11institutetext: LIX, CNRS UMR 7161, École Polytechnique,
Institut Polytechnique de Paris,
1 rue Honoré d’Estienne d’Orves
91120 Palaiseau Cedex 22institutetext: Inria
22email: {maxime.bombar<EMAIL_ADDRESS>
# Right-hand side decoding of Gabidulin codes and applications

Maxime Bombar 1122 Alain Couvreur 2211

###### Abstract

We discuss the decoding of Gabidulin and interleaved Gabidulin codes. We give
the full presentation of a decoding algorithm for Gabidulin codes, which as
Loidreau’s seminal algorithm consists in localizing errors in the spirit of
Berlekamp–Welch algorithm for Reed–Solomon codes. On the other hand this
algorithm consists in acting on codewords on the right while Loidreau’s
algorithm considers an action on the left. This right–hand side decoder was
already introduced by the authors in a previous work for cryptanalytic
applications. We give a generalised version which applies to non–full length
Gabidulin codes. Finally, we show that this algorithm turns out to provide a
very clear and natural approach for the decoding of interleaved Gabidulin
codes.

###### Keywords:

Gabidulin codes Decoding Interleaved codes

## Introduction

Rank metric codes have been introduced in [7] by Gabidulin and have found
applications in cryptography [8, 6, 1, 11, 16, 3, 2], in network
communications [18] or in data storage [15]. Compared to the Hamming world,
only few families of codes endowed with the rank metric are known to have
efficient decoding algorithms. Gabidulin codes are the rank-metric analogue of
Reed-Solomon codes and are somehow optimal because they reach the rank-metric
Singleton bound and come with efficient decoders up to the unique decoding
radius $\frac{n-k}{2}$. However, there exist no known decoder beyond this
bound, even probabilistic ones. More, there exist families of Gabidulin codes
that cannot be list decoded in polynomial time [14]. Nonetheless, if we
consider $u$ codewords in parallel, it is possible to overcome this
restriction and decode up to $\frac{u}{u+1}(n-k)$ rank errors with
overwhelming probability.

In the present article, we recall a right-hand side decoder for Gabidulin
codes recently introduced in [5] for cryptanalytic applications. While the
aforementioned reference restricted to the case of full length Gabidulin codes
(i.e. n=m), in the present article we extend it to handle Gabidulin codes of
any length $n\leqslant m$.

Next, we show how this decoder can be used to provide a simple decoder for
$u$-interleaved Gabidulin codes. We claim that the use of this algorithm
provides a much simpler point of view on the decoding of interleaved Gabidulin
codes because it only involves solving an overdetermined linear system. In
particular, this algorithm is very similar to the decoder for Interleaved
Reed-Solomon codes presented in [4], and its decoding radius is given by
comparing the number of equations to the number of unknowns. Moreover, it
permits to clarify a cryptographic attack based on the decoding of interleaved
Gabidulin codes and provides a very simple explanation of the decoding
failures.

##  1 Notations and Prerequisites

In this article, $q$ is a prime power and $k,m,n,u$ are non negative integers.
$\mathbb{F}_{q}$ denotes the finite field with $q$ elements, and for a non
negative integer $\ell$, $\mathbb{F}_{q^{\ell}}$ is the algebraic extension of
$\mathbb{F}_{q}$ of degree $\ell$. The space of $m\times n$ matrices with
entries in a field $\mathbb{K}$ is denoted by $\mathcal{M}_{m\times
n}(\mathbb{K})$. Lower case bold face letters such as $\boldsymbol{x}$
represent vectors, and upper case bold face letters such as $\boldsymbol{X}$
denote matrices.

###  1.1 Rank metric codes

Given a vector $\boldsymbol{x}\in\mathbb{F}_{q^{m}}^{n}$, the column support
(or support) of $\boldsymbol{x}$, denoted $\textrm{Supp}(\boldsymbol{x})$ is
the $\mathbb{F}_{q}$-vector subspace of $\mathbb{F}_{q^{m}}$ spanned by the
entries of $\boldsymbol{x}$:

$\textrm{Supp}(\boldsymbol{x})=\mathbf{Span}_{\mathbb{F}_{q}}\\{x_{1},\dots,x_{n}\\}.$

We consider also another notion of support, namely the row support. Let
$\mathscr{B}$ be a basis of the extension field
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. Then we define the extension of
$\boldsymbol{x}$ with respect to $\mathscr{B}$ as the matrix
$\textrm{Ext}_{\mathscr{B}}(\boldsymbol{x})\in\mathcal{M}_{m\times
n}(\mathbb{F}_{q})$ whose columns are the decompositions of the entries of
$\boldsymbol{x}$ in the basis $\mathscr{B}$. The row space of
$\textrm{Ext}_{\mathscr{B}}(\boldsymbol{x})$ is called the row support of
$\boldsymbol{x}$, i.e.

$\textrm{RowSupp}(\boldsymbol{x})\stackrel{{\scriptstyle\textrm{def}}}{{=}}\\{\boldsymbol{y}\textrm{Ext}_{\mathscr{B}}(\boldsymbol{x})\mid\boldsymbol{y}\in\mathbb{F}_{q}^{m}\\}\subset\mathbb{F}_{q}^{n}.$

It is a vector subspace of $\mathbb{F}_{q}^{n}$. Notice that the above
definition does not depend on the choice of the basis $\mathscr{B}$. The rank
weight $\mathbf{rank}_{q}(\boldsymbol{x})$ (or rank) of $\boldsymbol{x}$ is
the rank of $\textrm{Ext}_{\mathscr{B}}(\boldsymbol{x})$ with respect to any
basis $\mathscr{B}$. The rank distance or distance between two vectors
$\boldsymbol{x},\boldsymbol{y}\in\mathbb{F}_{q^{m}}^{n}$ is defined as
$d(\boldsymbol{x},\boldsymbol{y})=\mathbf{rank}_{q}(\boldsymbol{x}-\boldsymbol{y}).$
In this article, we consider only $\mathbb{F}_{q^{m}}$-linear codes: a code
$\mathscr{C}$ of length $n$ and dimension $k$ is an
$\mathbb{F}_{q^{m}}$-subspace of $\mathbb{F}_{q^{m}}^{n}$ whose dimension, as
an $\mathbb{F}_{q^{m}}$-vector space, is $k$. The minimum distance of
$\mathscr{C}$ is defined as

$d_{min}(\mathscr{C})=\min\\{\mathbf{rank}_{q}(\boldsymbol{c})\mid\boldsymbol{c}\in\mathscr{C},\boldsymbol{c}\neq
0\\}.$

###  1.2 Gabidulin codes and $q$-polynomials

A $q$–polynomial is a polynomial of the form

$P(X)=p_{0}X+p_{1}X^{q}+\dots+p_{r}X^{q^{r}},\quad p_{r}\neq 0.$

The integer $r$ is called the $q$-degree of $P$ and is denoted by
$\deg_{q}(P)$. A $q$-polynomial $P$ induces an $\mathbb{F}_{q}$-linear map
$\mathbb{F}_{q^{m}}\to\mathbb{F}_{q^{m}}$ whose kernel has dimension bounded
by $\deg_{q}(P)$. The rank of a $q$-polynomial will denote the rank of the
induced linear map. Let $\mathcal{L}$ be the space of $q$-polynomials and
given a positive integer $k<m$, we denote by $\mathcal{L}_{<k}$ (resp.
$\mathcal{L}_{\leqslant k}$) the space of $q$-polynomials of $q$-degree less
than (resp. less than or equal to) $k$. Equipped with the addition and the
composition law, $\mathcal{L}$ is a non commutative ring which is left and
right euclidean [10] and the two-sided ideal $(X^{q^{m}}-X)$ is the kernel of
the canonical map

$\mathcal{L}\rightarrow\textrm{Hom}_{\mathbb{F}_{q}}(\mathbb{F}_{q^{m}},\mathbb{F}_{q^{m}}),$

inducing an isomorphism

$\mathcal{L}/(X^{q^{m}}-X)\simeq\textrm{Hom}_{\mathbb{F}_{q}}(\mathbb{F}_{q^{m}},\mathbb{F}_{q^{m}}).$

Let $n\leqslant m$, $k\leqslant n$ and
$\boldsymbol{g}=(g_{1},\dots,g_{n})\in\mathbb{F}_{q^{m}}^{n}$ whose entries
are linearly independent. The Gabidulin code of dimension $k$ and evaluation
vector $\boldsymbol{g}$ is defined as

$\mathscr{G}_{k}(\boldsymbol{g})\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left\\{(P(g_{1}),\dots,P(g_{n}))\mid
P\in\mathcal{L}_{<k}\right\\}\subset\mathbb{F}_{q^{m}}^{n}.$

Notice that the canonical map

$\left\\{\begin{array}[]{ccc}\mathcal{L}_{<k}&\longrightarrow&\mathscr{G}_{k}(\boldsymbol{g})\\\
P&\longmapsto&(P(g_{1}),\dots,P(g_{n})).\end{array}\right.$

is rank preserving, which allows to identify $\mathscr{G}_{k}(\boldsymbol{g})$
with $\mathcal{L}_{<k}$. It is well known that Gabidulin codes are Maximum
Rank Distance (MRD), which means that they reach the rank metric analogue of
the Singleton bound

$d_{min}(\mathcal{L}_{<k})=n-k+1.$

Moreover, Gabidulin codes come with efficient decoders able to correct errors
up to the unique decoding radius $\frac{n-k}{2}$. However, contrary to Reed-
Solomon codes, there exists families of Gabidulin codes that cannot be list
decoded in polynomial time beyond this bound [14].

Following [5, § 1], to any class $P\in\mathcal{L}/(X^{q^{m}}-X)$, corresponds
an adjoint $P^{\vee}$ defined as follows:

$\text{for}\quad P(X)=\sum_{i=0}^{m-1}p_{i}X^{q^{i}}\qquad\text{and}\qquad
P^{\vee}(X)=\sum_{i=0}^{m-1}p^{q^{m-i}}_{i}X^{q^{m-i}}.$

This corresponds to the usual notion of the adjoint endomorphism with respect
to the non degenerate bilinear form on $\mathbb{F}_{q^{m}}$:
$(x,y)\longmapsto\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(xy).$

##  2 Right hand side decoding algorithm

Let $k<n\leqslant m,\leavevmode\nobreak\
\boldsymbol{g}=(g_{1},\dots,g_{n})\in\mathbb{F}_{q^{m}}^{n}$ whose entries are
linearly independent, and let
$\mathscr{C}\stackrel{{\scriptstyle\textrm{def}}}{{=}}\mathscr{G}_{k}(\boldsymbol{g})$.
In [12], Loidreau introduced a Berlekamp-Welch-like decoding algorithm, which
can decode up to $\lfloor\frac{n-k}{2}\rfloor$ rank errors. This algorithm
works on the left and can be applied to Gabidulin codes of arbitrary length
$n\leqslant m$. Indeed, representing vectors in $\mathbb{F}_{q^{m}}^{n}$ as
matrices, the left–hand side decoding consists in acting on matrices on the
left, which is possible whatever the length $n$ (which corresponds to the
number of columns of the corresponding matrices).

In [5], the authors proposed to work on the right-hand side instead, which was
useful to provide an attack on the code-based encryption scheme Ramesses [11].
However, their decoding algorithm was only considered in the case where $n=m$
(which was enough for cryptanalysis). The right-hand side algorithm is not
completely straightforward when $n<m$. In particular, one can no longer
transpose the matrices representing codewords.

In the present section, we recall a self-contained presentation of the right-
hand side version, and prove that restricting $n$ to be maximal is
unnecessary. In particular, we show how the right-hand side decoding algorithm
applied to any $[n,k]$ Gabidulin code can correct up to
$\left\lfloor\frac{n-k}{2}\right\rfloor$ errors.

###  2.1 $n=m$

Suppose we receive a word
$\boldsymbol{y}=\boldsymbol{c}+\boldsymbol{e}\in\mathbb{F}_{q^{m}}^{n}$ where
$\boldsymbol{c}\in\mathscr{C}\stackrel{{\scriptstyle\textrm{def}}}{{=}}\mathscr{G}_{k}(\boldsymbol{g})$
and $\boldsymbol{e}\in\mathbb{F}_{q^{m}}^{n}$ has rank
$t\leqslant\frac{n-k}{2}$. By linear interpolation, there exist three
$q$-polynomials $C\in\mathcal{L}_{<k}$ and $Y,E\in\mathcal{L}_{<m}$ such that

$Y=C+E,$

and $Y$ is known to the receiver (See for instance [19, Chapter 3]). Since
$n=m$, $(g_{1},\dots,g_{n})$ forms a basis of the extension field
$\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$. Therefore,
$\mathbf{rank}_{q}(E)=\mathbf{rank}_{q}(\boldsymbol{e})=t$. The core of the
algorithm relies in the following proposition:

###### Proposition 1

Let $E$ be a $q$-polynomial of rank $t$. There exists a unique monic
$q$-polynomial $\Lambda$ such that $\deg_{q}(\Lambda)\leqslant t$ and
$E\circ\Lambda=0$ modulo $(X^{q^{m}}-X)$.

