powers with rational exponents.
Using arithmetic partial derivatives, we define the arithmetic Jacobian
matrix~$\bf J_a$ of the vector ${\bf a}=(a_1,\dots,a_m)$ analogously to
the Jacobian matrix~$\bf J_f$ of a vector function~$\bf f$.
We introduce the concept of multiplicative independence of $\{a_1,\dots,a_m\}$ and show that
$\bf J_a$ plays in it a similar role as $\bf J_f$ does in functional independence.
We also present a kind of arithmetic implicit function theorem and show that $\bf J_a$
applies to it somewhat analogously as $\bf J_f$ applies to the ordinary implicit function
theorem.
\end{abstract}

\section{Introduction}

Let $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$, $\mathbb{N}$,
and~$\mathbb{P}$ stand for the set of real numbers, rational numbers,
integers, nonnegative integers, and primes, respectively.

If $a\in\mathbb{R}$, there
may be a sequence of rational numbers $(\nu_p(a))_{p\in\mathbb{P}}$
with only finitely many nonzero terms satisfying
\begin{eqnarray}
\label{a}
a=(\mathrm{sgn}\,a)\prod_{p\in\mathbb{P}}p^{\nu_p(a)},
\end{eqnarray}
where sgn is the sign function. We let $\mathbb{R}'$
and~$\mathbb{R}'_+$ denote the set
of all such real numbers and the subset consisting of its positive elements,
respectively. Formula~(\ref{a}) is also valid for~$a=0$, as we define $\nu_p(0)=0$
for all $p\in\mathbb{P}$. 
If $\nu_p(a)\ne 0$, we say that $p$
\emph{divides}~$a$ and denote $p\mid a$. Otherwise, we denote $p\nmid a$.

\begin{proposition}
Let $a\in\mathbb{R}'$ and $V_a=\{\nu_p(a)\mid p\in\mathbb{P}\}$. Then
\begin{itemize}
\item[{\rm (a)}] $a\in\mathbb{Z}$ if and only if
$V_a\subset\mathbb{N}$;
\item[{\rm (b)}] $a\in\mathbb{Q}$ if and only if $V_a\subset\mathbb{Z}$.
\end{itemize}
\end{proposition}
\begin{proof}
Simple and omitted.
\end{proof}

\begin{proposition}
\label{uniq}
If $a\in\mathbb{R}'$, then the sequence $(\nu_p(a))_{p\in\mathbb{P}}$
is unique.
\end{proposition}
\begin{proof}
This is well known if $a\in\mathbb{Q}$. Otherwise, see \cite[Lemma~1]{UA}.
\end{proof}

We define the {\em arithmetic derivative} of~$a\in\mathbb{R}'$ by 
$$
a'=
a\sum_{p\in\mathbb{P}}\frac{\nu_p(a)}{p}=\sum_{p\in\mathbb{P}}a'_p,
$$
where
\begin{eqnarray}
\label{partder}
a'_p=\frac{\nu_p(a)}{p}a
\end{eqnarray}
is the {\em arithmetic partial derivative} of~$a$ with respect to~$p$.
For the background and history of these concepts, see, e.g., 
\cite{Ba, UA, Ko, HMT}. These references mainly concern the arithmetic derivative in
$\mathbb{N}$, $\mathbb{Z}$, or~$\mathbb{Q}$, but most of the results can be
extended to~$\mathbb{R}'$ in an obvious way, see \cite[Section~9]{UA}.

