# Bounded reduction of orthogonal matrices over polynomial rings

Pavel Gvozdevsky  Chebyshev Laboratory, St. Petersburg State University, 14th
Line V.O., 29B, Saint Petersburg 199178 Russia

###### Abstract.

We prove that a matrix from the split orthogonal group over a polynomial ring
with coefficients in a small-dimensional ring can be reduced to a smaller
matrix by a bounded number of elementary orthogonal transformations. The bound
is given explicitly. This result is an effective version of the early
stabilisation of the orthogonal $K_{1}$ functor proven by Suslin and Kopeiko
in [39]. Since the similar effective results for special linear and symplectic
groups are obtained by Vaserstein in [44], the present paper closes the
problem for split classical groups.

###### Keywords:

Split orthogonal group, surjective stability of $K_{1}$, bounded reduction,
polynomial rings

###### 2020 Mathematics Subject Classification:

19B14(primary), 19B10, and 20G35(secondary)

##  1\. Introduction

This paper deals with the last remaining case of bounded reduction in split
classical groups over polynomial rings. Namely we study the case of orthogonal
groups.

Chevalley groups over certain rings have bounded width with respect to the
elementary generators. For example this holds for Dedekind domains of
arithmetic type, see [4], [5], [23], [40], [41],[42],[7],[8]. Results on such
bounded generation are of great value, for example they are connected to the
congruence subgroup property, see [21],[24], and to Margulis–Zimmer
conjecture, see [30]. However bounded generation occurs very rarely in the
sense that classes of rings for which it is known to hold are pretty narrow.
Nevertheless, for some applications it is enough to have a weaker result, such
as: bounded length of conjugates of elementary generators (see [37]), bounded
length of commutators (see [31],[36],[6]), or bounded generation with respect
to a larger set of generators. Bounded reduction is a variation of the last
property.

A given Chevalley group $G$ over a given ring is said to have bounded
reduction if any element of $G$ can be decomposed as a product of bounded
number of elementary generators and one (not necessarily elementary) element
from a certain subsystem subgroup. In other words, it means that one can
reduce any element to the subsystem subgroup by bounded number of elementary
transformations. Without requirement for the number of elementary
transformations to be bounded this property is called the surjective stability
of the $K_{1}$-functor. In papers [34], [35], [25], [26], [27], and [13] this
problem is considered for rings that satisfy certain conditions on stable
rank, absolute stable rank, or other similar conditions. Actually, from the
proofs of the theorems in these papers one can recover the bound on the
required number of elementary transformations, despite the fact that this
bound is not stated in papers explicitly. Therefore, these are results on
bounded reduction.

However, conditions on stable rank are still very strong. Even though small
Jacobson dimension implies small stable rank, rings with large Jacobson
dimension usually fail to have small stable rank. In the present paper, we
consider another important class of rings. Namely we take a polynomial ring in
arbitrary number of variables with coefficients in a small-dimensional ring.
Here we use Krull dimension because the techniques require for dimension to
behave well with respect to adding an independent variable.

Without the bound on the number of elementary transformations similar result
for classical groups is known. This is so called early surjective stability of
the $K_{1}$-functor. For the special linear group this was proved by Suslin in
[38]. Similar result for the orthogonal group follows from [39], and for the
symplectic group it is proven in [17], see also [12], [18]. Note that if the
ring of coefficient is a Dedekind domain or a smooth algebra over a field,
then this result for all Chevalley groups follows from the homotopy invariance
of the non-stable $K_{1}$-functor, see [1],[33],[32].

In the case of special linear and symplectic groups, there are similar results
for Laurent polynomial rings, see [19], [20].

In the paper [44], Vaserstein obtained the effective version of the Suslin
result, i.e. he proved the bounded reduction for the special linear group over
a polynomial ring, and gave this bound explicitly. From this result he deduced
that the elementary subgroup of the general linear group over an arbitrary
finitely generated commutative ring has Kazhdan’s property (T).

In [28], the basic connection between bounded generation and property (T) has
been established and used to estimate the Kazhdan Constants for
$\mathop{\mathrm{SL}}\nolimits_{n}(\mathbb{Z})$. Later the bounds for these
constants were improved in [16]. In order to deduce property (T) from the
Vaserstein’s result one needs to refer to [29].

In fact, property (T) for Chevalley groups and groups similar to them has
already been studied by other methods, see [9],[10]. However, we believe that
the bounded reduction has an independent value, and we aim to study this
question for other Chevalley groups. It was noted in the concluding remarks of
[44] that the bounded reduction for the symplectic group follows formally from
the case of special linear group. Therefore, as we said, the last remaining
classical case is the one of orthogonal groups. The main result of the present
paper is the following theorem.

######  Theorem 1.1.

Let $C$ be a commutative Noetherian ring and $\dim C=D<\infty$. Let
$A=C[x_{1},\ldots,x_{n}]$. Then for any $r\geqslant\max(3,D+2)$, every matrix
from the split orthogonal group $O(2r,A)$ resp. $O(2r+1,A)$ can be reduced to
the subgroup

$\begin{pmatrix}1&&\\\ &O(2r-2,A)&\\\
&&1\end{pmatrix}\qquad\text{resp.}\qquad\begin{pmatrix}1&&\\\ &O(2r-1,A)&\\\
&&1\end{pmatrix}$

by multiplication from the left by

$\displaystyle
N=n(11r-7)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+8r-2\right)+10r-10$
resp. $\displaystyle
N=n(12r-8)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+9r-2\right)+11r-9$

elementary orthogonal transvections.