The goal is now to compute this right annihilator $\Lambda$. It satisfies the
equation

$Y\circ\Lambda=C\circ\Lambda+E\circ\Lambda\equiv
C\circ\Lambda\mod(X^{q^{m}}-X),$

which yields a non linear system of $n$ equations whose unknowns are the
$k+t+1$ coefficients of $C$ and $\Lambda$.

$\left\\{\begin{array}[]{l}(Y\circ\Lambda)(g_{i})=C\circ\Lambda(g_{i})\\\
\deg_{q}\Lambda\leqslant t\\\ \deg_{q}C\leqslant k-1.\end{array}\right.$ (1)

In order to linearize the system, set $N=C\circ\Lambda$ and consider instead

$\left\\{\begin{array}[]{l}(Y\circ\Lambda)(g_{i})=N(g_{i})\\\
\deg_{q}\Lambda\leqslant t\\\ \deg_{q}N\leqslant k+t-1,\end{array}\right.$ (2)

whose unknowns are the $k+2t+1$ coefficients of $N$ and $\Lambda$. The
relationships between those two systems are specified in the following
proposition.

###### Proposition 2

  * $\bullet$

Any solution $(\Lambda,C)$ of (1) gives a solution $(\Lambda,N=C\circ\Lambda)$
of (2).

  * $\bullet$

Assume that $E$ is of rank $t\leqslant\lfloor\frac{n-k}{2}\rfloor$. If
$(\Lambda,N)$ is a nonzero solution of (2) then $N=C\circ\Lambda$ where
$C=Y-E$ is the interpolating $q$–polynomial of the codeword.

Therefore, it is possible to recover the codeword $C$ from any non zero
solution $(\Lambda,N)$ of (2) by computing the right hand side euclidean
division of $N$ by $\Lambda$ which can be done efficiently.

###### Remark 1

Actually, the previous system is only linear over $\mathbb{F}_{q}$, not over
$\mathbb{F}_{q^{m}}$. To address this issue, one can use the adjunction
operation. Details can be found in [5].

###  2.2 $n<m$

Assume now that $n$ is not maximal, and consider a received word
$\boldsymbol{y}=\boldsymbol{c}+\boldsymbol{e}$, where
$\boldsymbol{c}=(C(g_{1}),\dots,C(g_{n}))$ for some $q$–polynomial $C$ of
$q$–degree $<k$ and $\boldsymbol{e}\in\mathbb{F}_{q^{m}}^{n}$ has rank $t$
whose value is discussed further.

As in the previous section, our first objective is to reformulate the decoding
problem in terms of $q$–polynomials instead of vectors. Here lies the first
issue. Indeed, since $\boldsymbol{y}$ has length $n<m$ there is not a unique
$q$–polynomial $Y$ in $\mathcal{L}/(X^{q^{m}}-X)$ such that
$\boldsymbol{y}=(Y(g_{1}),\dots,Y(g_{n}))$. For this reason, when choosing
such an arbitrary interpolator $Y$ for $\boldsymbol{y}$, one can define
$E\stackrel{{\scriptstyle\textrm{def}}}{{=}}Y-C$ and get a new $q$–polynomial
formulation of the decoding problem as

$Y=C+E,$

but here there is no longer any reason that $E$ would have rank $t$, we only
know that the vector $(E(g_{1}),\dots,E(g_{n}))$ has rank $t$. In terms of
linear operators, this means that the restriction of $E$ to the span $V$ of
$g_{1},\dots,g_{n}$ over $\mathbb{F}_{q}$ has rank $t$.

To fix this issue we proceed as follows. First we choose $Y$ as the
interpolator of lowest degree by choosing the unique monic interpolator of
degree $<n$. Since $\deg_{q}(C)<k<n$, this entails that $\deg_{q}(E)<n$. Next
we will change the interpolating polynomials $Y$ and $E$ in order $E$ to have
rank $t$. This should be done without knowing the error. We need a slight
generalization of Proposition 1 which we prove here for the sake of
completeness.

###### Proposition 3

There exists a $q$–polynomial $G$ of $q$–degree $\leqslant m-n$ whose image
equals the $\mathbb{F}_{q}$–span of $g_{1},\dots,g_{n}$.

###### Proof

Let $V$ denote the $\mathbb{F}_{q}$–span of $g_{1},\dots,g_{n}$. By
interpolation, it is well-known that there exists $G_{0}$ of $q$–degree
$\leqslant m-n$ whose kernel equals the $(m-n)$–dimensional space $V^{\perp}$
for the inner product
$(x,y)\mapsto\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(xy)$. Then, the
$q$–polynomial $G_{1}\stackrel{{\scriptstyle\textrm{def}}}{{=}}X^{q^{n}}\circ
G_{0}$ has the same kernel $V$ and is in the span of the $q$–monomials
$X^{q^{n}},\dots,X^{q^{m}}$. Then, let
$G\stackrel{{\scriptstyle\textrm{def}}}{{=}}G_{1}^{\vee}$ be the adjoint as
introduced in § 1.2. It has $q$–degree $\leqslant m-n$ and, by adjunction
properties, satisfies $\text{Im}(G)=\text{Im}(G_{1}^{\vee})={\rm
Ker}(G_{1})^{\perp}=V$.

Let $G$ be the $q$–polynomial of degree $\leqslant m-n$ given by Proposition 3
and consider $Y\circ G$ instead of $Y$, then we get a new problem which is

$Y\circ G=C\circ G+E\circ G.$ (3)

First, since $\boldsymbol{y}$ and $\boldsymbol{g}$ are known, the
$q$–polynomials $Y,G$ are computable using interpolation. Then, the
$q$–polynomial $C\circ G$ has $q$–degree $<k+m-n$ and hence corresponds to a
codeword of a Gabidulin code of dimension $k+m-n$. Finally, $E\circ G$ has
rank $t$. Indeed, as mentioned earlier, $t$ is the rank of the restriction of
$E$ to the span of $g_{1},\dots,g_{n}$, which is precisely the image of $G$.
Thus, the reformulated problem (3) can be regarded as correcting a rank $t$
error in a Gabidulin code of length $m$ and dimension $k+m-n$. Using our
right-hand decoding algorithm it is hence possible to correct an amount of
errors up to

$t=\frac{m-(k+m-n)}{2}=\frac{n-k}{2}\cdot$

###### Remark 2

The previous results may be interpreted in terms of decoding a length $m$
Gabidulin code which was column–punctured on the right at $\delta=m-n$
positions (see [17, § 2.3] for a definition of column–puncturing). Similarly,
this can be unterstood in terms of decoding a length $m$ Gabidulin code under
$\delta$ rank erasures and $t$ rank errors. In this situation we recover the
usual fact that $2t+\delta\leqslant n-k$.

##  3 Decoding interleaved Gabidulin codes

###  3.1 Interleaving

Interleaving a code $\mathscr{C}$ consists in considering several codewords of
$\mathscr{C}$ at the same time, corrupted by errors having the same support.
In the Hamming metric, interleaved Reed-Solomon codes have been extensively
studied and come with efficient probabilistic decoders allowing to correct
uniquely almost all error patterns slightly beyond the error capability of the
code. See [4] for further reference. In the rank metric, interleaved Gabidulin
codes have been introduced by Loidreau and Overbeck in [13]. Let
$\boldsymbol{g}$ be an evaluation vector, and let $u\in\mathbb{N}$. The
$u$-interleaved Gabidulin code of evaluation vector $\boldsymbol{g}$ and
dimension $k$ is

$I\mathscr{G}_{u,k}(\boldsymbol{g})\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left\\{\begin{pmatrix}\mathbf{c^{(1)}}\\\
\vdots\\\ \mathbf{c^{(u)}}\end{pmatrix}\mid\mathbf{c^{(i)}}\in
Gab_{k}(\mathbf{g})\right\\}.$

###### Remark 3

For $u=1$, we recover usual Gabidulin codes.

Each codeword $\boldsymbol{C}\in I\mathscr{G}_{u,k}(\boldsymbol{g})$ is the
evaluation of a column vector of $q$-polynomials of bounded $q$-degrees on
$\boldsymbol{g}$:

$\boldsymbol{C}=(\boldsymbol{\Gamma}(\boldsymbol{g}_{1}),\dots,\boldsymbol{\Gamma}(\boldsymbol{g}_{n})),\quad\boldsymbol{\Gamma}=\begin{pmatrix}C^{(1)}\\\
\vdots\\\ C^{(u)}\end{pmatrix}\text{ where }C^{(i)}\in\mathcal{L}_{<k}.$

Using the inverse extension map, each $\boldsymbol{\Gamma}(g_{i})$ can be
interpreted as an element of $\mathbb{F}_{q^{mu}}$, and
$I\mathscr{G}_{u,k}(\boldsymbol{g})$ is then a code of length $n$ and
dimension $k$ over $\mathbb{F}_{q^{mu}}$. Moreover, they are known to be MRD
(see [19, Lemma 2.17]), so the error correction capability of
$I\mathscr{G}_{u,k}(\boldsymbol{g})$ is $\frac{n-k}{2}$. However, their
specific structure allows to design efficient algorithms being able to
uniquely decode $I\mathscr{G}_{u,k}(\boldsymbol{g})$ up to
$\frac{u}{u+1}(n-k)>\frac{n-k}{2}$ for $u>1$, with very high probability [13,
20].

In this section we show how to use the right-hand side variant of the
Berlekamp-Welch algorithm introduced before to decode $u$-interleaved
Gabidulin codes, up to $\frac{u}{u+1}(n-k)$. Since this is beyond the error
capability of the code, this algorithm might fail but the probability of
failure is very low.

###  3.2 Error model

Similarly to the Hamming metric, we consider a channel model where errors
happen in burst. In this model, the transmitted codeword is a matrix
$\boldsymbol{C}\in\mathcal{M}_{u\times n}(\mathbb{F}_{q^{m}})$ representing
$u$ codewords of $\mathscr{G}_{k}(\boldsymbol{g})$ in parallel, and the error
is a matrix $\boldsymbol{E}\in\mathcal{M}_{u\times n}(\mathbb{F}_{q^{m}})$ of
$\mathbb{F}_{q}$-rank $t$, i.e. such that the matrix of $\mathcal{M}_{um\times
n}(\mathbb{F}_{q})$ obtained from $\boldsymbol{E}$ by extending every row in a
basis of $\mathbb{F}_{q^{m}}/\mathbb{F}_{q}$ is of rank $t$. The receiver then
gets a word $\boldsymbol{Y}=\boldsymbol{C}+\boldsymbol{E}$, and the goal is to
recover $\boldsymbol{C}$.

In the Hamming metric, the receiver gets $u$ noisy
$\boldsymbol{y}^{(i)}=\boldsymbol{c}^{(i)}+\boldsymbol{e}^{(i)}$ such that
$\boldsymbol{c}^{(i)}$ are codewords of some code $\mathscr{C}$ (e.g. a Reed-
Solomon code) and all the $\boldsymbol{e}^{(i)}$ have the same support of
cardinality $t$.

In the current setting, each row of $\boldsymbol{Y}$ is of the form
$\boldsymbol{y}^{(i)}=\boldsymbol{c}^{(i)}+\boldsymbol{e}^{(i)}$ where
$\boldsymbol{c}^{(i)}\in\mathscr{G}_{k}(\boldsymbol{g})$. The following
proposition whose proof is straightforward justifies the term burst rank-
errors.

###### Proposition 4

The row support of each $\boldsymbol{e}^{(i)}$ is contained in the
$\mathbb{F}_{q}$-row space of $E$ which is of dimension $t$.

###### Remark 4

In this article the error model consists in considering error vectors
$\boldsymbol{e}^{(i)}$ sharing the same row support. It seems to be the most
natural error model when considering the code regarded as a code over
$\mathbb{F}_{q^{mu}}$, and it is the one used in most references. On the other
hand, one may consider another error model where the errors share a common
column support. In the latter case, the usual left–hand side decoder can be
used, see for instance [13].