Let ${\bf f}=(f_1,\dots,f_m): E\to\mathbb{R}^m$
be a continuously differentiable function, where $E\subseteq\mathbb{R}^n$ is a connected open set.
Its {\em Jacobian matrix} at ${\bf t}=(t_1,\dots,t_n)\in E$ is defined by
$$
{\bf J_f}({\bf t})=\left(
\begin{array}{cccc}
(f_1)'_{t_1}(\bf{t})&(f_1)'_{t_2}(\bf{t})&\dots&(f_1)'_{t_n}(\bf{t})
\\
(f_2)'_{t_1}(\bf{t})&(f_2)'_{t_2}(\bf{t})&\dots&(f_2)'_{t_n}(\bf{t})
\\
&\vdots&&
\\
(f_m)'_{t_1}(\bf{t})&(f_m)'_{t_2}(\bf{t})&\dots&(f_m)'_{t_n}(\bf{t})
\end{array}
\right),
$$
where $(f_i)'_{t_j}=\partial f_i/\partial t_j$. If $m=n$, then $\det{\bf J_f}({\bf x})$ is the
{\em Jacobian determinant} (or, more briefly,
the {\em Jacobian}) of~$\bf f$.

Let $a_1,\dots,a_m\in\mathbb{R}'_+$ (actually, we could study~$\mathbb{R}'$ instead
of~$\mathbb{R}'_+$, which, however, would not benefit us in any significant way), and
denote
\begin{eqnarray}
\label{p}
P=\{p_1,\dots,p_n\}=\{p\in\mathbb{P}\mid
\exists a_i: p\mid a_i\}
\end{eqnarray}
and
\begin{eqnarray}
\label{alpha}
\alpha_{ij}=\nu_{p_j}(a_i),\quad i=1,\dots,m,\,j=1,\dots,n.
\end{eqnarray}
Then
\begin{eqnarray}
\label{ai}
a_i=\prod_{p\in\mathbb{P}}p^{\nu_p(a_i)}=
p_1^{\alpha_{i1}}p_2^{\alpha_{i2}}\cdots p_n^{\alpha_{in}},\quad
i=1,\dots,m.
\end{eqnarray}
Further, let
\begin{eqnarray}
\label{alphavec}
{\bf a}=\left(
\begin{array}{c}
a_1\\a_2\\\vdots\\a_m
\end{array}
\right),
\qquad
\boldsymbol{\alpha}_i=\left(
\begin{array}{c}
\alpha_{i1}
\\
\alpha_{i2}
\\
\vdots
\\
\alpha_{in}
\end{array}
\right),\quad
i=1,\dots,m,
\end{eqnarray}
and
\begin{eqnarray}
\label{as}
{\bf A_a}=
\left(
\begin{array}{cccc}
\alpha_{11}&\alpha_{12}&\dots&\alpha_{1n}
\\
\alpha_{21}&\alpha_{22}&\dots&\alpha_{2n}
\\
&\vdots&&
\\
\alpha_{m1}&\alpha_{m2}&\dots&\alpha_{mn}
\end{array}
\right)=
\left(
\begin{array}{c}
\boldsymbol{\alpha}_1^T
\\
\boldsymbol{\alpha}_2^T
\\
\vdots
\\
\boldsymbol{\alpha}_m^T
\end{array}
\right).
\end{eqnarray}

We define the \emph{arithmetic Jacobian matrix} of~$\bf a$ by
$$
{\bf J_a}=\left(
\begin{array}{cccc}
(a_1)'_{p_1}&(a_1)'_{p_2}&\dots&(a_1)'_{p_n}
\\
(a_2)'_{p_1}&(a_2)'_{p_2}&\dots&(a_2)'_{p_n}
\\
&\vdots&&
\\
(a_m)'_{p_1}&(a_m)'_{p_2}&\dots&(a_m)'_{p_n}
\end{array}
\right)
$$
and, if $m=n$, the \emph{arithmetic Jacobian
determinant} (or, more briefly, the \emph{arithmetic Jacobian}) of~$\bf a$ by
$$
\det{\bf J_a}=\left|
\begin{array}{cccc}
(a_1)'_{p_1}&(a_1)'_{p_2}&\dots&(a_1)'_{p_m}
\\
(a_2)'_{p_1}&(a_2)'_{p_2}&\dots&(a_2)'_{p_m}
\\
&\vdots&&
\\
(a_m)'_{p_1}&(a_m)'_{p_2}&\dots&(a_m)'_{p_m}
\end{array}
\right|.
$$