Therefore, on one hand this theorem is an effective version of the result from
[39], and on the other hand, an extension of [44] to the orthogonal group.

Very soon we reduce Theorem 1.1 to Theorem 4.1, which deals with an arbitrary
column vector rather than a column of a particular matrix. So we make a
convention that the word "column" means an arbitrary column vector.

In fact, Theorem 4.1 below is something stronger than just bonded reduction.
It is easy to see that we can deduce from it the following corollary, which is
a special case of Theorem 7.10 of [39].

######  Corollary 1.2.

Let $C$ be a commutative Noetherian ring and $\dim C=D<\infty$,
$A=C[x_{1},\ldots,x_{n}]$. Let $V$ be an $A$-module equipped with a quadratic
form $q$. Assume that an orthogonal sum of $(V,q)$ with a hyperbolic plane is
a free module with a split quadratic form of rank $2r$ or $2r+1$ with
$r\geqslant\max(3,D+2)$. Then $V$ is a free module and $q$ is split.

The paper is organised as follows. In Section 2, we give all necessary
preliminaries and introduce basic notation. In Section 3, we introduce the new
notion of an absolute flexible stable rank, and prove some of its properties.
In Sections 4, 5, and 6 we give the proof of the main result.

##  2\. Preliminaries and notation

### Rings, ideals and dimensions

By a ring we always mean associative and commutative ring with unity.

If $R$ is a ring, then by $R^{*}$ we denote the set of invertible elements in
$R$.

For the elements $r_{1}$,$\ldots$,$r_{k}\in R$, we denote by $\langle
r_{1},\ldots,r_{k}\rangle$ the ideal in $R$ generated by these elements.

In the present paper we use three different notions of a ring dimension.

  * •

By $\dim R$ we denote Krull dimension of the ring $R$. That is the supremum of
the lengths of all chains of prime ideals.

  * •

By $\operatorname{Jdim}R=\dim\operatorname{Max}(R)$ we denote the dimension of
the maximal spectrum $\operatorname{Max}(R)$ of the ring $R$. It is equal to
the supremum of the lengths of all chains of such prime ideals that coincide
with its Jacobson radical.

  * •

By $\operatorname{BSdim}R$ we denote the Bass–Serre dimension of a ring $R$.
That is the minimal $\delta$ such that $\operatorname{Max}(R)$ is a finite
union of irreducible Noetherian subspaces of dimension not greater than
$\delta$.

Obviously, for a Noetherian ring $R$ we have

$\operatorname{BSdim}R\leqslant\operatorname{Jdim}R\leqslant\dim R\text{.}$

The following property of Bass–Serre dimension is well known; see Lemma 4.17
in [2].

######  Lemma 2.1.

Let $R$ be a ring with $\operatorname{BSdim}R=\delta<\infty$. Then it has a
finite collection $P_{1}$,$\ldots$,$P_{m}$ of maximal ideals, such that for
any element $s\in R\setminus\bigcup_{i}P_{i}$ we have
$\operatorname{BSdim}R/(s)<\delta$. In case where $\delta=0$, this means that
$s\in R^{*}$.

### Even split orthogonal group

Let $R$ be a commutative ring and $V$ be a free $R$-module of rank $2r$ with
basis $e_{1}$,$\ldots$,$e_{r}$,$e_{-r}$,$\ldots$,$e_{-1}$; this ordering of
the basis elements will be used throughout the paper. Consider the quadratic
form $q$ on the module $V$ defined as follows

$q(x)=\sum_{i=1}^{r}x_{i}x_{-i}\text{,}$

where $x_{i}$ are the coordinates of $x$ in our basis.

The group of automorphism of $V$ preserving the form $q$ is called the even
split orthogonal group and denoted by $O(2r,R)$. We identify its elements with
their matrices in the basis
$e_{1}$,$\ldots$,$e_{r}$,$e_{-r}$,$\ldots$,$e_{-1}$.

For an element $\xi\in R$ and indices $i,j=1,\ldots,-1$ with $i\neq\pm j$, we
set

$T_{i,j}(\xi)=T_{-j,-i}(-\xi)=e+\xi e_{i,j}-\xi e_{-j,-i}\text{.}$

Such matrices are called elementary orthogonal transvections. It is easy to
see that $T_{i,j}(\xi)\in O(2r,R)$.

The subgroup of $O(2r,R)$, generated by all elementary orthogonal
transvections is called the elementary subgroup and denoted by $EO(2r,R)$.

For any $k\in\mathbb{N}$, we denote by $EO(2r,R)^{\leqslant N}$ the subset of
$EO(2r,R)$ consisting of matrices that can be expressed as the product of no
more than $N$ elementary orthogonal transvections.