###  3.3 Right-hand side decoding of interleaved Gabidulin codes

Let $\boldsymbol{Y}=\boldsymbol{C}+\boldsymbol{E}\in\mathcal{M}_{u\times
n}(\mathbb{F}_{q^{m}})$ be a received word. By linear interpolation of each
row of $\boldsymbol{Y}$, there exist $u$ triple of $q$-polynomials
$(Y_{i},C_{i},E_{i})$ such that

$Y_{i}=C_{i}+E_{i},$

and $\deg_{q}(C_{i})<k$. Since the errors have the same support of dimension
$t$, there exists a $q$-polynomial $\Lambda$ with $\deg_{q}(\Lambda)\leqslant
t$ that locates all the errors. More specifically, Proposition 4 induces the
following lemma:

###### Lemma 1

Denoting by $E_{i}$ the interpolator $q$-polynomial of $\boldsymbol{e}^{(i)}$,
there exists $\Lambda\in\mathcal{L}_{\leqslant t}$ such that

$E_{i}\circ\Lambda=0\mod(X^{q^{m}}-X),\quad\forall i\in\\{1,\ldots,u\\}.$

Lemma 1 yields the following non-linear system of $u\times n$ equations

$\left\\{\begin{array}[]{l}(Y_{i}\circ\Lambda)(g_{j})=(C_{i}\circ\Lambda)(g_{j})\\\
\deg_{q}\Lambda\leqslant t\\\ \deg_{q}C_{i}\leqslant k-1,\quad\text{ for
}i\in\\{1,\dots,u\\}.\end{array}\right.$ (4)

which can be linearized into the following system, setting
$N_{i}\stackrel{{\scriptstyle\textrm{def}}}{{=}}C_{i}\circ\Lambda$:

$\left\\{\begin{array}[]{l}(Y_{i}\circ\Lambda)(g_{j})=N_{i}(g_{j})\\\
\deg_{q}\Lambda\leqslant t\\\ \deg_{q}N_{i}\leqslant k+t-1,\quad\text{ for
}i\in\\{1,\dots,u\\}.\end{array}\right.$ (5)

This system has $u\times n$ equations, and $t+1+u(k+t)$ unknowns, and
therefore one can expect to retrieve $(\Lambda,N_{1},\ldots,N_{u})$ whenever
$t\leqslant\frac{u}{u+1}(n-k)$. Since $N_{i}=C_{i}\circ\Lambda$, the codewords
$C_{1},\dots,C_{u}$ can then be recovered by computing euclidean division on
the right.

###### Remark 5

The decoding algorithm mentioned in [13, 20] can be re-interpreted in terms of
the aforementioned decoder. The present section permits in particular to shed
light on the fact that previous algorithms are actually very comparable to
Loidreau’s original algorithm when acting on the right instead of acting on
the left.

###  3.4 Application to cryptography: Liga encryption scheme

Let $\mathbb{F}_{q^{m}},\mathbb{F}_{q^{mu}}$ be two algebraic extensions of
the finite field $\mathbb{F}_{q}$. In [6], Faure and Loidreau introduced a
rank metric encryption scheme with small key size. The originality of the
cryptosystem was to base the security on the hardness of decoding a (public)
Gabidulin code beyond the unique decoding radius. Indeed, the public key was
of the form $\boldsymbol{k}_{pub}=\boldsymbol{x}\boldsymbol{G}+\boldsymbol{z}$
where $\boldsymbol{G}$ is a generator matrix of a public $[n,k]$ Gabidulin
code over $\mathbb{F}_{q^{m}}$ and $\boldsymbol{x}\in\mathbb{F}_{q^{mu}}^{k}$,
$\boldsymbol{z}\in\mathbb{F}_{q^{mu}}^{n}$ together with
$t\stackrel{{\scriptstyle\textrm{def}}}{{=}}\mathbf{rank}_{q}(z)>\frac{n-k}{2}$
form the secret key.

However, it was shown in [9] that an attacker could easily compute $u$ noisy
codewords of the Gabidulin code generated by $\boldsymbol{G}$ using the
$\mathbb{F}_{q^{m}}$–linearity of the trace map
$\text{Tr}_{\mathbb{F}_{q^{mu}}/\mathbb{F}_{q^{m}}}$, and then recover the
secret providing that $t\leqslant\frac{u}{u+1}(n-k)$ (which was always the
case to resist other attacks). This really amounts to decoding the public key
with a decoder of $u$–interleaved Gabidulin codes. In order to repair the
scheme, the authors of Liga proposed instead to base the security on the
hardness of decoding $u$–interleaved Gabidulin codes. Indeed, they proved that
by reducing the rank of $\boldsymbol{z}$ over $\mathbb{F}_{q^{m}}$ (while
keeping its rank weight $t$ over $\mathbb{F}_{q}$ higher than the unique
decoding radius), it was no longer possible to recover the secret key. More
precisely, denoting by
$\zeta\stackrel{{\scriptstyle\textrm{def}}}{{=}}\mathbf{rank}_{q^{m}}(\boldsymbol{z})$
this rank, they proved by a careful analysis of known interleaved decoders
that a condition for making the decoder to fail was $\zeta<\frac{t}{n-k-t}$.
In particular, in Liga they proposed to set $\zeta=2$.

Using our decoder, we propose a new interpretation of this condition. Indeed,
let $\boldsymbol{Y}=\boldsymbol{C}+\boldsymbol{E}\in\mathcal{M}_{u\times
n}(\mathbb{F}_{q^{m}})$ be a noisy codeword of an $u$-interleaved Gabidulin
code. The results of Section 3.3 can be strengthen as follows: If some rows of
$\boldsymbol{E}$ share a linear dependency, then the equations in system (5)
are no longer independent. In particular, if $\zeta\leqslant u$ denotes the
rank of $\boldsymbol{E}$ over $\mathbb{F}_{q^{m}}$, one can refine the
reasoning and deduce an equivalent linear system with $\zeta\times n$
independent equations for $t+1+\zeta(k+t)$ unkowns. Therefore, when
$t>\frac{\zeta}{\zeta+1}(n-k)$, there are more unknowns than equations and the
decoder fails. This inequality is exactly the condition
$\zeta<\frac{t}{n-k-t}$ from Liga.

## Conclusion

We presented a full version of a right-hand side decoding algorithm for
Gabidulin codes. This algorithm is close to a verbatim translation of its
well–known left–hand counterpart. However, compared to its left–hand
counterpart, it was unclear how to apply it to non full length Gabidulin
codes. This issue has been addressed in the present article. Moreover, we
claim that this algorithm is of interest for various applications. First, it
provides a very natural approach for the decoding of interleaved Gabidulin
codes. It is actually very comparable to the algorithm proposed by Loidreau
and Overbeck [13] but the strong connection with a Berlekamp–Welch like
decoder was not that clear in the aforementioned reference. Second, this
right-hand side decoder already appeared to provide an interesting tool for
cryptanalytic applications.

## References

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  * [3] Aragon, N., Gaborit, P., Hauteville, A., Ruatta, O., Zémor, G.: Ranksign – a signature proposal for the NIST’s call. First round submission to the NIST post-quantum cryptography call (Nov 2017), https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/round-1/submissions/RankSign.zip
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  * [6] Faure, C., Loidreau, P.: A new public-key cryptosystem based on the problem of reconstructing p-polynomials. In: Coding and Cryptography, International Workshop, WCC 2005, Bergen, Norway, March 14-18, 2005. Revised Selected Papers. pp. 304–315 (2005)
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  * [11] Lavauzelle, J., Loidreau, P., Pham, B.D.: RAMESSES, a Rank Metric Encryption Scheme with Short Keys (Jan 2020), https://hal.archives-ouvertes.fr/hal-02426624, working paper or preprint
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  * [13] Loidreau, P., Overbeck, R.: Decoding rank errors beyond the error-correction capability. In: Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory, ACCT-10. pp. 168–190 (2006)
  * [14] Raviv, N., Wachter-Zeh, A.: Some Gabidulin codes cannot be list decoded efficiently at any radius. In: 2015 IEEE International Symposium on Information Theory (ISIT). pp. 6–10 (2015)
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  * [16] Renner, J., Puchinger, S., Wachter-Zeh, A.: LIGA: A cryptosystem based on the hardness of rank–metric list and interleaved decoding. Des. Codes Cryptogr. 89, 1279–1319 (2021)
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Neutrino spin oscillations in gravitational fields in noncommutative higher
dimensions

S.A. Alavi† ,T. Fallahi.Serish

_Department of physics, Hakim Sabzevari University, P.O. Box 397_

_Sabzevar, Iran_

†<EMAIL_ADDRESS>alaviag.gmail.com_

###### Abstract

Investigation of neutrino spin oscillation in the gravitational fields of
black holes(BH) is one of the interesting topics in neutrino physics. On the
other hand, in recent years, many studies have been devoted to the exploration
of different physical phenomena in higher dimensions. Noncommutative geometry
has also been in the focus of researchers in the past years to explore deeper
and more accurate the structure of space time. In this work, the neutrino spin
oscillation in the noncommutative higher dimensions gravitational fields of
Schwarzschild and Reissner-Nordstrom metrics are studied. The effects of
noncommutativity of space are calculated and its role in different dimensions
are discussed. Finally upper bounds on noncommutativity parameter are
obtained.

##  1 Introduction

Since the first time in 1930 that Pauli postulated the existence of neutrinos
to establish the conservation of energy, until now the discussion and research
about this particle and its properties [1] has made us understand matter
better but the field of neutrino physics still surrounded by more questions
than answers. What is the mechanism behind neutrino masses? Which neutrino is
the lightest?. How many flavors of neutrinos are there? Do neutrinos interact
with matter or among themselves in a way that differs from the Standard Model
prediction? Are neutrinos connected to the dark matter or the dark energy
content of the universe? Are neutrinos their own antiparticles?. Do neutrinos
violate the symmetries of physics?. Besides those theoretical questions, some
experimental challenges exist too. At present, the study of neutrinos from
different sources and at different energies using cutting-edge technologies is
entering new eras of innovation. Now neutrino is an important basis of various
astrophysical and cosmological researches. On the other hand, since the
Einstein’s proposal of general relativity (GR) in 1915, a lot of research has
been devoted to unify (GR) and Electromagnetism as two fundamental
interactions in nature. In addition there were some efforts in 1920s, to unify
these interactions through Kaluza Klein theory, that was a classical unified
field theory built in five-dimensional space-time[2, 3]. Recently, motivated
by string theory as a requirement for describing a consistent theory of
quantum gravity, extra dimensions have been the subject of much attention.
Besides string theory, there are some other theories proposing the necessity
of extra dimensions. Large extra dimensions, mostly motivated by the ADD
model, by Arkani-Hamed, Dimopoulos, and Dvali together with Antoniadis in
Refs. 4, 5 to solve the hierarchy problem. While the electromagnetic, weak and
strong interactions differ by just six orders of magnitude, the gravitational
interaction falls apart by further thirty-three orders[5]. Warped extra
dimensions were proposed by the Randall-Sundrum (RS) model [2, 6], in which
our observable universe is modeled as a four dimensional hyper surface, known
as the 3-brane, embedded in a five dimensional space, usually called the bulk.
The novel idea of the Brane world is that all the gauge interactions,
described by the Standard Model, are confined to live in the 3-brane while the
gravitational interaction can spread into the fifth dimension of the space.
Universal extra dimensions, proposed and first studied in Ref. 7. Motivation
by different sources, the formulation of physical theories in more than four
space-time dimensions is a subject that has received increasing attention in
recent years. On the other hand, it is broadly accepted that the idea of
continuous space-time should break down at very short distances. It has been
shown that the noncommutative geometry naturally appears in string theory with
a non-zero antisymmetric B field. Therefore reformulations of theories on
noncommutative spaces may play an important role in explaining the properties
of nature at the Planck scale.

Furthermore, the interaction of neutrinos with gravitational fields is one of
the interesting phenomena that can lead to the transition between different
states of helicity (spin oscillation). It also can lead to neutrino flavor
oscillations i.e., neutrinos of one type can be converted into another type if
there is a mixing between different neutrino eigenstates. Examples of this
oscillations type are $\nu_{e}\rightarrow\nu_{\mu}$ or
$\nu_{\mu}\rightarrow\nu_{\tau}$ conversions that are the possible
explanations of the solar and atmospheric neutrino problems. The
electromagnetic interactions are of great importance in studying of neutrino
spin and spin-flavor oscillations. For example, the neutrino interaction with
an external electromagnetic field provides one of the mechanisms for the
mixing between different helicity eigenstates. It is also interesting to study
the influence of gravitational fields on neutrino oscillations. Despite the
fact that, the gravitational interaction is relatively weak compared to
electromagnetic interaction, there are rather strong gravitational fields in
the Universe i.e., gravitational fileds of black holes. Neutrino oscillations
in gravitational fields are of interest in astro-particle physics and
cosmology. The influence of the gravitational interaction on neutrino
oscillations has been studied in many publications (see, e.g., Refs. 8, 9, 10,
11).

In this paper we study neutrino spin oscillations in noncommutative higher
dimensional gravitational fields. We investigate how the noncommuatativity of
space affects the neutrino spin oscillations in three and higher dimensions.
We also provide upper bounds on noncommutative parameter $\theta$.