Let $\bf f$ be as above. The functions
$f_1,\dots,f_m$ are functionally independent (i.e., there is no function
$\phi:\mathbb{R}^m\to\mathbb{R}$ such that $\nabla\phi(\bf f(t))\ne 0$
and $\phi(f_1({\bf t}),\dots,f_m({\bf t}))=0$ for all ${\bf t}\in E$) if and only if
$m\le n$ and ${\rm rank}\,{\bf J_f(t)}=m$ for all ${\bf t}\in E$.
(See, e.g.,~\cite{Ne}.) In Section~\ref{mlinindep}, we will define the concept
of multiplicative independence of the numbers $a_1,\dots,a_m$ and study the role
of~${\bf J_a}$ there.

We outline the implicit function theorem~\cite[Theorem~9.28]{Ru}.
Assuming $m<n$, write $\bf t=(x,y)$, where ${\bf x}\in\mathbb{R}^m$ and
${\bf y}\in\mathbb{R}^{n-m}$. Let
${\bf a}\in\mathbb{R}^m$ and ${\bf b}\in\mathbb{R}^{n-m}$ satisfy
$({\bf a,b})\in E$. Define the function
$\boldsymbol{\phi}(\bf x)=\bf f(x,b)$, where $\bf x$ is ``close to''~$\bf a$. If it satisfies
$\det{\bf J}_{\boldsymbol{\phi}}({\bf a,b})\ne 0$ and if $\bf y$ is
``close to''~$\bf b$, then the equation $\bf f(x,y)=0$ is uniquely ``solvable''
with respect to~$\bf x$.
We will in Section~\ref{implicit} present a theorem where the arithmetic Jacobian matrix and determinant
play a somewhat similar role. Section~\ref{remarks} is devoted to some concluding remarks.

\section{Multiplicative independence}
\label{mlinindep}

Let ${\rm seq}\,\mathbb{Q}$ be the set of infinite 
sequences of rational numbers with finitely many nonzero terms.
Consider the mapping
$$
\boldsymbol{\nu}: \mathbb{R}'_+\to{\rm seq}\,\mathbb{Q}:
\boldsymbol{\nu}(a)=(\nu_p(a))_{p\in\mathbb{P}}.
$$
Defining in~${\rm seq}\,\mathbb{Q}$ addition and scalar multiplication in the
ordinary way, this set becomes a vector space
over~$\mathbb{Q}$. On the other hand, defining in~$\mathbb{R}'_+$
addition as the ordinary multiplication and scalar multiplication
as the ordinary exponentation, also $\mathbb{R}'_+$ becomes a vector
space over~$\mathbb{Q}$. Then $\boldsymbol{\nu}$ is an isomorphism,
and we can identify $\mathbb{R}'_+$
with~${\rm seq}\,\mathbb{Q}$. Because linear
independence is a well-defined concept in~${\rm seq}\,\mathbb{Q}$,
we may so define linear independence in~$\mathbb{R}'_+$. However,
we find the term ``multiplicative independence'' more appropriate.
(We quote this term from Pong~\cite{Po}, who studied this concept
in an Abelian group.)
So we say that a set
\begin{eqnarray}
\label{s}
S=\{a_1,\dots,a_m\}\subset\mathbb{R}'_+
\end{eqnarray}
is (and the numbers
$a_1,\dots,a_m$ are) {\em multiplicatively independent}
if the set $\{\boldsymbol{\nu}(a_1),\dots,\boldsymbol{\nu}(a_m)\}$
is linearly independent. Otherwise, $S$ is
(and $a_1,\dots,a_m$ are) {\em multiplicatively dependent}.