### Odd split orthogonal group

Now let $V$ be a free $R$-module of rank $2r+1$ with basis
$e_{1}$,$\ldots$,$e_{r}$,$e_{0}$,$e_{-r}$,$\ldots$,$e_{-1}$, and the quadratic
form $q$ on the module $V$ defined as follows

$q(x)=x_{0}^{2}+\sum_{i=1}^{r}x_{i}x_{-i}\text{.}$

The group of automorphism of $V$ preserving the form $q$ is called the odd
split orthogonal group and denoted by $O(2r+1,R)$.

In the odd case, we have two types of elementary orthogonal transvections.
First type is the same as in the even case:

$T_{i,j}(\xi)=T_{-j,-i}(-\xi)=e+\xi e_{i,j}-\xi e_{-j,-i}\qquad
i,j=1,\ldots,-1\text{,}\;i\neq\pm j\text{,}$

and we refer to them as long root transvections. Second type is called short
root transvections, and those are the matrices

$T_{i,0}(\xi)=e+2\xi e_{i,0}-\xi e_{0,-i}-\xi^{2}e_{i,-i}\qquad i=\pm
1,\ldots\pm r\text{.}$

######  Remark.

The terms "long root" and "short root" come from the terminology for Chevalley
groups, which are defined by root systems; see, for example, [15].

The even split orthogonal group corresponds to the root system

$D_{n}=\\{e_{i}\pm e_{j}\colon i\neq j,1\leqslant i,j\leqslant n\\}\text{,}$

in which all the roots have the same length.

However, the odd split orthogonal group corresponds to the root system

$B_{n}=\\{e_{i}\pm e_{j}\colon i\neq j,1\leqslant i,j\leqslant
n\\}\cup\\{e_{i}\colon 1\leqslant i\leqslant n\\}\text{,}$

in which there are long roots (of length $\sqrt{2}$) and short roots (of
length 1).

As in the even case, we denote by $EO(2r+1,R)$ the subgroup of $O(2r+1,R)$,
generated by all elementary orthogonal transvections, and by
$EO(2r+1,R)^{\leqslant N}$, the subset of $EO(2r+1,R)$ consisting of matrices
that can be expressed as product of no more than $N$ elementary orthogonal
transvections.

Obviously, the group $O(2r,R)$ embeds into $O(2r+1,R)$ as the stabiliser of
$e_{0}$.

### Matrix transpose with respect to antidiagonal

Since we use weird numeration of rows and columns in orthogonal matrices, we
need a notation for the matrix transpose with respect to antidiagonal. Such
transpose of a matrix $g$ we denote by $g^{\reflectbox{t}}$.

By $g^{t}$ we denote regular transpose of the matrix $g$.

### Hyperbolic embedding

Consider the map

$H\colon\mathop{\mathrm{GL}}\nolimits(r,R)\to O(2r,R)\qquad
g\mapsto\begin{pmatrix}g&\\\ &(g^{-1})^{\reflectbox{t}}\end{pmatrix}\text{,}$

This map is called the hyperbolic embedding.

Note that $H$ takes elementary transvections in
$\mathop{\mathrm{GL}}\nolimits(r,R)$ to elementary orthogonal transvections.
Here by elementary transvections in $\mathop{\mathrm{GL}}\nolimits(r,R)$ we
mean the matrices

$t_{i,j}(\xi)=e+\xi e_{i,j}\text{,}$

where $\xi\in R$, $i,j=1,\ldots,r$, $i\neq j$.

In the odd case, we denote by $H$ the composition

$\textstyle{\mathop{\mathrm{GL}}\nolimits(r,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H}$$\textstyle{O(2r,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{O(2r+1,R)\text{.}}$

### ASR-condition

Recall that a commutative ring $R$ satisfies the absolute stable rank
condition $\mathrm{ASR}_{d}$ if for any row $(b_{1},\ldots,b_{d})$ with
coordinates in $R$, there exist elements $c_{1}$,$\ldots$,$c_{d-1}\in R$ such
that every maximal ideal of $R$ containing the ideal $\langle
b_{1}+c_{1}b_{d},\ldots,b_{d-1}+c_{d-1}b_{d}\rangle$ contains already the
ideal $\langle b_{1},\ldots,b_{d}\rangle$. This notion was introduced in [11]
and used in [34], [35] and then in [25],[26], [27], and [13] to study
stability problems.

If we assume that a row $(b_{1},\ldots,b_{d})$ is unimodular, then the
absolute stable rank condition boils down to the usual stable rank condition
$\mathrm{SR}_{d}$ (see [3],[43]).

Absolute stable rank satisfies the usual properties, namely for every ideal
$I\unlhd R$ condition $\mathrm{ASR}_{d}$ for $R$ implies $\mathrm{ASR}_{d}$
for the quotient $R/I$, and if $d\geqslant d^{\prime}$, then
$\mathrm{ASR}_{d^{\prime}}$ implies $\mathrm{ASR}_{d}$. Finally, it is well
known that if the maximal spectrum of $R$ is a Noetherian space of dimension
$\operatorname{Jdim}R=d-2$, then both conditions $\mathrm{ASR}_{d}$ and
$\mathrm{SR}_{d}$ are satisfied (see [11],[22],[34]).