##  2 Neutrino spin oscillations in noncommutative gravitational fields

The noncommutative space coordinates $x^{\mu}$ satisfy in the following
commutation relations:

$[x^{\mu},x^{\nu}]=i\theta^{\mu\nu},[x^{\mu},p^{\nu}]=i\delta^{\mu\nu},[p^{\mu},p^{\nu}]=0.$
(2.1)

The simplest case is that $\theta^{\mu\nu}$ be a constant antisymmetric tensor
with real elements which have square length dimension. To study gravitational
fields in non-commutative spaces, it is not necessary to change the part of
the Einstein tensor in the field equations, and noncommutative effects only
act on the matter source. noncommutativity of space implies to replace the
point-like structures by smeared objects. So we change the point distribution
like source to spread objects. The spread effect is implemented mathematically
as follows: Dirac delta function everywhere is replcced by a Gaussian
distribution of minimum width $\sqrt{\theta}$. Therefore, we choose the mass
and charge densities of a static, spherically symmetric, smeared, particle-
like gravitational source as (see, Refs. 12, 13, 14, 15):

$\rho_{\theta}(r)=\frac{M}{(4\pi\theta)^{\frac{3}{2}}}exp(-\frac{r^{2}}{4\theta}),$
(2.2)

$\rho_{Q}(r)=\frac{Q}{(4\pi\theta)^{\frac{3}{2}}}exp(-\frac{r^{2}}{4\theta}).$
(2.3)

A particle with mass M instead of being completely in one spot, spread out
through a region of space with the area $\theta$. For an observer at large
distance this smeared density looks like a small sphere of matter with radius
about $\sqrt{\theta}$ of mass $M$ and charge $Q$, so the metric becomes
Schwarzschild ($Q=0$) or RN black hole at large distances. solutions of
Einstein equations with such smeared sources give new kind of regular black
holes in four (Refs. 13, 14, 15, 16, 17, 18) and higher dimensions (Refs. 19,
20, 21, 22, 23, 24, 25, 26, 27).
Vierbein four-velocity $u^{a}=(u^{a},u_{1},u_{2},u_{3})$ and four-velocity of
the particle in its geodesic path $U^{\mu}$ are related by the following
relation[28, 29, 30, 31]:

$u^{a}=e^{a}_{\nu}U^{\mu},$ (2.4)

where $e^{a}_{\mu}$ are non-zero vierbein vectors and we can decompose the
four velocity vectors $U^{\mu}$ using spherical coordinates
${U^{\mu}}=(U^{0},U_{r},U_{\theta},U_{\varphi})$ which it’s components are
defined as follow:

$U^{0}=\frac{dt}{d\tau},U_{r}=\frac{dr}{d\tau},U_{\theta}=\frac{d\theta}{d\tau},U_{\varphi}=\frac{d\varphi}{d\tau}.$
(2.5)

$U^{\mu}$ is related to the world velocity of the particle through
$\overrightarrow{U}=\alpha\overrightarrow{v}$ where $\alpha=\frac{dt}{d\tau}$
and $\tau$ is the proper time. To study the spin oscillation of a particle in
a gravitational field we need to calculate
$G_{ab}=(\overrightarrow{E},\overrightarrow{B})$ which is the analog tensor of
the electromagnetic field and is defined as[28]:

$G_{ab}=e_{a\mu;\nu}e^{\mu}_{b}U^{\nu},G_{0i}=E_{i},G_{ij}=-\varepsilon_{ijk}B_{k}.$
(2.6)

$e_{a\mu;\nu}$ are the covariant derivatives of vierbein vectors:

$e_{a\mu;\nu}=\frac{\partial
e_{a\mu}}{\partial_{x^{\nu}}}-\Gamma^{\lambda}_{\mu\nu}e_{a\lambda},$ (2.7)

where $\Gamma^{\lambda}_{\mu\nu}$ is the Christopher symbol. Geodesic equation
of a particle in a gravitational field is:[32, 33]

$\frac{d^{2}x^{\mu}}{dq^{2}}+\Gamma^{\mu}_{\sigma\upsilon}\frac{dx^{\sigma}}{dq}\frac{dx^{\upsilon}}{dq}=0,$
(2.8)

where the variable $q$ parameterizes the particle’s world line.

Neutrino spin oscillations frequency is given by the expression
$\overrightarrow{\Omega}=\frac{\overrightarrow{G}}{\alpha}$ , where vector
$\overrightarrow{G}$ is defined in this way:

$\overrightarrow{G}=\frac{1}{2}(\overrightarrow{B}+\frac{1}{1+u^{0}}[\overrightarrow{E}\times\overrightarrow{u}]).$
(2.9)

##  3  Neutrino spin oscillations in noncommutative higher dimensional
Schwarzschild metric

In this section, by solving Einstein’s equations with $\rho_{\theta}(r)$ as a
source of matter and using the natural units $\hbar=c=1$, we study stationary,
spherically symmetric, asymptotically flat metrics in $(d+1)$ dimensions that
is described by the most general standard form of the space-time element:

$ds^{2}=A^{2}(r)dt^{2}-A^{-2}(r)dr^{2}-r^{2}d\Omega^{2}_{d-1},$ (3.1)

where d is the dimension of space and $d\Omega^{2}_{d-1}$ denotes the
differential solid angle in d dimensions . For the Schwarzschild black hole
with mass $M$ in three dimensional space ($d=3$) we have:

$ds^{2}=A^{2}(r)dt^{2}-A^{-2}(r)dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (3.2)

with :

$A(r)=1-\frac{2MG}{r}$ (3.3)

where $M=\frac{r_{g}}{2}$ and $r_{g}$ is the gravitational radius. the tt
component of this metric in the non-commutative space is defined as follows
[12]:

$A(r)=\sqrt{1-\frac{4GM}{\pi^{\frac{1}{2}}r}\gamma(\frac{3}{2},\frac{r^{2}}{4\theta})}$
(3.4)

$G$ is the Newton’s constant with dimension $L^{d-1}$ and we define the
dimensionless parameter $\eta$ as:

$\eta=\frac{M}{\sqrt{\theta}}$ (3.5)

By setting $Q=0$ in the time component of the RN metric in NC higher
dimensions in the Ref. 26, the time component of the Schwarzschild metric is
obtained, so for higher dimensions, the above metric is given as follows:

$A(r)=\sqrt{1-\frac{4GM}{\pi^{\frac{d-2}{2}}r^{d-2}}\gamma(\frac{d}{2},\frac{r^{2}}{4\theta})}$
(3.6)

.

where
$\gamma(\frac{d}{2},\frac{r^{2}}{4\theta})=\int^{\frac{r^{2}}{4\theta}}_{0}t^{\frac{d}{2}-1}e^{-t}dt$.
We have plotted $A(r)$ component of the Schwarzschild metric in three
dimensions as a function of x for different values of $\eta$ in Fig. 1. The
location of the even horizon is calculated by the equation $A(r)=0$. The
existence of two Intersections points with the horizontal axis indicates the
formation of a black hole with two event horizons and presence of one
intersection shows a degenerate horizon.

Figure 1: The $A(r)$ component of the Schwarzschild metric vs.
$x=\frac{r}{\sqrt{\theta}}$ for different values of $\eta$ in 3 dimensions.

According to Eq.(3.5) $\eta$ is inversely proportional to the noncommutative
parameter $\theta$. In NC space for $\eta=\frac{M}{\sqrt{\theta}}>1.9$ there
are two distinct horizons shown by dash in Fig .1 and for
$\eta=\frac{M}{\sqrt{\theta}}=1.9$ there is a degenerate horizon shown by dot
. The case $\eta\longrightarrow\infty$ shows the metric in commutative space.
As observed from the figure, by increasing the noncommutative parameter
$\theta$ (derasing $\eta$), the metric becomes flat in shorter distances which
indicates that, the non-commutativity of space has an opposite role to the
curvature of space.

Vaierbein Components of vector $u^{a}$ is defined as follows:

$u^{a}=(\alpha A,U_{r}A^{-1},U_{\theta}r,U_{\phi}rSin\theta),$ (3.7)

Since the schwarzschild metric s spherically symmetric, we can assume that the
neutrino moves in the equatorial plane $\theta=\frac{\pi}{2}$, i.e.
$d\theta=0$. The magnetic field is given by
$\vec{B}=(U_{\phi}Cos\theta,-U_{\phi}ASin\theta,U_{\theta}A)$ and for the
orbits that lie in the equatorial plane we have $\nu_{\theta}=0$ and
$U_{\theta}=0$ . $\Omega_{1}$ and $\Omega_{3}$ depend on $\cos\theta$ and
$\nu_{\theta}$ so $\Omega_{1}=\Omega_{3}=0$. The four-vector vaierbein, the
electric-gravity and magneto-gravity fields take the following forms
respectively[28, 29]:

$u^{a}=(\alpha A,0,0,\alpha\nu_{\varphi}r)$ (3.8)

,

$E=(E_{0},0,0),B=(0,-\nu_{\varphi}A\alpha,0),$ (3.9)

$E_{0}$ , $\nu_{\varphi}$ and $\alpha^{-1}$ are defined as follows:[32]

$v_{\varphi}=\frac{d\varphi}{dt}=\sqrt{\frac{F(r)}{2r}},$ (3.10)

$E_{0}=\frac{1}{A^{2}(r)}(1-\frac{rF(r)}{2A^{2}(r)}),$ (3.11)

$\alpha^{-1}=\frac{d\tau}{dt}=\sqrt{A^{2}(r)-\frac{1}{2}rF(r)},$ (3.12)

where $F(r)=\frac{dA^{2}(r)}{dr}$.
We can obtain $\Omega_{2}$ which is the only nonzero component of the neutrino
spin oscillations frequency in Schwarzschild metric by inserting time metric
component A(r) in Eqs. (3.8) to (3.12), then by substituting Eq. (2.9) in
$\overrightarrow{\Omega}=\frac{\overrightarrow{G}}{\alpha}$, we have :

$\Omega_{2}=-\frac{\nu_{\varphi}}{2}\alpha^{-1}$ (3.13)

Figure 2: Neutrino spin oscillations frequency for Schwarzschild metric vs.
$x$ for different values of $\eta$ in 3 dimension.

We have plotted $2|\Omega_{2}|M$ versus $x$ in Fig. 2 in three dimensions for
the Schwarzschild metric with different values of $\eta$. By reducing the
value of $\eta$ which corresponds to increasing the noncommutative parameter,
The peak of the curve occurs in shorter distances and it is observed that
there is no big difference in the maximum of the spin oscillation frequency
with the increasing of the noncommutative effects. It is also seen that, as
$\eta$ decreases, the spin oscillation tends to zero faster. This is in
agreement with the result we got before: the non-commutativity of space has an
opposite role to the curvature of space.

Figure 3: Neutrino spin oscillations frequency in Schwarzschild metric vs. $x$
for different values $\eta$ in 5 dimension.

In Fig. 3, we have shown the oscillation frequency of neutrino spin for the
Schwarzschild metric in five dimensions with different $\eta$ and we observe
that, increasing the non-commutativity of space in higher dimensions reduces
the peak of the spin oscillation frequency. It can also be checked that, by
increasing the non-commutativity of space, the spin oscillations tend to zero,
faster than commutative space which confirms the result we got earlier for the
case of three dimensions.

Figure 4: Neutrino spin oscillations frequency in Schwarzschild metric vs. $x$
for different values of $\eta$ in 7 dimensions.

In Fig. 4, it can be seen that the frequency peak of the neutrino spin
oscillation is decreased further with increasing the noncommutative effects
and by comparing Figs. 3,4, we observe that the reduce of the peak related to
the increasing of the noncommutative effects is bigger in seven dimension than
five dimension. This mean that by increasing the dimension of space, the
effects of non-commutativity of space will become more obvious. In addition,
in seven dimension compared to five dimension, the peaks of the curves occur
at shorter distances. Furthermore, by increasing the noncommutativity of
space, the ampitude of oscillations tends to zero more rapidly in agreement
with the situation in three and five dimensions.

Figure 5: Neutrino spin oscillations frequency in Schwarzschild(Sh) metric in
9 dimensions  Figure 6: Neutrino spin oscillations frequency in
Schwarzschild(Sh) metric in 11 dimensions

In Figs 5 and 6, frequencies of neutrino spin oscillations vs. $x$ in nine and
eleven dimensions are plotted respectively. Similar to other dimensions, we
observe that, the reduction of $\eta$ or equivalently increasing the
noncommutative parameter leads to the reduction of the maximum value of the
oscillation frequency. It is also observe that, by increasing the
noncommutative effects (decreasing $\eta$), the spin oscillations tend to zero
faster. So we conclude that, in each dimension, the noncommutativity of space
plays an opposite character to the curvature of space.