\begin{proposition}
\label{linindep}
Let $S$ be as in~{\rm (\ref{s})} and
$\boldsymbol{\alpha}_1,\dots,\boldsymbol{\alpha}_m$ as 
in~{\rm (\ref{alphavec})}. The following conditions are equivalent:
\begin{itemize}
\item[{\rm (a)}] The set~$S$ is multiplicatively independent.
\item[{\rm (b)}] The only rational numbers $\lambda_1,\dots,\lambda_m$
satisfying $a_1^{\lambda_1}\cdots a_m^{\lambda_m}=1$ are
$\lambda_1=\dots=\lambda_m=0$.
\item[{\rm (c)}] The set $\{\boldsymbol{\alpha}_1,\dots,\boldsymbol{\alpha}_m\}$
is linearly independent in the vector
space~$\mathbb{Q}^n$, where $n$ is as in~{\rm (\ref{p})}.
\end{itemize}
\end{proposition}
\begin{proof}
Simple and omitted.
\end{proof}

We now present such properties of the arithmetic Jacobian determinant that
have relevance to multiplicative independence.

\begin{proposition}
\label{jprop}
Let $\bf a$ be as in~{\rm (\ref{alphavec})}. If $n=m$ in~{\rm (\ref{p})}, then
$$
\det{\bf J_a}=\frac{a_1a_2\cdots a_m}{p_1p_2\cdots p_m}
\det{\bf A_a}.
$$
\end{proposition}
\begin{proof}
By~(\ref{partder}),
$$
\det{\bf J_a}=\left|
\begin{array}{cccc}
\frac{\alpha_{11}}{p_1}a_1&\frac{\alpha_{12}}{p_2}a_1&\dots&
\frac{\alpha_{1m}}{p_m}a_1
\\
\frac{\alpha_{21}}{p_1}a_2&\frac{\alpha_{22}}{p_2}a_2&\dots&
\frac{\alpha_{2m}}{p_m}a_2
\\
&\vdots&&
\\
\frac{\alpha_{m1}}{p_1}a_m&\frac{\alpha_{m2}}{p_2}a_m&\dots&
\frac{\alpha_{mm}}{p_m}a_m
\end{array}
\right|.
$$
Take the factor~$a_1$ from the first row, $a_2$ from the second one, etc.
Further, take the factor $1/p_1$ from the remaining first column, $1/p_2$ from
the second one, etc. We obtain
$$
\det{\bf J_a}=a_1a_2\cdots a_m\left|
\begin{array}{cccc}
\frac{\alpha_{11}}{p_1}&\frac{\alpha_{12}}{p_2}&\dots&
\frac{\alpha_{1m}}{p_m}
\\
\frac{\alpha_{21}}{p_1}&\frac{\alpha_{22}}{p_2}&\dots&
\frac{\alpha_{2m}}{p_m}
\\
&\vdots&&
\\
\frac{\alpha_{m1}}{p_1}&\frac{\alpha_{m2}}{p_2}&\dots&
\frac{\alpha_{mm}}{p_m}
\end{array}
\right|=
\frac{a_1a_2\cdots a_m}{p_1p_2\cdots p_m}\left|
\begin{array}{cccc}
\alpha_{11}&\alpha_{12}&\dots&\alpha_{1m}
\\
\alpha_{21}&\alpha_{22}&\dots&\alpha_{2m}
\\
&\vdots&&
\\
\alpha_{m1}&\alpha_{m2}&\dots&\alpha_{mm}
\end{array}
\right|.
$$
\end{proof}

\begin{corollary}
\label{jcor}
If $\bf a$ is as in~{\rm(\ref{alphavec})}, then
${\rm rank}\,{\bf J_a}={\rm rank}\,{\bf A_a}$.
\end{corollary}
\begin{proof}
Given $I\subseteq\{1,\dots,m\}$ and $J\subseteq\{1,\dots,n\}$ with
equal number of elements, let
${\bf J_a}(I,J)$ and ${\bf A_a}(I,J)$ denote the submatrix of
$\bf J_a$ and respectively~$\bf A_a$ with row indices in~$I$ and
column indices in~$J$. By Proposition~\ref{jprop}, these matrices
are either both nonsingular or both singular. Since rank is the largest
dimension of a nonsingular square submatrix, the claim therefore follows.
\end{proof}