##  3\. Absolute flexible stable rank

In this section, we introduce a new type of stable rank condition called the
absolute flexible stable rank. Here is the definition.

######  Definition 3.1.

A commutative ring $A$ satisfies the absolute flexible stable rank condition
$\mathrm{AFSR}_{d}$ if for any row $(b_{1},\ldots,b_{d})$ with coordinates in
$A$, there exists an element $c_{1}\in A$ such that for any invertible element
$\varepsilon_{1}\in A^{*}$, there exists $c_{2}\in A$ such that for any
$\varepsilon_{2}\in A^{*}$, $\ldots$, there exists $c_{d-1}\in A$ such that
for any $\varepsilon_{d-1}\in A^{*}$, every maximal ideal of $A$ containing
the ideal $\langle
b_{1}+\varepsilon_{1}c_{1}b_{d},\ldots,b_{d-1}+\varepsilon_{d-1}c_{d-1}b_{d}\rangle$
contains already the ideal $\langle b_{1},\ldots,b_{d}\rangle$.

One can think of it as follows. Two players are playing a game. Player 1
chooses a row $(b_{1},\ldots,b_{d})$ with coordinates in $A$. Then they take
turns starting with Player 2. Player 2 in his $i$-th turn chooses element
$c_{i}\in A$; after that Player 1 in his turn chooses invertible element
$\varepsilon_{i}\in A^{*}$. Player 2 wins if after $d$ turns every maximal
ideal of $A$ containing the ideal $\langle
b_{1}+\varepsilon_{1}c_{1}b_{d},\ldots,b_{d-1}+\varepsilon_{d-1}c_{d-1}b_{d}\rangle$
contains already the ideal $\langle b_{1},\ldots,b_{d}\rangle$. A commutative
ring $A$ satisfies the absolute flexible stable rank condition
$\mathrm{AFSR}_{d}$ if Player 2 has a winning strategy.

Now we show that the condition $\mathrm{AFSR}_{d}$ holds for small-dimensional
rings. That generalises the result of [11].

######  Lemma 3.2.

Let $A$ be a commutative ring. Assume that $\operatorname{Max}(A)$ is
Noetherian and $\operatorname{Jdim}A\leqslant d-2$. Then $A$ satisfies
$\mathrm{AFSR}_{d}$.

###### Proof.

Following [11], for any ideal $I\unlhd A$ we denote by $J(I)$ the intersection
of all the maximal ideals containing $I$, and we set $J=\\{I\unlhd A\mid
J(I)=I\\}$. The ideals from $J$ are in correspondence with the closed subsets
of $\operatorname{Max}(A)$.

For an ideal $I\unlhd A$, a prime ideal $P\in J$ is called a component of $I$
if $P$ is minimal among the primes of $J$ that contains $I$. The assumption
for $\operatorname{Max}(A)$ to be Noetherian implies that any ideal has only
finitely many components.

Now we show that making his $i$-th turn Player 2 can guarantee that any
component of ideal $I=\langle
b_{1}+\varepsilon_{1}c_{1}b_{d},\ldots,b_{i-1}+\varepsilon_{i-1}c_{i-1}b_{d}\rangle$
(for $i=1$ set $I=0$) that contains $b_{i}+\varepsilon_{i}c_{i}b_{d}$ also
contains $b_{d}$, and that doing so is a winning strategy.

When the $i$-th turn of Player 2 starts, the ideal $I$ is already fixed and
has finitely many component. Let $X$ be the set of components of $I$ that does
not contain $b_{d}$. All Player 2 has to do is to guarantee that
$b_{i}+\varepsilon_{i}c_{i}b_{d}$ is not contained in the union of ideals from
$X$.

Set $Y=\\{P\in X\mid b_{i}\in P\\}$. Since the components of $I$ cannot
contain each other, we can choose for any $\mathfrak{A}$,$\mathfrak{B}\in X$
an element $z(\mathfrak{A},\mathfrak{B})\in\mathfrak{A}\setminus\mathfrak{B}$.
Then to achive his goal Player 2 can set

$c_{i}=\sum_{\mathfrak{B}\in Y}\prod_{\mathfrak{A}\in
X\setminus\\{\mathfrak{B}\\}}z(\mathfrak{A},\mathfrak{B})\text{.}$

Let us prove that Player 2 won. Let
$b_{i}^{\prime}=b_{i}+\varepsilon_{i}c_{i}b_{d}$, $i=1,\ldots,d-1$. Assume
that some maximal ideal $\mathfrak{M}$ contains $\langle
b_{1}^{\prime},\ldots,b_{d-1}^{\prime}\rangle$, but does not contain $b_{d}$.
We claim that in this case there exists a chain of primes in $J$

$P_{0}\subsetneq P_{1}\subsetneq\ldots\subsetneq P_{d-1}=\mathfrak{M}\text{,}$

which contradicts the assumption on $\operatorname{Jdim}A$.