Figure 7: Neutrino spin oscillations frequency in Schwarzschild(Sh) metric in
13 dimensions

##  4 Neutrino spin oscillation in noncommutative higher dimensional Reissner-
Nordström(RN) Metric

The Reissner-Nordström metric describes a black hole with charge Q and mass M.
The time component of its metric in the three dimensional non-commutative
space is given by[13, 15] :

$A(r)=(1-\frac{4GM}{r\sqrt{\pi}}\gamma(\frac{3}{2},\frac{r^{2}}{4\theta})+\frac{Q^{2}}{\pi
r^{2}}\gamma^{2}(\frac{1}{2},\frac{r^{2}}{4\theta})-\frac{Q^{2}}{\pi
r\sqrt{2\theta}}\gamma(\frac{1}{2},\frac{r^{2}}{2\theta}))^{\frac{1}{2}},$
(4.1)

and in higher dimensions [20, 22] :

$A(r)=(1-\frac{4GM}{r^{d-2}\pi^{\frac{d-2}{2}}}\gamma(\frac{d}{2},\frac{r^{2}}{4\theta})+\frac{(d-2)Q^{2}G}{\pi^{d-2}r^{2d-4}}(\gamma^{2}(\frac{d}{2}-1,\frac{r^{2}}{4\theta})$
(4.2)
$-\frac{2^{\frac{8-3d}{2}}r^{d-2}}{(d-2)\theta^{\frac{d-2}{2}}}\gamma(\frac{d}{2}-1,\frac{r^{2}}{2\theta})))^{\frac{1}{2}},$

By inserting Eq. (4.1) and Eq. 4.2 in spin Oscillation equations introduced in
the previous section , i.e. Eqs. (3.8),(3.9),(3.10),(3.11),(3.12) and by using
Eq. (2.9), we get $\Omega_{2}$ in three and extra dimensions.

Fig. 8 shows the changes of the RN metric in three dimensions versus x for
three values of $\eta$.

$\eta\longrightarrow\infty$ represents the case of commutative space. The
intersections points with the x axis indicate the formation of a black hole
and the existence of event horizons. The metric approaches the metric of the
flat space in long distances, and we notice that this process happens faster
in the NC space than the commutative space.

Figure 8: The metric Rn vs. $x$ for different values of $\eta$ in 3 dimension.

In Figures (9)-(11), $2|\Omega_{2}|M$ in three, five and seven dimensional RN
metric are represented as functions of x for different values of $\eta$
respectively. The maximum values (peaks) reduce by decreasing $\eta$ and the
peaks of the frequencies take place in smaller x as the value of NC parameter
increase. As we get away from the horizon the values of frequencies go faster
to zero by increasing noncommutative effects (parameter), which means that, in
long distances , the dominate contribution belongs to commutative space, but
in short distance(small x) in higher dimensions , the contributions of
noncommutative space become stronger.

Figure 9: $2|\Omega_{2}|M$ in RN metric vs. $x$ for different values of $\eta$
in 3 dimension. Figure 10: $2|\Omega_{2}|M$ in RN metric vs. $x$ for different
values of $\eta$ in 5 dimension. Figure 11: $2|\Omega_{2}|M$ in Rn metric vs.
$x$ for different values of $\eta$ in 7 dimension. Figure 12: Neutrino spin
oscillations frequency in RN metric for 9 dimension($d$)  Figure 13: Neutrino
spin oscillations frequency in RN metric for 11 dimension($d$)

we have also plotted $2|\Omega_{2}|M$ in nine, eleven and thirteen dimensions
in Figures (12), (13) and (14) respectively. It can be seen that the peaks
decrease as the noncommutative parameter increases. By comparing the curves,
it is observed that the difference between neutrino spin oscillations
frequencies in non-commutative spaces compared to commutative cases, become
smaller.

Figure 14: Neutrino spin oscillations frequency in RN metric for 13
dimension($d$)

Now, let us study the effects of blak hole charge on the neutrino spin
oscillations in nonommutative higher dimensional spaces. To proceed further,
we define a new variable $\beta=\frac{Q}{\sqrt{\theta}}$. Figs .15, shows that
in all dimensions, black hole charge has a negative effects on the spin
oscillations i.e., by increasing the charge of the black hole in all
dimensions, the amplitude of spin oscillations decreases. It also shows that
in noncommutative spaces, by increasing the dimension of space, the effects of
black hole charge decreases which is the same as commutative spaces [27].

Figure 15: The effects of the BH charge on neutrino spin oscillation frequency
for RN metric vs. $x$ for certain value of $\eta$ in different dimensions.

Figs .16 and 17, indicates that in an arbitrary dimension in noncommutative
case, by increasing the noncommutativity of space, the black hole charge shows
itself more.

Figure 16: The effects of the BH charge on neutrino spin oscillation frequency
in RN metric vs. $x$ for different values of $\eta$ in 5 dimension Figure 17:
The effects of the BH charge on neutrino spin oscillation frequency in RN
metric vs. $x$ for different values of $\eta$ in 7 dimension

In Fig. 18, $2|\Omega_{2}|M$ is shown for two values of $\eta=500$ and
$\eta=1000$ in 5 dimensions and in Fig. 19 for $\eta=3000$ and $\eta=3500$ in
7 dimension. By increasing the noncommutative effects which occur in smaller
$\eta$, and by increasing the charge of the black hole, we can see a decrease
in the peak of the spin oscillations curve in both figures 18 and 19, although
we see a smaller difference in 7 dimensions (Fig. 19).

Figure 18: The effects of the black hole charge on neutrino spin oscillation
frequency for RN metric vs. $x$ in five dimensions and for different values of
$\eta$ and $\beta$ in 5 dimension  Figure 19: The effects of the black hole
charge on neutrino spin oscillation frequency for RN metric vs. $x$ with
different values of $\eta$ and $\beta$ in 7 dimension

Finally we can impose some upper bounds on noncommutativity parameter
$\theta$. Using the fact that if the metric curve does not touch the x-axis
for some ranges of $\eta$ values, so no black hole forms in this range, it is
possible to find an upper bound on $\sqrt{\theta}$, which has an inverse
relation with $\eta$. Upper bound on noncommutative parameter have shown in
Table. 1 for higher dimensions in RN and Schwarzschild metrics respectively.

Table 1: Upper bound on $\sqrt{\theta}$ in RN and Schwarzschild(Sh) metrics
for different dimensions($d$).

As it is observed, more stringent bound on $theta$ are obtained in higher
dimensions. It is worth mentioning that a lower bound for NC parameter
$\theta$ has been obtained in some papers through the study of black holes
thermodynamics in NC spaces [12, 13, 14]. it is also interesting to note that
an upper bound on the dynamical noncommutative parameter has been obtained in
Ref.34.

##  5 Conclusion

In this paper, we have investigated neutrino spin oscillations in
gravitational fields in noncommutative higher dimensional spaces. We have
analyzed the dependence of the oscillation frequency on the orbital radius for
both Schwarzschild and RN metrics. It is shown that in both metrics the
maximum of the frequencies decraese by increasing the noncommutativity of
space and the peaks occur in smaller distances compared to the three
dimensional case. It is also observed that for long distances the main
contribution to the neutrino spin transition probability belongs to three
dimension. We have also studied the effects of the black hole charge on the
frequency of spin oscillations in noncommutative higher dimensions. It is
shown that in all dimensions, by Increasing the noncommutative effects the
role of black hole charge on spin oscillations increases.
a). In any arbitrary dimension, the peak of the spin oscillations freuency
decreases with increasing noncommutativity of space.
b). In each dimension, by increasing the noncommutativity of space, the peak
is shifted towards smaller r.
c). In each dimension, as the noncommutativity of space increases, the
oscillations tend to zero faster.
As an important result, in every dimension, the non-commutativity of space
plays a role contrary to the curvature of space and reduces gravity.

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# On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of
several variables in a complete non-Archimedean valued field

Khodr Shamseddine  Department of Physics and Astronomy, University of
Manitoba, Winnipeg, Manitoba R3T 2N2, Canada<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{N}$ be a non-Archimedean ordered field extension of the real
numbers that is real closed and Cauchy complete in the topology induced by the
order, and whose Hahn group is Archimedean. In this paper, we first review the
properties of weakly locally uniformly differentiable (WLUD) functions, $k$
times weakly locally uniformly differentiable (WLUDk) functions, and WLUD∞
functions [1, 12] at a point or on an open subset of $\mathcal{N}$. Then we
show under what conditions a WLUD∞ function at a point $x_{0}\in\mathcal{N}$
is analytic in an interval around $x_{0}$, that is, it has a convergent Taylor
series at any point in that interval.

We generalize the concepts of WLUDk and WLUD∞ to functions from
$\mathcal{N}^{n}$ to $\mathcal{N}$, for any $n\in\mathbb{N}$. Then we
formulate conditions under which a WLUD∞ function at a point
$\boldsymbol{x_{0}}\in\mathcal{N}^{n}$ is analytic at that point.

###### Key words and phrases:

Taylor Series Expansion; Analytic Functions; non-Archimedean Analysis; non-
Archimedean Valued Fields

###### 2020 Mathematics Subject Classification:

41A58, 32P05, 12J25, 26E20, 46S10

This research was funded by the Natural Sciences and Engineering Council of
Canada (NSERC, Grant # RGPIN/4965-2017)

##  1\. Introduction

Let $\mathcal{N}$ be a non-Archimedean ordered field extension of $\mathbb{R}$
that is real closed and complete in the order topology and whose Hahn group
$S_{\mathcal{N}}$ is Archimedean, i.e. (isomorphic to) a subgroup of
$\mathbb{R}$. Recall that $S_{\mathcal{N}}$ is the set of equivalence classes
under the relation $\sim$ defined on
$\mathcal{N}^{*}:=\mathcal{N}\setminus\\{0\\}$ as follows: For
$x,y\in\mathcal{N}^{*}$, we say that $x$ is of the same order as $y$ and write
$x\sim y$ if there exist $n,m\in\mathbb{N}$ such that $n|x|>|y|$ and
$m|y|>|x|$, where $|\cdot|$ denotes the ordinary absolute value on
$\mathcal{N}$: $|x|=\max\left\\{x,-x\right\\}$. $S_{\mathcal{N}}$ is naturally
endowed with an addition via $[x]+[y]=[x\cdot y]$ and an order via $[x]<[y]$
if $|y|\ll|x|$ (which means $n|y|<|x|$ for all $n\in\mathbb{N}$), both of
which are readily checked to be well-defined. It follows that
$(S_{\mathcal{N}},+,<)$ is an ordered group, often referred to as the Hahn
group or skeleton group, whose neutral element is $[1]$, the class of $1$.

The theorem of Hahn [3] provides a complete classification of non-Archimedean
ordered field extensions of $\mathbb{R}$ in terms of their skeleton groups. In
fact, invoking the axiom of choice it is shown that the elements of our field
$\mathcal{N}$ can be written as (generalized) formal power series (also called
Hahn series) over its skeleton group $S_{\mathcal{N}}$ with real coefficients,
and the set of appearing exponents forms a well-ordered subset of
$S_{\mathcal{N}}$. That is, for all $x\in\mathcal{N}$, we have that
$x=\sum_{q\in S_{\mathcal{N}}}a_{q}d^{q}$; with $a_{q}\in\mathbb{R}$ for all
$q$, $d$ a positive infinitely small element of $\mathcal{N}$, and the support
of $x$, given by $\mbox{supp}(x):=\\{q\in S_{\mathcal{N}}:a_{q}\neq 0\\}$,
forming a well-ordered subset of $S_{\mathcal{N}}$.

We define for $x\neq 0$ in $\mathcal{N}$,
$\lambda(x)=\min\left(\mbox{supp}(x)\right)$, which exists since
$\mbox{supp}(x)$ is well-ordered. Moreover, we set $\lambda(0)=\infty$. Given
a nonzero $x=\sum_{q\in\mbox{supp}(x)}a_{q}d^{q}$, then $x>0$ if and only if
$a_{\lambda(x)}>0$.

The smallest such field $\mathcal{N}$ is the Levi-Civita field $\mathcal{R}$,
first introduced in [5, 6]. In this case $S_{\mathcal{R}}=\mathbb{Q}$, and for
any element $x\in\mathcal{R}$, supp$(x)$ is a left-finite subset of
$\mathbb{Q}$, i.e. below any rational bound $r$ there are only finitely many
exponents in the Hahn representation of $x$. The Levi-Civita field
$\mathcal{R}$ is of particular interest because of its practical usefulness.
Since the supports of the elements of $\mathcal{R}$ are left-finite, it is
possible to represent these numbers on a computer. Having infinitely small
numbers allows for many computational applications; one such application is
the computation of derivatives of real functions representable on a computer
[13, 15], where both the accuracy of formula manipulators and the speed of
classical numerical methods are achieved. For a review of the Levi-Civita
field $\mathcal{R}$, see [11] and references therein.