\begin{theorem}
\label{rankthm}
Let $S$, $\bf a$, and~$P$ be as in~{\rm (\ref{s}), (\ref{alphavec})},
and~{\rm (\ref{p})}, respectively. The set~$S$ is multiplicatively
independent if and only if
${\rm rank}\,{\bf J}_{\bf a}=m$.
\end{theorem}
\begin{proof}
Apply Proposition~\ref{linindep} and Corollary~\ref{jcor}.
(Note that ${\rm rank}\,{\bf J}_{\bf a}=m$ implies~$m\le n$.)
\end{proof}

\begin{proposition}
Let $\mathbf a$ and $P$ be as in~{\rm (\ref{alphavec})}
and~{\rm (\ref{p})}, respectively,
let $0\ne x_1,\dots,x_m\in\mathbb{Q}$, and
denote ${\bf a^x}=(a_1^{x_1},\dots,a_m^{x_m})$. Then
$$
{\rm rank}\,{\bf J_{a^x}}={\rm rank}\,\bf J_a.
$$
\end{proposition}
\begin{proof}
In general, we have
$$
(a^x)'_p=\frac{\nu_p(a^x)}{p}a^x=\frac{x\nu_p(a)}{p}a^x=
xa^{x-1}\frac{\nu_p(a)}{p}a=xa^{x-1}a'_p
$$
for all $a\in\mathbb{R}'_+$, $x\in\mathbb{Q}$, $p\in\mathbb{P}$. Consequently,
\begin{eqnarray*}
{\bf J_{a^x}}=
\left(
\begin{array}{cccc}
(a_1^{x_1})'_{p_1}&(a_1^{x_1})'_{p_2}&\dots&(a_1^{x_1})'_{p_n}
\\
(a_2^{x_2})'_{p_1}&(a_2^{x_2})'_{p_2}&\dots&(a_2^{x_2})'_{p_n}
\\
&\vdots&&
\\
(a_m^{x_m})'_{p_1}&(a_m^{x_m})'_{p_2}&\dots&(a_m^{x_m})'_{p_n}
\end{array}
\right)=\qquad\qquad\qquad\qquad
\\
\left(
\begin{array}{cccc}
x_1a_1^{x_1-1}(a_1)'_{p_1}&x_1a_1^{x_1-1}(a_1)'_{p_2}&\dots&x_1a_1^{x_1-1}(a_1)'_{p_n}
\\
x_2a_2^{x_2-1}(a_2)'_{p_1}&x_2a_2^{x_2-1}(a_2)'_{p_2}&\dots&x_2a_2^{x_2-1}(a_2)'_{p_n}
\\
&\vdots&&
\\
x_ma_m^{x_m-1}(a_m)'_{p_1}&x_ma_m^{x_m-1}(a_m)'_{p_2}&\dots&x_ma_m^{x_m-1}(a_m)'_{p_n}
\end{array}
\right)=\bf DJ_a,
\end{eqnarray*}
where
$$
{\bf D}={\rm diag}\,(x_1a_1^{x_1-1},\dots,x_ma_m^{x_m-1}).
$$
Since $\bf D$ is invertible, the claim follows.
\end{proof}

\section{An arithmetic implicit function theorem}
\label{implicit}

In this section, we establish a kind of arithmetic implicit function theorem
where the arithmetic Jacobian matrix and determinant apply. 
The problem is that arithmetic differentiation
operates on numbers, not on functions; so, we must include variables in this attempt.

Let $a_1,\dots,a_m,b_1,\dots,b_r\in\mathbb{R}'_+$. We consider
the equation
\begin{eqnarray}
\label{impleq}
a_1^{x_1}\cdots a_m^{x_m}=b_1^{y_1}\cdots b_r^{y_r},
\end{eqnarray}
where $x_1,\dots,x_m\in\mathbb{Q}$ are variables and
$y_1,\dots,y_r\in\mathbb{Q}\setminus\{0\}$ are constants.
Factorizing $b_1,\dots,b_r$ as in~(\ref{a}), we can reduce~(\ref{impleq}) to
\begin{eqnarray}
\label{implequ}
a_1^{x_1}\cdots a_m^{x_m}=q_1^{z_1}\cdots q_s^{z_s},
\end{eqnarray}
where $q_1,\dots,q_s\in\mathbb{P}$ are distinct and
$z_1,\dots,z_s\in\mathbb{Q}\setminus\{0\}$ are constants.
We define $P$ and~$\alpha_{ij}$ by (\ref{p}) and~(\ref{alpha}), respectively;
then (\ref{ai}) holds. We also denote
$$
Q=\{q_1,\dots,q_s\}.
$$