We build this chain from the end in such a way that $\langle
b_{1}^{\prime},\ldots,b_{i}^{\prime}\rangle\leqslant P_{i}$ but
$b_{i+1}^{\prime}\notin P_{i}$. We set $P_{d-1}=\mathfrak{M}$. Then for
$i<d-1$ we choose $P_{i}$ to be the component of $\langle
b_{1}^{\prime},\ldots,b_{i}^{\prime}\rangle$ that is contained in $P_{i+1}$,
such a component exists by Zorn’s lemma. Since $P_{i}\subseteq\mathfrak{M}$,
it follows that $b_{d}\notin P_{i}$, and by construction of
$b_{i+1}^{\prime}$, this implies that $b_{i+1}^{\prime}\notin P_{i}$.

The fact that $b_{i+1}^{\prime}\notin P_{i}$ guarantees that the inclusions in
the chain are proper. ∎

The following lemma shows how one can use the $\mathrm{AFSR}$ condition.

######  Lemma 3.3.

Let $A$ be a commutative ring, and $S$ be a multiplicative system in $A$.
Assume that the localisation $A[S^{-1}]$ satisfies $\mathrm{AFSR}_{d}$. Then
for any row $(b_{1},\ldots,b_{d})$ with coordinates in $A[S^{-1}]$ and for any
$s\in S$, there exist $c_{1}$,$\ldots$,$c_{d-1}\in sA$ such that such that
every maximal ideal of $A[S^{-1}]$ containing the ideal $\langle
b_{1}+c_{1}b_{d},\ldots,b_{d-1}+c_{d-1}b_{d}\rangle$ contains already the
ideal $\langle b_{1},\ldots,b_{d}\rangle$.

###### Proof.

Just let Player 1 choose his invertible element in such a way that the
resulting coefficient be in $sA$. ∎

##  4\. Reduction of Theorem 1.1 to propositions 4.2 and 4.3

We denote by $\operatorname{IUm}_{2r}A$ the set isotropic unimodular columns
of over a ring $A$:

$\operatorname{IUm}_{2r}A=\\{b\in A^{2r}\colon b\text{ is unimodular,
}q(b)=0\\}\text{.}$

Similarly, for the odd case we denote

$\operatorname{IUm}_{2r+1}A=\\{b\in
A^{2r+1}\colon(b_{1},\ldots,b_{r},2b_{0},b_{-r},\ldots,b_{-1})\text{ is
unimodular, }q(b)=0\\}\text{.}$

We have111We need to clarify why in the odd case the first column of an
orthogonal matrix remains unimodular after multiplying the middle element by
two. That is because the symmetric bilinear form associated to $q$ is
$Q(x,y)=q(x+y)-q(x)-q(y)=2x_{0}y_{0}+\sum_{i=1}^{r}(x_{i}y_{-i}+x_{-i}y_{i})$.
If $b$ and $c$ are the first and the last columns of some matrix in
$O(2r+1,A)$, then
$b_{1}c_{-1}+\ldots+b_{b}c_{-r}+(2b_{0})c_{0}+b_{-r}c_{r}+\ldots+b_{-1}c_{1}=Q(b,c)=Q(e_{1},e_{-1})=1$.
So the row $(b_{1},\ldots,b_{r},2b_{0},b_{-r},\ldots,b_{-1})$ is unimodular.
$O(2r,A)e_{1}\leqslant\operatorname{IUm}_{2r}A$ and
$O(2r+1,A)e_{1}\leqslant\operatorname{IUm}_{2r+1}A$, where $e_{1}$ is the
first column of the identity matrix.

If the first column of a matrix $\alpha\in O(2r,A)$ resp. $O(2r+1,A)$
coincides with $e_{1}$, then $\alpha$ automatically has shape

$\begin{pmatrix}1&*&*\\\ &\beta&*\\\ &&1\end{pmatrix}\text{,}$

where $\beta\in O(2r-2,A)$ resp. $O(2r-1,A)$. Hence it can be reduced to

$\begin{pmatrix}1&*&*\\\ &\beta&\\\ &&1\end{pmatrix}\text{,}$

which automatically should be

$\begin{pmatrix}1&&\\\ &\beta&\\\ &&1\end{pmatrix}\text{,}$

by multiplication from the left by $2r-2$ resp $2r-1$ elementary orthogonal
transvections.

Therefore, theorem 1.1 follows trivially from the following result.

######  Theorem 4.1.

Under the condition of Theorem 1.1, for every column
$b\in\operatorname{IUm}_{2r}A$ resp. $\operatorname{IUm}_{2r+1}A$ we have

$e_{1}\in EO(2r,A)^{\leqslant N}b\qquad\text{resp.}\qquad e_{1}\in
EO(2r+1,A)^{\leqslant N}b\text{,}$

where

$\displaystyle
N=n(11r-7)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+8r-2\right)+8r-8$
resp. $\displaystyle
N=n(12r-8)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+9r-2\right)+9r-8\text{.}$

Consider the lexicographic order on the monomials in variables
$x_{1}$,$\ldots$,$x_{n}$. That is the order where $x_{1}^{k_{1}}\ldots
x_{n}^{k_{n}}$ is bigger than $x_{1}^{l_{1}}\ldots x_{n}^{l_{n}}$ if for some
$m$ we have $k_{i}=l_{i}$ for $i<m$, and $k_{m}>l_{m}$. A polynomial in
$A=C[x_{1},\ldots,x_{n}]$ is called lexicographically monic if its leading
coefficient in lexicographic order is equal to one.