In the wider context of valuation theory, it is interesting to note that the
topology induced by the order on $\mathcal{N}$ is the same as the valuation
topology $\tau_{v}$ introduced via the ultrametric
$\Lambda:\mathcal{N}\times\mathcal{N}\rightarrow\mathbb{R}$, given by
$\Lambda(x,y)=\exp{(-\lambda(x-y))}$. It follows therefore that the field
$\mathcal{N}$ is just a special case of the class of fields discussed in [9].
For a general overview of the algebraic properties of formal power series
fields, we refer to the comprehensive overview by Ribenboim [8], and for an
overview of the related valuation theory the book by Krull [4]. A thorough and
complete treatment of ordered structures can also be found in [7]. A more
comprehensive survey of all non-Archimedean fields can be found in [2].

##  2\. Weak Local Uniform Differentiability and Review of Recent Results

Because of the total disconnectedness of the field $\mathcal{N}$ in the order
topology, the standard theorems of real calculus like the intermediate value
theorem, the inverse function theorem, the mean value theorem, the implicit
function theorem and Taylor’s theorem require stronger smoothness criteria of
the functions involved in order for the theorems to hold. In this section we
will present one such criterion: the so-called ‘weak local uniform
differentiability’, we will review recent work based on that smoothness
criterion and then present new results.

In [1], we focus our attention on $\mathcal{N}$-valued functions of one
variable. We study the properties of weakly locally uniformly differentiable
(WLUD) functions at a point $x_{0}\in\mathcal{N}$ or on an open subset $A$ of
$\mathcal{N}$. In particular, we show that WLUD functions are $C^{1}$, they
include all polynomial functions, and they are closed under addition,
multiplication and composition. Then we generalize the definition of weak
local uniform differentiability to any order. In particular, we study the
properties of WLUD2 functions at a point $x_{0}\in\mathcal{N}$ or on an open
subset $A$ of $\mathcal{N}$; and we show that WLUD2 functions are $C^{2}$,
they include all polynomial functions, and they are closed under addition,
multiplication and composition. Finally, we formulate and prove an inverse
function theorem as well as a local intermediate value theorem and a local
mean value theorem for these functions.

Here we only recall the main definitions and results (without proofs) in [1]
and refer the reader to that paper for the details.

######  Definition 1.

Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow\mathcal{N}$, and let
$x_{0}\in A$ be given. We say that $f$ is weakly locally uniformly
differentiable (abbreviated as WLUD) at $x_{0}$ if $f$ is differentiable in a
neighbourhood $\Omega$ of $x_{0}$ in $A$ and if for every $\epsilon>0$ in
$\mathcal{N}$ there exists $\delta>0$ in $\mathcal{N}$ such that
$(x_{0}-\delta,x_{0}+\delta)\subset\Omega$, and for every
$x,y\in(x_{0}-\delta,x_{0}+\delta)$ we have that $\left\lvert
f(y)-f(x)-f^{\prime}(x)(y-x)\right\rvert\leq\epsilon\left\lvert
y-x\right\rvert$. Moreover, we say that $f$ is WLUD on $A$ if $f$ is WLUD at
every point in $A$.

We extend the WLUD concept to higher orders of differentiability and we define
WLUDk as follows.

######  Definition 2.

Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow\mathcal{N}$, let
$x_{0}\in A$, and let $k\in\mathbb{N}$ be given. We say that $f$ is WLUDk at
$x_{0}$ if $f$ is $k$ times differentiable in a neighbourhood $\Omega$ of
$x_{0}$ in $A$ and if for every $\epsilon>0$ in $\mathcal{N}$ there exists
$\delta>0$ in $\mathcal{N}$ such that
$(x_{0}-\delta,x_{0}+\delta)\subset\Omega$, and for every
$x,y\in(x_{0}-\delta,x_{0}+\delta)$ we have that

$\left|f(y)-\sum\limits_{j=0}^{k}\frac{f^{(j)}(x)}{j!}(y-x)^{j}\right|\leq\epsilon\left|y-x\right|^{k}.$

Moreover, we say that $f$ is WLUDk on $A$ if $f$ is WLUDk at every point in
$A$. Finally, we say that $f$ is WLUD∞ at $x_{0}$ (respectively, on $A$) if
$f$ is WLUDk at $x_{0}$ (respectively, on $A$) for every $k\in\mathbb{N}$.

######  Theorem 1 (Inverse Function Theorem).

Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow\mathcal{N}$ be WLUD
on $A$, and let $x_{0}\in A$ be such that $f^{\prime}(x_{0})\neq 0$. Then
there exists a neighborhood $\Omega$ of $x_{0}$ in $A$ such that

  1. (1)

$\left.f\right|_{\Omega}$ is one-to-one;

  2. (2)

$f(\Omega)$ is open; and

  3. (3)

$f^{-1}$ exists and is WLUD on $f(\Omega)$ with
$(f^{-1})^{\prime}=1/\left(f^{\prime}\circ f^{-1}\right)$.

######  Theorem 2 (Local Intermediate Value Theorem).

Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow\mathcal{N}$ be WLUD
on $A$, and let $x_{0}\in A$ be such that $f^{\prime}(x_{0})\neq 0$. Then
there exists a neighborhood $\Omega$ of $x_{0}$ in $A$ such that for any $a<b$
in $f(\Omega)$ and for any $c\in(a,b)$, there is an
$x\in\left(\min\left\\{f^{(-1)}(a),f^{(-1)}(b)\right\\},\max\left\\{f^{(-1)}(a),f^{(-1)}(b)\right\\}\right)$
such that $f(x)=c$.

######  Theorem 3 (Local Mean Value Theorem).

Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow\mathcal{N}$ be WLUD2
on $A$, and let $x_{0}\in A$ be such that $f^{\prime\prime}(x_{0})\neq 0$.
Then there exists a neighborhood $\Omega$ of $x_{0}$ in $A$ such that $f$ has
the mean value property on $\Omega$. That is, for every $a,b\in\Omega$ with
$a<b$, there exists $c\in(a,b)$ such that

$f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}.$

As in the real case, the mean value property can be used to prove other
important results. In particular, while L’Hôpital’s rule does not hold for
differentiable functions on $\mathcal{N}$, we prove the result under similar
conditions to those of the local mean value theorem.

######  Theorem 4 (L’Hôpital’s Rule).

Let $A\subset\mathcal{N}$ be open, let $f,g:A\rightarrow\mathcal{N}$ be WLUD2
on $A$, and let $a\in A$ be such that $f^{\prime\prime}(a)\neq 0$ and
$g^{\prime\prime}(a)\neq 0$. Furthermore, suppose that $f(a)=g(a)=0$, that
there exists a neighborhood $\Omega$ of $a$ in $A$ such that
$g^{\prime}(x)\neq 0$ for every $x\in\Omega\setminus\\{a\\}$, and that
$\lim\limits_{x\rightarrow a}f^{\prime}(x)/g^{\prime}(x)$ exists. Then

$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow
a}\frac{f^{\prime}(x)}{g^{\prime}(x)}.$

In [12], we formulate and prove a Taylor theorem with remainder for WLUDk
functions from $\mathcal{N}$ to $\mathcal{N}$. Then we extend the concept of
WLUD to functions from $\mathcal{N}^{n}$ to $\mathcal{N}^{m}$ with
$m,n\in\mathbb{N}$ and study the properties of those functions as we did for
functions from $\mathcal{N}$ to $\mathcal{N}$. Then we formulate and prove the
inverse function theorem for WLUD functions from $\mathcal{N}^{n}$ to
$\mathcal{N}^{n}$ and the implicit function theorem for WLUD functions from
$\mathcal{N}^{n}$ to $\mathcal{N}^{m}$ with $m<n$ in $\mathbb{N}$.

As in the real case, the proof of Taylor’s theorem with remainder uses the
mean value theorem. However, in the non-Archimedean setting, stronger
conditions on the function are needed than in the real case for the
formulation of the theorem.

######  Theorem 5.

(Taylor’s Theorem with Remainder) Let $A\subseteq\mathcal{N}$ be open, let
$k\in\mathbb{N}$ be given, and let $f:A\rightarrow\mathcal{N}$ be WLUDk+2 on
$A$. Assume further that $f^{(j)}$ is WLUD2 on $A$ for $0\leq j\leq k$. Then,
for every $x\in A$, there exists a neighborhood $U$ of $x$ in $A$ such that,
for any $y\in U$, there exists $c\in\left[\min(y,x),\max(y,x)\right]$ such
that

(1)
$f(y)=\sum_{j=0}^{k}\frac{f^{(j)}\left(x\right)}{j!}\left(y-x\right)^{j}+\frac{f^{(k+1)}(c)}{(k+1)!}\left(y-x\right)^{k+1}.$

Before we define weak local uniform differentiability for functions from
$\mathcal{N}^{n}$ to $\mathcal{N}^{m}$ and then state the inverse function
theorem and the implicit function theorem, we introduce the following
notations.

######  Notation 1.

Let $A\subset\mathcal{N}^{n}$ be open, let $\boldsymbol{x_{0}}\in A$ be given,
and let $\boldsymbol{f}:A\rightarrow\mathcal{N}^{m}$ be such that all the
first order partial derivatives of $\boldsymbol{f}$ at $\boldsymbol{x_{0}}$
exist. Then $\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x_{0}})$ denotes the
linear map from $\mathcal{N}^{n}$ to $\mathcal{N}^{m}$ defined by the $m\times
n$ Jacobian matrix of $\boldsymbol{f}$ at $\boldsymbol{x_{0}}$:

$\begin{pmatrix}\boldsymbol{f}^{1}_{1}(\boldsymbol{x_{0}})&\boldsymbol{f}^{1}_{2}(\boldsymbol{x_{0}})&\ldots&\boldsymbol{f}^{1}_{n}(\boldsymbol{x_{0}})\\\
\boldsymbol{f}^{2}_{1}(\boldsymbol{x_{0}})&\boldsymbol{f}^{2}_{2}(\boldsymbol{x_{0}})&\ldots&\boldsymbol{f}^{2}_{n}(\boldsymbol{x_{0}})\\\
\vdots&\vdots&\ddots&\vdots\\\
\boldsymbol{f}^{m}_{1}(\boldsymbol{x_{0}})&\boldsymbol{f}^{m}_{2}(\boldsymbol{x_{0}})&\ldots&\boldsymbol{f}^{m}_{n}(\boldsymbol{x_{0}})\end{pmatrix}$

with $\boldsymbol{f}^{i}_{j}(\boldsymbol{x_{0}})=\frac{\partial
f_{i}}{\partial x_{j}}(\boldsymbol{x_{0}})$ for $1\leq i\leq m$ and $1\leq
j\leq n$. Moreover, if $m=n$ then the determinant of the $n\times n$ matrix
$\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x_{0}})$ is denoted by
$J\boldsymbol{f}(\boldsymbol{x_{0}})$.

######  Definition 3 (WLUD).

Let $A\subset\mathcal{N}^{n}$ be open, let
$\boldsymbol{f}:A\to\mathcal{N}^{m}$, and let $\boldsymbol{x_{0}}\in A$ be
given. Then we say that $\boldsymbol{f}$ is weakly locally uniformly
differentiable (WLUD) at $\boldsymbol{x_{0}}$ if $\boldsymbol{f}$ is
differentiable in a neighborhood $\Omega$ of $\boldsymbol{x_{0}}$ in $A$ and
if for every $\epsilon>0$ in $\mathcal{N}$ there exists $\delta>0$ in
$\mathcal{N}$ such that
$B_{\delta}(\boldsymbol{x_{0}}):=\left\\{\boldsymbol{t}\in\mathcal{N}:\left|\boldsymbol{t}-\boldsymbol{x_{0}}\right|<\delta\right\\}\subset\Omega$,
and for all $\boldsymbol{x},\boldsymbol{y}\in B_{\delta}(\boldsymbol{x_{0}})$
we have that

$\left|\boldsymbol{f}(\boldsymbol{y})-\boldsymbol{f}(\boldsymbol{x})-\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x})(\boldsymbol{y}-\boldsymbol{x})\right|\leq\epsilon|\boldsymbol{y}-\boldsymbol{x}|.$

Moreover, we say that $\boldsymbol{f}$ is WLUD on $A$ if $\boldsymbol{f}$ is
WLUD at every point in $A$.