\begin{theorem}
If %Equation
{\rm (\ref{implequ})} has a solution, then
\begin{eqnarray}
\label{subset}
Q\subseteq P.
\end{eqnarray} 
If {\rm (\ref{subset})} holds and
\begin{eqnarray}
\label{rank}
{\rm rank}\,{\bf J_a}=n,
\end{eqnarray}
where $\bf a$ is as in~{\rm (\ref{alphavec})}, then {\rm (\ref{implequ})} has a solution.
\end{theorem}
\begin{proof}
First, assume that (\ref{implequ}) has a solution. Let $q_i\in Q$. Since $z_i\ne 0$,
$q_i$ divides the left-hand side of~(\ref{implequ}); so, $q_i=p_j$ for some~$j$,
and (\ref{subset}) follows.

Second, assume that (\ref{subset}) and~(\ref{rank}) hold. By reordering the indices of
$p_1,\dots,p_n$, we can write $Q=\{p_1,\dots,p_s\}$.
Let $\bf a$ and~$\bf A_a$ be as in (\ref{alphavec}) and~(\ref{as}). Then
$$
a_i^{x_i}=\prod_{j=1}^np_j^{\alpha_{ij}x_i},\quad i=1,\dots,m,
$$
and (\ref{implequ}) reads
$$
\prod_{i=1}^m\prod_{j=1}^np_j^{\alpha_{ij}x_i}=
\prod_{j=1}^sp_j^{z_j},
$$
i.e.,
\begin{eqnarray}
\label{prodeq}
\prod_{j=1}^np_j^{\sum_{i=1}^m\alpha_{ij}x_i}=
\prod_{j=1}^sp_j^{z_j}.
\end{eqnarray}
By Proposition~\ref{uniq}, this is equivalent to
\begin{eqnarray*}
\sum_{i=1}^m\alpha_{ij}x_i=z_j,\quad j=1,\dots,s, 
\\
\sum_{i=1}^m\alpha_{ij}x_i=0,\quad j=s+1,\dots,n.
\end{eqnarray*}
In matrix form, this reads
\begin{eqnarray}
\label{matrixeq}
{\bf A}^T_{\bf a}\bf x=z,
\end{eqnarray}
where $z_{s+1}=\dots=z_n=0$ and
$$
{\bf x}=\left(
\begin{array}{c}
x_1\\x_2\\\vdots\\x_m
\end{array}
\right),\quad
{\bf z}=\left(
\begin{array}{c}
z_1\\z_2\\\vdots\\z_n
\end{array}\right).
$$
By Corollary~\ref{jcor}
and~(\ref{rank}),
\begin{eqnarray}
\label{rankeqs}
{\rm rank}\,{\bf A}^T_{\bf a}=
{\rm rank}\,{\bf A_a}=
{\rm rank}\,{\bf J_a}=n.
\end{eqnarray}
Therefore, ${\bf A}^T_{\bf a}\mathbb{Q}^m=\mathbb{Q}^n$, which implies that
(\ref{matrixeq}) has a solution. (Note that $m\ge n$ by~(\ref{rankeqs}).)
\end{proof}

\begin{corollary}
Assume that $m=n$ in~${\rm (\ref{alpha})}$.
Then %Equation~
{\rm (\ref{implequ})} has a unique solution if and only if
{\rm (\ref{subset})} holds and $\det{\bf J_a}\ne 0$.
\end{corollary}
\begin{proof}
The claim follows from~(\ref{rankeqs}).
\end{proof}

\section{Concluding remarks}
\label{remarks}

We defined the arithmetic Jacobian matrix and multiplicative independence. We saw that the arithmetic Jacobian matrix relates
to multiplicative independence in the same way as the ordinary Jacobian matrix relates
to functional independence. We also noticed that the arithmetic Jacobian
matrix and determinant play a role in establishing a certain kind of implicit function theorem somewhat similarly as
the ordinary Jacobian matrix and determinant do in the ordinary implicit function
theorem. For this purpose, we needed to introduce extra variables.