Further we reduce Theorem 4.1 to the following two propositions.

######  Proposition 4.2.

Under the condition of Theorem 1.1, for every column
$b\in\operatorname{IUm}_{2r}A$ resp. $\operatorname{IUm}_{2r+1}A$ there exists
a column

$b^{\prime}\in EO(2r,A)^{\leqslant 11r-7}b\qquad\text{resp.}\qquad
EO(2r+1,A)^{\leqslant 12r-8}b\text{,}$

such that its entry $b^{\prime}_{-2}$ is lexicographically monic.

######  Proposition 4.3.

Let $B$ be a commutative ring such that $\operatorname{BSdim}B=\delta<\infty$.
Let $r\geqslant 3$, $A=B[y]$, $b\in\operatorname{IUm}_{2r}A$ resp.
$\operatorname{IUm}_{2r+1}A$ such that its entry $b_{-2}$ is monic. Then

$EO(2r,A)^{\leqslant N}b\cap B^{2r}\neq\varnothing\quad\text{resp.}\quad
EO(2r+1,A)^{\leqslant N}b\cap B^{2r+1}\neq\varnothing\text{,}$

where $N=\delta\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+8r-2\right)$
resp. $\delta\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+9r-2\right)$

First we need the following lemma.

######  Lemma 4.4.

Let $f\in C[x_{1},\ldots,x_{n}]$ be a lexicographically monic polynomial. Then
there exists an invertible change of variables

$x_{1},\ldots,x_{n}\leftrightarrow y_{1},\ldots,y_{n}\text{,}$

such that $f$ becomes monic in $y_{n}$.

###### Proof.

Take $K>\deg f$. Set $x_{i}=y_{i}+y_{n}^{K^{n-i}}$, $i=1$,$\ldots$,$n-1$, and
$x_{n}=y_{n}$. ∎

Now we deduce Theorem 4.1 from Propositions 4.2 and 4.3.

Take $b\in\operatorname{IUm}_{2r}A$ resp. $\operatorname{IUm}_{2r+1}A$. By
Proposition 4.2 there exists a column

$b^{\prime}\in EO(2r,A)^{\leqslant 11r-7}b\qquad\text{resp.}\qquad
EO(2r+1,A)^{\leqslant 12r-8}b\text{,}$

such that its entry $b^{\prime}_{-2}$ is lexicographically monic. Applying
Lemma 4.4, we change variables to $y_{1}$,$\ldots$,$y_{n}$ so that
$b^{\prime}_{-2}$ is now monic in $y_{n}$. Now we apply Proposition 4.3 to
$B=C[y_{1},\ldots,y_{n-1}]$. Note that $\operatorname{BSdim}B\leqslant\dim
B=D+n-1$. Hence we can obtain a column from

$\displaystyle EO(2r,A)^{\leqslant N_{1}^{\prime}}b^{\prime}\cap
B^{2r}\leqslant EO(2r,A)^{\leqslant N_{1}}b\cap B^{2r}$ resp. $\displaystyle
EO(2r+1,A)^{\leqslant N_{1}^{\prime}}b^{\prime}\cap B^{2r+1}\leqslant
EO(2r+1,A)^{\leqslant N_{1}}b\cap B^{2r+1}\text{,}$

where

$N_{1}^{\prime}=(D+n-1)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+8r-2\right)\quad\text{resp.}\quad(D+n-1)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+9r-2\right)\text{,}$

and $N_{1}=N_{1}^{\prime}+11r-7$ resp. $N_{1}^{\prime}+12r-8$.

Repeating this argument $n$ times we can obtain a column from

$EO(2r,A)^{\leqslant N_{n}}b\cap C^{2r}\qquad\text{resp.}\qquad
EO(2r+1,A)^{\leqslant N_{n}}b\cap C^{2r+1}\text{,}$

where

$\displaystyle
N_{n}=n(11r-7)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+8r-2\right)$
resp. $\displaystyle
N_{n}=n(12r-8)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+9r-2\right)\text{.}$

Since $\dim C\leqslant r-2$, the ring $C$ satisfies the condition
$\mathrm{ASR}_{r}$. Hence the proof of surjective stability of $K_{1}$ for
orthogonal groups given in [34] (Theorem 2.1), which is essentially a proof of
bounded reduction under the $\mathrm{ASR}$-condition, implies222Stein uses
action not on columns but on rows, and in the odd case there is a discrepancy
with the present paper about where the factor of two arises. So if our column,
say is $(v_{1},\ldots,v_{r},v_{0},v_{-r},\ldots,v_{-1})^{t}$, one has to apply
Stein’s proof to the row $(v_{1},\ldots,v_{r},2v_{0},v_{-r},\ldots,v_{-1})$.
Note that this row remains unimodular. that applying to our column $6r-6$
resp. $7r-7$ elementary orthogonal transvections we can make its first entry
equal to one. Then by $2r-2$ resp. $2r-1$ elementary orthogonal transvections
we can annihilate entries from $2$ to $-2$, and the last $-1$-th entry becomes
zero automatically because the column remains isotropic.