We show in [12] that if $\boldsymbol{f}$ is WLUD at $\boldsymbol{x_{0}}$
(respectively on $A$) then $\boldsymbol{f}$ is C1 at $\boldsymbol{x_{0}}$
(respectively on $A$). Thus, the class of WLUD functions at a point
$\boldsymbol{x_{0}}$ (respectively on an open set $A$) is a subset of the
class of $C^{1}$ functions at $\boldsymbol{x_{0}}$ (respectively on $A$).
However, this is still large enough to include all polynomial functions. We
also show in [12] that if $\boldsymbol{f},\boldsymbol{g}$ are WLUD at
$\boldsymbol{x_{0}}$ (respectively on $A$) and if $\alpha\in\mathcal{N}$ then
$\boldsymbol{f}+\alpha\boldsymbol{g}$ and $\boldsymbol{f}\cdot\boldsymbol{g}$
are WLUD at $\boldsymbol{x_{0}}$ (respectively on $A$). Moreover, we show that
if $\boldsymbol{f}:A\to\mathcal{N}^{m}$ is WLUD at $\boldsymbol{x_{0}}\in A$
(respectively on $A$) and if $\boldsymbol{g}:C\to\mathcal{N}^{p}$ is WLUD at
$\boldsymbol{f}(\boldsymbol{x_{0}})\in C$ (respectively on $C$), where $A$ is
an open subset of $\mathcal{N}^{n}$, $C$ an open subset of $\mathcal{N}^{m}$
and $\boldsymbol{f}(A)\subseteq C$, then $\boldsymbol{g}\circ\boldsymbol{f}$
is WLUD at $\boldsymbol{x_{0}}$ (respectively on $A$).

######  Theorem 6 (Inverse Function Theorem).

Let $A\subset\mathcal{N}$ be open, let
$\boldsymbol{g}:A\rightarrow\mathcal{N}^{n}$ be WLUD on $A$ and let
$\boldsymbol{t_{0}}\in A$ be such that
$J\boldsymbol{g}(\boldsymbol{t_{0}})\neq 0$. Then there is a neighborhood
$\Omega$ of $\boldsymbol{t_{0}}$ such that:

  1. (1)

$\boldsymbol{g}|_{\Omega}$ is one-to-one;

  2. (2)

$\boldsymbol{g}(\Omega)$ is open;

  3. (3)

the inverse $\boldsymbol{f}$ of $\boldsymbol{g}|_{\Omega}$ is WLUD on
$\boldsymbol{g}(\Omega)$; and
$\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x})=\left[\boldsymbol{D}\boldsymbol{g}(\boldsymbol{t})\right]^{-1}$
for $\boldsymbol{t}\in\Omega$ and
$\boldsymbol{x}=\boldsymbol{g}(\boldsymbol{t})$.

As in the real case, the inverse function theorem is used to prove the
implicit function theorem. But before we state the implicit function theorem,
we introduce the following notations.

######  Notation 2.

Let $A\subseteq\mathcal{N}^{n}$ be open and let
$\boldsymbol{\Phi}:A\rightarrow\mathcal{N}^{m}$ be WLUD on $A$. For
$\boldsymbol{t}=(t_{1},...,t_{n-m},t_{n-m+1},...,t_{n})\in A$, let

$\hat{\boldsymbol{t}}=(t_{1},...,t_{n-m})\text{ and
}\tilde{J}\boldsymbol{\Phi}(\boldsymbol{t})=\det\left(\dfrac{\partial(\Phi_{1},...,\Phi_{m})}{\partial(t_{n-m+1},...,t_{n})}\right).$

######  Theorem 7 (Implicit Function Theorem).

Let $\boldsymbol{\Phi}:A\rightarrow\mathcal{N}^{m}$ be WLUD on $A$, where
$A\subseteq\mathcal{N}^{n}$ is open and $1\leq m<n.$ Let
$\boldsymbol{t_{0}}\in A$ be such that
$\boldsymbol{\Phi}(\boldsymbol{t_{0}})=\boldsymbol{0}$ and
$\tilde{J}\boldsymbol{\Phi}(\boldsymbol{t_{0}})\neq 0$. Then there exist a
neighborhood $U$ of $\boldsymbol{t_{0}}$, a neighborhood $R$ of
$\hat{\boldsymbol{t_{0}}}$ and $\boldsymbol{\phi}:R\rightarrow\mathcal{N}^{m}$
that is WLUD on $R$ such that

$\tilde{J}\boldsymbol{\Phi}(\boldsymbol{t})\neq 0\text{ for all
}\boldsymbol{t}\in U,$

and

$\\{\boldsymbol{t}\in
U:\boldsymbol{\Phi}(\boldsymbol{t})=\boldsymbol{0}\\}=\\{(\hat{\boldsymbol{t}},\boldsymbol{\phi}(\hat{\boldsymbol{t}})):\hat{\boldsymbol{t}}\in
R\\}.$

##  3\. New Results

This paper is a continuation of the work done in [1, 12]. In the following
section, we will generalize in Definition 5 and Definition 6 the concepts of
WLUDk and WLUD∞ to functions from $\mathcal{N}^{n}$ to $\mathcal{N}$; and we
will formulate (in Theorem 8 and Theorem 9 and their proofs) conditions under
which a WLUD∞ $\mathcal{N}$-valued function at a point $x_{0}\in\mathcal{N}$
or a WLUD∞ $\mathcal{N}$-valued function at a point
$\boldsymbol{x_{0}}\in\mathcal{N}^{n}$ will be analytic at that point.

######  Theorem 8.

Let $A\subseteq\mathcal{N}$ be open, let $x_{0}\in A$, and let
$f:A\rightarrow\mathcal{N}$ be WLUD∞ at $x_{0}$. For each $k\in\mathbb{N}$,
let $\delta_{k}>0$ in $\mathcal{N}$ correspond to $\epsilon=1$ in Definition
2. Assume that

$\limsup_{j\rightarrow\infty}\left(\frac{-\lambda\left(f^{(j)}(x_{0})\right)}{j}\right)<\infty\text{
and }\limsup_{k\rightarrow\infty}\lambda\left(\delta_{k}\right)<\infty.$

Then there exists a neighborhood $U$ of $x_{0}$ in $A$ such that, for any
$x,y\in U$, we have that

$f(y)=\sum_{j=0}^{\infty}\frac{f^{(j)}\left(x\right)}{j!}\left(y-x\right)^{j}.$

That is, the Taylor series
$\sum\limits_{j=0}^{\infty}\frac{f^{(j)}\left(x\right)}{j!}\left(y-x\right)^{j}$
converges in $\mathcal{N}$ to $f(y)$; and hence $f$ is analytic in $U$.

###### Proof.

Let

$\lambda_{0}=\limsup_{j\rightarrow\infty}\left(\frac{-\lambda\left(f^{(j)}(x_{0})\right)}{j}\right).$

Then $\lambda_{0}\in\mathbb{R}$ and $\lambda_{0}<\infty$; and, by [9, Page
59], we have that
$\sum\limits_{j=0}^{\infty}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}$
converges in $\mathcal{N}$ for all $x\in\mathcal{N}$ satisfying
$\lambda(x-x_{0})>\lambda_{0}$.

For all $k\in\mathbb{N}$, we have that
$(x_{0}-\delta_{k},x_{0}+\delta_{k})\subset A$, $f$ is $k$ times
differentiable on $(x_{0}-\delta_{k},x_{0}+\delta_{k})$, and

$\left|f(x)-\sum\limits_{j=0}^{k}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}\right|\leq\left|x-x_{0}\right|^{k}\text{
for all }x\in(x_{0}-\delta_{k},x_{0}+\delta_{k}).$

Since
$\limsup\limits_{k\rightarrow\infty}\lambda\left(\delta_{k}\right)<\infty$,
there exists $t>0$ in $\mathbb{Q}$ such that
$\limsup\limits_{k\rightarrow\infty}\lambda\left(\delta_{k}\right)<t<\infty$.
Thus, there exists $N\in\mathbb{N}$ such that

(2) $\lambda(\delta_{k})<t\text{ for all }k>N.$

Let $\delta>0$ in $\mathcal{N}$ be such that
$\lambda(\delta)>\max\\{\lambda_{0},t,0\\}$; this is possible since
$\max\\{\lambda_{0},t,0\\}<\infty$. It follows from (2) that
$\lambda(\delta)>\lambda(\delta_{k})$ and hence $0<\delta\ll\delta_{k}$ for
all $k>N$. Thus, $(x_{0}-\delta,x_{0}+\delta)\subset A$, $f$ is infinitely
often differentiable on $(x_{0}-\delta,x_{0}+\delta)$, and

(3)
$\left|f(x)-\sum\limits_{j=0}^{k}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}\right|\leq\left|x-x_{0}\right|^{k}\forall\
x\in(x_{0}-\delta,x_{0}+\delta)\text{ and }\forall\ k>N.$

Moreover, for all $x\in(x_{0}-\delta,x_{0}+\delta)$, we have that
$\lambda(x-x_{0})\geq\lambda(\delta)>\lambda_{0}$ and hence
$\sum\limits_{j=0}^{\infty}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}$
converges in $\mathcal{N}$. Let $U=(x_{0}-\delta,x_{0}+\delta)$.

First we show that

$f(x)=\sum_{j=0}^{\infty}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}\text{
for all }x\in U.$

Let $x\in U$ be given. Taking the limit in (3) as $k\rightarrow\infty$, we
get:

$0\leq\lim_{k\rightarrow\infty}\left|f(x)-\sum\limits_{j=0}^{k}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}\right|\leq\lim_{k\rightarrow\infty}\left|x-x_{0}\right|^{k},$

from which we obtain

$0\leq\left|f(x)-\lim_{k\rightarrow\infty}\sum\limits_{j=0}^{k}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}\right|\leq\lim_{k\rightarrow\infty}\left|x-x_{0}\right|^{k}.$

Since $\lambda(x-x_{0})\geq\lambda(\delta)>0$, we obtain that
$\lim\limits_{k\rightarrow\infty}\left|x-x_{0}\right|^{k}=0$. It follows that

$0\leq\left|f(x)-\sum\limits_{j=0}^{\infty}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}\right|\leq
0$

from which we infer that
$f(x)=\sum\limits_{j=0}^{\infty}\frac{f^{(j)}\left(x_{0}\right)}{j!}\left(x-x_{0}\right)^{j}$
or, equivalently,

(4)
$f(x)=\sum\limits_{l=0}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{l!}\left(x-x_{0}\right)^{l}.$

Since the convergence of the Taylor series above is in the order (valuation)
topology, we will show that the derivatives of $f$ at $x$ to any order are
obtained by differentiating the power series in Equation (4) term by term.
That is, for all $j\in\mathbb{N}$,

(5)
$f^{(j)}(x)=\sum_{l=j}^{\infty}l(l-1)\ldots(l-j+1)\frac{f^{(l)}\left(x_{0}\right)}{l!}\left(x-x_{0}\right)^{l-j}.$

First note that, since $\lambda\left(l(l-1)\ldots(l-j+1)\right)=0$, it follows
that
$\sum_{l=j}^{\infty}l(l-1)\ldots(l-j+1)\frac{f^{(l)}\left(x_{0}\right)}{l!}\left(x-x_{0}\right)^{l-j}$
converges in $\mathcal{N}$ for all $j\in\mathbb{N}$. Using induction on $j$,
it suffices to show that

$f^{\prime}(x)=\sum_{l=1}^{\infty}l\frac{f^{(l)}\left(x_{0}\right)}{l!}\left(x-x_{0}\right)^{l-1}=\sum_{l=1}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{(l-1)!}\left(x-x_{0}\right)^{l-1}.$

Let $h\in\mathcal{N}$ be such that $x+h\in U$. We will show that

$\lim_{h\rightarrow
0}\left\\{\frac{f(x+h)-f(x)}{h}\right\\}=\sum_{l=1}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{(l-1)!}\left(x-x_{0}\right)^{l-1}.$

Thus,

$\displaystyle\lim_{h\rightarrow
0}\left\\{\frac{f(x+h)-f(x)}{h}\right\\}=\lim_{h\rightarrow
0}\left\\{\sum\limits_{l=0}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{l!}\frac{\left(x+h-x_{0}\right)^{l}-\left(x-x_{0}\right)^{l}}{h}\right\\}$
$\displaystyle=$ $\displaystyle\lim_{h\rightarrow
0}\left\\{\sum\limits_{l=1}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{l!}\frac{\left(x+h-x_{0}\right)^{l}-\left(x-x_{0}\right)^{l}}{h}\right\\}$
$\displaystyle=$ $\displaystyle\lim_{h\rightarrow
0}\left\\{\sum\limits_{l=1}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{l!}\left[(x+h-x_{0})^{l-1}+(x+h-x_{0})^{l-2}(x-x_{0})+\cdots+(x-x_{0})^{l-1}\right]\right\\}$
$\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{l!}\left[l(x-x_{0})^{l-1}\right]$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{(l-1)!}\left(x-x_{0}\right)^{l-1}.$

Now let $y\in U$ be given. Then

$\displaystyle f(y)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{l!}\left(y-x_{0}\right)^{l}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\frac{f^{(l)}\left(x_{0}\right)}{l!}\left[(y-x)+(x-x_{0})\right]^{l}$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{j=0}^{l}\frac{f^{(l)}\left(x_{0}\right)}{l!}\left(\begin{array}[]{c}l\\\
j\end{array}\right)(y-x)^{j}(x-x_{0})^{l-j}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{j=0}^{l}\frac{l(l-1)\ldots(l-j+1)}{j!}\frac{f^{(l)}\left(x_{0}\right)}{l!}(x-x_{0})^{l-j}(y-x)^{j}.$

Since convergence in the order topology (valuation topology) entails absolute
convergence, we can interchange the order of the summations in the last
equality [10, 14]. We get:

$\displaystyle f(y)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{1}{j!}\left[\sum_{l=j}^{\infty}l(l-1)\ldots(l-j+1)\frac{f^{(l)}\left(x_{0}\right)}{l!}(x-x_{0})^{l-j}\right](y-x)^{j}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{f^{(j)}(x)}{j!}(y-x)^{j}$

where we made use of Equation (5) in the last step. ∎

Replacing $m$ by $1$ in Definition 3, then the $m\times n$ matrix
$\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x})$ is replaced by the gradient of
$\boldsymbol{f}$ at $\boldsymbol{x}$: $\boldsymbol{\nabla}f(\boldsymbol{x})$,
and we readily obtain the definition of a WLUD $\mathcal{N}$-valued function
at a point $\boldsymbol{x_{0}}$ or on an open subset $A$ of $\mathcal{N}^{n}$.