Another functional interpretation of arithmetic differentiation may be obtained as follows:
If $a\in\mathbb{R}'_+$ and $p\in\mathbb{P}$, then
there are unique $\alpha\in\mathbb{Q}$ and
$\tilde{a}\in\mathbb{R}'_+$ such that $a=\tilde{a}p^\alpha$
and $p\nmid \tilde{a}$; in fact, $\alpha=\nu_p(a)$. Further, $a'_p=\alpha\tilde{a}p^{\alpha-1}$.
So, in studying arithmetic partial derivatives of first order, the primes behave,
in a certain sense, like variables, and the rational numbers like functions.

Since the isomorphism~$\boldsymbol{\nu}$ brings the vector space structure
to~$\mathbb{R}'_+$, we can define all linear algebra concepts there.
In particular, we have already studied the ``arithmetic inner product'',
see~\cite{HMMT}. (In that paper, we considered only $\mathbb{Q}_+$
but every argument applies also for~$\mathbb{R}'_+$.)

\section{Acknowledgment}

We thank the referee whose comments significantly improved our paper.

\begin{thebibliography}{9}

\bibitem{Ba} E.~J.~Barbeau, Remarks on an arithmetic derivative,
\emph{Canad. Math. Bull.}~\textbf{4} (1961), 117--122.

\bibitem{HMMT} P.~Haukkanen, M.~Mattila, J.~K.~Merikoski, and
T.~Tossavainen, Perpendicularity in an Abelian group, {\em Internat. J. Math.
Math. Sci.}~\textbf{2013}, Article ID 983607, 8~pp.

\bibitem{HMT} P.~Haukkanen, J.~K.~Merikoski, and T.~Tossavainen, On arithmetic
partial differential equations, {\em J. Integer Sequences}~\textbf{19} (2016),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL19/Tossavainen/tossa6.html}
{Article~16.8.6}.

\bibitem{Ko} J.~Kovi\v c, The arithmetic derivative and antiderivative,
\emph{J. Integer Sequences}~\textbf{15} (2012),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL15/Kovic/kovic4.html}{Article~12.3.8}.

\bibitem{Ne} W.~F.~Newns, Functional dependence, {\em Amer. Math.
Monthly}~\textbf{74} (1967), 911--920.

\bibitem{Po} W.~Y.~Pong, Applications of differential algebra to algebraic
independence of arithmetic functions, {\em Acta Arith.}~\textbf{172} (2016), 149--173.

\bibitem{Ru} W.~Rudin, {\em Principles of Mathematical Analysis}, Third Edition,
McGraw-Hill, 1976.

\bibitem{UA} V.~Ufnarovski and B.~\AA hlander, How to differentiate a number,
\emph{J. Integer Sequences}~\textbf{6} (2003), 
\href{http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html}{Article 03.3.4}.

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\noindent 2010 {\it Mathematics Subject Classification:}
Primary 11C20; Secondary 11A25, 15B36.

\noindent {\it Keywords:} 
arithmetic derivative, arithmetic partial derivative, Jacobian matrix,
Jacobian determinant, implicit function theorem, multiplicative independence.

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\noindent (Concerned with sequences
\seqnum{A000040} and
\seqnum{A003415}.)

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\vspace*{+.1in}
\noindent
Received  January 27 2017;
revised versions received July 11 2017; August 1 2017.
Published in {\it Journal of Integer Sequences}, September 8 2017.

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