Therefore,

$e_{1}\in EO(2r,A)^{\leqslant N}b\qquad\text{resp.}\qquad e_{1}\in
EO(2r+1,A)^{\leqslant N}b\text{,}$

where

$\displaystyle
N=n(11r-7)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+8r-2\right)+8r-8$
resp. $\displaystyle
N=n(12r-8)+\left(nD+\tfrac{r(r-1)}{2}\right)\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+9r-2\right)+9r-8\text{.}$

##  5\. Obtaining a monic polynomial

In this section, we give the proof of Proposition 4.2. First we need some
preparation.

Recall that the height $\operatorname{ht}_{A}(P)$ of a prime ideal $P$ in a
ring $A$ is the is the largest number $h$ (or $\infty$ if such a number does
not exist) such that there exists a chain of different prime ideals

$P_{0}\subsetneq P_{1}\subsetneq\ldots\subsetneq P_{n}=P\text{.}$

The height $\operatorname{ht}_{A}(I)$ of an arbitrary ideal $I\unlhd A$ is
defined by

$\operatorname{ht}_{A}(I)=\inf\\{\operatorname{ht}_{A}(P)\colon P\text{ is
prime, }I\leqslant P\\}\text{.}$

######  Lemma 5.1.

Let $C$ be a Noetherian ring, $A=C[x_{1},\ldots,x_{n}]$, $I\unlhd A$ and
$\operatorname{ht}_{A}(I)\geqslant\dim(C)+1$. Then the ideal $I$ contains a
lexicographically monic polynomial.

###### Proof.

Let $J$ be an ideal in $B=C[x_{2},\ldots,x_{n}]$ consisting of leading terms
of polynomials in $I$. Here we consider $A$ as $B[x_{1}]$. By Lemma 10.4 of
[45] we have

$\operatorname{ht}_{B}(J)\leqslant\operatorname{ht}_{A}(I)\leqslant\dim(C)+1\text{.}$

In case $n=1$ where $B=C$, this implies that $1\in J$. Hence $I$ contains a
monic polynomial. That is the base of induction. For the induction step, note
that by induction hypothesis $J$ contains a lexicographically monic
polynomial. Thus $I$ contains a polynomial $f$ with leading term in $x_{1}$
being lexicographically monic, but this means exactly that $f$ is
lexicographically monic itself. ∎

######  Lemma 5.2.

Let $C$ be a Noetherian ring, $A=C[x_{1},\ldots,x_{n}]$. Let $S$ be a
multiplicative system of lexicographically monic polynomials in $A$. Then we
have $\dim A[S^{-1}]\leqslant\dim C$.

###### Proof.

Assume the converse. Then there exists a chain

$P_{0}\subsetneq P_{1}\subsetneq\ldots\subsetneq P_{k}$

of prime ideals in $A[S^{-1}]$, where $k>\dim C$. Hence we have

$P_{0}\cap A\subsetneq P_{1}\cap A\subsetneq\ldots\subsetneq P_{k}\cap
A\text{.}$

Thus $\operatorname{ht}(P_{k}\cap A)\geqslant k\geqslant\dim C+1$, and by
Lemma 5.1 the ideal $P_{k}\cap A$ contains a lexicographically monic
polynomial. Hence $P_{k}=A[S^{-1}]$. This is a contradiction.

∎

Now we recall a definition from [44].

######  Definition 5.3.

Let $A$ be an associative ring with 1, $s$ be a central element of $A$,
$r\geqslant 2$, $v\in A^{r-1}$ (a column over $A$), $u\in\vphantom{,}^{r-1}A$
(a row over $A$). We define an $r$ by $r$ matrix over $A$ by

$\mu(u,s,v)=\begin{pmatrix}1_{r-1}+vsu&vs^{2}\\\
-uvu&1-uvs\end{pmatrix}\text{.}$

This matrix is invertible with $\mu(u,s,v)^{-1}=\mu(u,s,-v)$. If $s\in A^{*}$,
then

$\mu(u,s,v)=\begin{pmatrix}1_{r-1}&0\\\
-u/s&1\end{pmatrix}\begin{pmatrix}1_{r-1}&vs^{2}\\\
0&1\end{pmatrix}\begin{pmatrix}1_{r-1}&0\\\ u/s&1\end{pmatrix}$

The following lemma was proved in [44].

######  Lemma 5.4.

(Lemma 2.2 in [44]) When $r\geqslant 3$, the matrix $\mu(u,s,v)$ is a product
of $7r-3$ elementary transvections in $\mathop{\mathrm{GL}}\nolimits(r,R)$.

Now we are ready to prove Proposition 4.2.

Let $S$ be a multiplicative system of lexicographically monic polynomials in
$A$. It follows from Lemma 5.2 and 3.2 that the ring $A[S^{-1}]$ satisfies
$\mathrm{AFSR}_{r}$, and so does any quotient of $A[S^{-1}]$.

### Even case

Take $b\in\operatorname{IUm}_{2r}A$. We need to obtain a lexicographically
monic polynomial by $11r-7$ elementary orthogonal transvections. We perform
the following steps.

Step 1. Make the row $(b_{2},\ldots,b_{-1})$ unimodular in $A[S^{-1}]$ by
$r-1$ transvections.