######  Definition 4.

Let $A\subset\mathcal{N}^{n}$ be open, let $f:A\to\mathcal{N}$, and let
$\boldsymbol{x_{0}}\in A$ be given. Then we say that $f$ is WLUD at
$\boldsymbol{x_{0}}$ if $f$ is differentiable in a neighborhood $\Omega$ of
$\boldsymbol{x_{0}}$ in $A$ and if for every $\epsilon>0$ in $\mathcal{N}$
there exists $\delta>0$ in $\mathcal{N}$ such that
$B_{\delta}(\boldsymbol{x_{0}})\subset\Omega$, and for all
$\boldsymbol{x},\boldsymbol{y}\in B_{\delta}(\boldsymbol{x_{0}})$ we have that

$\left|f(\boldsymbol{y})-f(\boldsymbol{x})-\boldsymbol{\nabla}f(\boldsymbol{x})\cdot(\boldsymbol{y}-\boldsymbol{x})\right|\leq\epsilon|\boldsymbol{y}-\boldsymbol{x}|.$

Moreover, we say that $f$ is WLUD on $A$ if $f$ is WLUD at every point in $A$.

Using Defintion 2 and Definition 4, the natural way to define $k$ times weak
local uniform differentiability (WLUDk) at a point $\boldsymbol{x_{0}}$ or on
an open subset $A$ of $\mathcal{N}^{n}$ is as follows.

######  Definition 5.

Let $A\subset\mathcal{N}^{n}$ be open, let $f:A\to\mathcal{N}$, and let
$\boldsymbol{x_{0}}\in A$ be given. Then we say that $f$ is WLUDk at
$\boldsymbol{x_{0}}$ if $f$ is $k$-times differentiable in a neighborhood
$\Omega$ of $\boldsymbol{x_{0}}$ in $A$ and if for every $\epsilon>0$ in
$\mathcal{N}$ there exists $\delta>0$ in $\mathcal{N}$ such that
$B_{\delta}(\boldsymbol{x_{0}})\subset\Omega$, and for all
$\boldsymbol{\xi},\boldsymbol{\eta}\in B_{\delta}(\boldsymbol{x_{0}})$ we have
that

$\left|f(\boldsymbol{\eta})-f(\boldsymbol{\xi})-\sum_{j=1}^{k}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{\xi})\cdot\nabla\right]^{j}f(\boldsymbol{\xi})\right|\leq\epsilon|\boldsymbol{\eta}-\boldsymbol{\xi}|^{k},$

where

$\displaystyle\left[(\boldsymbol{\eta}-\boldsymbol{\xi})\cdot\nabla\right]^{j}f(\boldsymbol{\xi})$
$\displaystyle=$
$\displaystyle\left.\left[(\eta_{1}-\xi_{1})\frac{\partial}{\partial
x_{1}}+\cdots+(\eta_{n}-\xi_{n})\frac{\partial}{\partial
x_{n}}\right]^{j}f(\boldsymbol{x})\right|_{\boldsymbol{x}=\boldsymbol{\xi}}$
$\displaystyle=$
$\displaystyle\sum_{l_{1},\ldots,l_{j}=1}^{n}\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{\xi}}\prod_{m=1}^{j}\left(\eta_{l_{m}}-\xi_{l_{m}}\right)\right).$

Moreover, we say that $f$ is WLUDk on $A$ if $f$ is WLUDk at every point in
$A$.

######  Definition 6.

Let $A\subset\mathcal{N}^{n}$ be open, let $f:A\to\mathcal{N}$, and let
$\boldsymbol{x_{0}}\in A$ be given. Then we say that $f$ is WLUD∞ at
$\boldsymbol{x_{0}}$ if $f$ is WLUDk at $\boldsymbol{x_{0}}$ for every
$k\in\mathbb{N}$. Moreover, we say that $f$ is WLUD∞ on $A$ if $f$ is WLUD∞ at
every point in $A$.

Now we are ready to state and prove the analog of Theorem 8 for functions of
$n$ variables.

######  Theorem 9.

Let $A\subseteq\mathcal{N}^{n}$ be open, let $\boldsymbol{x_{0}}\in A$, and
let $f:A\rightarrow\mathcal{N}$ be WLUD∞ at $\boldsymbol{x_{0}}$. For each
$k\in\mathbb{N}$, let $\delta_{k}>0$ in $\mathcal{N}$ correspond to
$\epsilon=1$ in Definition 5. Assume that

$\displaystyle\limsup_{{\tiny\begin{array}[]{l}j\rightarrow\infty\\\
l_{1}=1,\ldots,n\\\ \vdots\\\
l_{j}=1,\ldots,n\end{array}}}\left(\frac{-\lambda\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_{0}}}\right)}{j}\right)$
$\displaystyle<$ $\displaystyle\infty$ $\displaystyle\text{ and
}\limsup_{k\rightarrow\infty}\lambda\left(\delta_{k}\right)$ $\displaystyle<$
$\displaystyle\infty.$

Then there exists a neighborhood $U$ of $\boldsymbol{x_{0}}$ in $A$ such that,
for any $\boldsymbol{\eta}\in U$, we have that

$f(\boldsymbol{\eta})=f(\boldsymbol{x_{0}})+\sum_{j=1}^{\infty}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}}).$

###### Proof.

Let

$\lambda_{0}=\limsup_{{\tiny\begin{array}[]{l}j\rightarrow\infty\\\
l_{1}=1,\ldots,n\\\ \vdots\\\
l_{j}=1,\ldots,n\end{array}}}\left(\frac{-\lambda\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_{0}}}\right)}{j}\right).$

Then $\lambda_{0}\in\mathbb{R}$ and $\lambda_{0}<\infty$.

For all $k\in\mathbb{N}$, we have that
$B_{\delta_{k}}(\boldsymbol{x_{0}})\subset A$, $f$ is $k$ times differentiable
on $B_{\delta_{k}}(\boldsymbol{x_{0}})$, and

$\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_{0}})-\sum_{j=1}^{k}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right|\leq|\boldsymbol{\eta}-\boldsymbol{x_{0}}|^{k}\text{
for all }\boldsymbol{\eta}\in B_{\delta_{k}}(\boldsymbol{x_{0}}).$

Since
$\limsup\limits_{k\rightarrow\infty}\lambda\left(\delta_{k}\right)<\infty$,
there exists $t>0$ in $\mathbb{Q}$ such that
$\limsup\limits_{k\rightarrow\infty}\lambda\left(\delta_{k}\right)<t<\infty$.
Thus, there exists $N\in\mathbb{N}$ such that

(8) $\lambda(\delta_{k})<t\text{ for all }k>N.$

Let $\delta>0$ in $\mathcal{N}$ be such that
$\lambda(\delta)>\max\\{\lambda_{0},t,0\\}$. It follows from (8) that
$\lambda(\delta)>\lambda(\delta_{k})$ and hence $0<\delta\ll\delta_{k}$ for
all $k>N$. Thus, $B_{\delta}(\boldsymbol{x_{0}})\subset A$, $f$ is infinitely
often differentiable on $B_{\delta}(\boldsymbol{x_{0}})$, and

(9)
$\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_{0}})-\sum_{j=1}^{k}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right|\leq|\boldsymbol{\eta}-\boldsymbol{x_{0}}|^{k}\
\forall\boldsymbol{\eta}\in B_{\delta}(\boldsymbol{x_{0}})\text{ and }\forall
k>N.$

Let $U=B_{\delta}(\boldsymbol{x_{0}})$; and let $\boldsymbol{\eta}\in U$ be
given. Then we have that
$\lambda(|\boldsymbol{\eta}-\boldsymbol{x_{0}}|)\geq\lambda(\delta)>\lambda_{0}$.
We will show first that
$\sum_{j=1}^{\infty}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})$
converges in $\mathcal{N}$. Since
$\lambda(|\boldsymbol{\eta}-\boldsymbol{x_{0}}|)>\lambda_{0}$, there exists
$q>0$ in $\mathbb{Q}$ such that
$\lambda(|\boldsymbol{\eta}-\boldsymbol{x_{0}}|)-q>\lambda_{0}$. Hence there
exists $M\in\mathbb{N}$ such that

$\lambda(|\boldsymbol{\eta}-\boldsymbol{x_{0}}|)-q>\frac{-\lambda\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_{0}}}\right)}{j}$

for all $j>M$ and for $l_{1}=1,\ldots,n$, $l_{2}=1,\ldots,n$, …,
$l_{j}=1,\ldots,n$. It follows that

$\displaystyle\lambda\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_{0}}}\prod_{m=1}^{j}\left(\eta_{l_{m}}-x_{0,l_{m}}\right)\right)$
$\displaystyle\geq$
$\displaystyle\lambda\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_{0}}}|\boldsymbol{\eta}-\boldsymbol{x_{0}}|^{j}\right)$
$\displaystyle=$
$\displaystyle\lambda\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_{0}}}\right)+j\lambda\left(|\boldsymbol{\eta}-\boldsymbol{x_{0}}|\right)$
$\displaystyle>$ $\displaystyle jq$

for all $j>M$ and for $l_{1}=1,\ldots,n$, $l_{2}=1,\ldots,n$, …,
$l_{j}=1,\ldots,n$. Thus,

$\displaystyle\lambda\left(\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right)$
$\displaystyle=$
$\displaystyle\lambda\left(\sum_{l_{1},\ldots,l_{j}=1}^{n}\left(\left.\frac{\partial^{j}f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_{0}}}\prod_{m=1}^{j}\left(\eta_{l_{m}}-x_{0,l_{m}}\right)\right)\right)$
$\displaystyle>$ $\displaystyle jq$

for all $j>M$; and hence

$\displaystyle\lim_{j\rightarrow\infty}\lambda\left(\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right)$
$\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}\lambda\left(\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right)$
$\displaystyle\geq$ $\displaystyle q\lim_{j\rightarrow\infty}j=\infty.$

Thus,

$\lim_{j\rightarrow\infty}\left(\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right)=0$

and hence
$\sum_{j=1}^{\infty}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})$
converges in $\mathcal{N}$; that is,

$\lim\limits_{k\rightarrow\infty}\sum\limits_{j=1}^{k}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\text{
exists in }\mathcal{N}.$

Taking the limit in (9) as $k\rightarrow\infty$, we get:

$0\leq\lim_{k\rightarrow\infty}\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_{0}})-\sum_{j=1}^{k}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right|\leq\lim_{k\rightarrow\infty}|\boldsymbol{\eta}-\boldsymbol{x_{0}}|^{k},$

from which we obtain

$0\leq\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_{0}})-\lim_{k\rightarrow\infty}\sum_{j=1}^{k}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right|\leq\lim_{k\rightarrow\infty}|\boldsymbol{\eta}-\boldsymbol{x_{0}}|^{k}.$

Since $\lambda(|\boldsymbol{\eta}-\boldsymbol{x_{0}}|)\geq\lambda(\delta)>0$,
we obtain that
$\lim\limits_{k\rightarrow\infty}\left|\boldsymbol{\eta}-\boldsymbol{x_{0}}\right|^{k}=0$.
It follows that

$0\leq\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_{0}})-\sum_{j=1}^{\infty}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}})\right|\leq
0$

from which we infer that

$f(\boldsymbol{\eta})=f(\boldsymbol{x_{0}})+\sum_{j=1}^{\infty}\frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_{0}})\cdot\nabla\right]^{j}f(\boldsymbol{x_{0}}).$

∎

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