Let $\mathfrak{A}=\langle b_{-r},\ldots,b_{-1}\rangle\unlhd A[S^{-1}]$. Since
$A[S^{-1}]/\mathfrak{A}$ satisfies $\mathrm{AFSR}_{r}$ and the row
$(b_{1},\ldots,b_{r})$ is unimodular in $A[S^{-1}]/\mathfrak{A}$, it follows
from Lemma 3.3 that there exist $c_{2}$,$\ldots$,$c_{r}\in A$ such that the
row $(b_{2}+c_{2}b_{1},\ldots,b_{r}+c_{r}b_{1})$ is unimodular in
$A[S^{-1}]/\mathfrak{A}$. Thus by applying the transvections $T_{i,1}(c_{i})$
for $i=2$,$\ldots$,$r$, we make the row $(b_{2},\ldots,b_{r})$ unimodular in
$A[S^{-1}]/\mathfrak{A}$ without changing the ideal $\mathfrak{A}$. Thus the
row $(b_{2},\ldots,b_{-1})$ becomes unimodular in $A[S^{-1}]$.

Step 2. Make the row $(b_{1},b_{-r}\ldots,b_{-1})$ unimodular in $A[S^{-1}]$
by $r-1$ transvections.

Since the row $(b_{2},\ldots,b_{-1})$ unimodular in $A[S^{-1}]$, it follows
that the ideal generated by $(b_{2},\ldots,b_{-1})$ in $A$ contains a
lexicographically monic polynomial. So for some $f_{2}$,$\ldots$,$f_{-1}\in
A$, the polynomial

$\sum_{i=2}^{-1}f_{i}b_{i}$

is lexicographically monic. Multiplying polynomials $f_{i}$ by a large enough
power of $x_{1}$, we may assume that the polynomial

$b_{1}+\sum_{i=2}^{-1}f_{i}b_{i}$

is also lexicographically monic.

Let us now apply the transvections $T_{1,i}(f_{i})$ for $i=2$,$\ldots$,$r$.
Then the ideal generated by the new $b_{1}$ and old $b_{-r}$,$\ldots$,$b_{-1}$
contains a lexicographically monic polynomial. However, these transvections do
not change the ideal generated by $b_{-r}$,$\ldots$,$b_{-1}$. Hence we
actually achieve that the ideal generated by new
$b_{1}$,$b_{-r}$,$\ldots$,$b_{-1}$ contains a lexicographically monic
polynomial. Thus the row $(b_{1},b_{-r}\ldots,b_{-1})$ becomes unimodular in
$A[S^{-1}]$.

Step 3. Make the row $(b_{1},b_{-r}\ldots,b_{-2})$ unimodular in $A[S^{-1}]$
by $7r-3$ transvections.

Let $\mathfrak{A}=\langle b_{-r},\ldots,b_{-1}\rangle\unlhd A[S^{-1}]$. Since
$b_{1}$ is invertible in $A[S^{-1}]/\mathfrak{A}$, it follows that there exist
$\xi_{2}$,$\ldots$,$\xi_{r}\in A[S^{-1}]$ such that
$b_{i}-\xi_{i}b_{1}\in\mathfrak{A}$ for $i=2,\ldots,r$. Let $s$ be a common
denominator of $\xi_{i}$. So we have $\xi_{i}=\tfrac{u_{i}}{s}$. Set

$\widetilde{b}_{-1}=b_{-1}+\sum_{i=2}^{r}\xi_{i}b_{-i}\text{.}$

Since $A[S^{-1}]$ satisfies $\mathrm{AFSR}_{r}$, it follows from Lemma 3.3
that there exist $c_{-2}$,$\ldots$,$c_{-r}\in s^{2}A$ such that every maximal
ideal of $A[S^{-1}]$ containing the ideal $\langle
b_{-r}+c_{-r}\widetilde{b}_{-1},\ldots b_{-2}+c_{-2}\widetilde{b}_{-1}\rangle$
contains already the ideal $\langle
b_{-r},\ldots,b_{-2},\widetilde{b}_{-1}\rangle=\mathfrak{A}$. Let
$c_{i}=-s^{2}v_{i}$.

Now let $u=(u_{r},\ldots,u_{2})$, and $v=(v_{-r},\ldots,v_{-2})^{t}$. We claim
that if we multiply the column $b$ by the matrix
$H(\mu(u,s,v)^{\reflectbox{t}})$, then we make the row
$(b_{1},b_{-r}\ldots,b_{-2})$ unimodular in $A[S^{-1}]$. Let us prove that.

Assume that some maximal ideal $\mathfrak{M}$ of the ring $A[S^{-1}]$ contains
the new elements $b_{1}$,$b_{-r}$,$\ldots$,$b_{-2}$. In terms of the old
elements that would be

$\displaystyle(1-uvs)b_{1}+s^{2}\left(\sum_{k=2}^{r}v_{-k}b_{k}\right)=b_{1}+s^{2}\left(\sum_{k=2}^{r}v_{-k}(b_{k}-\xi_{k}b_{1})\right)\in\mathfrak{M}\text{,}$
$\displaystyle
b_{i}-v_{i}\left(s\left(\sum_{k=2}^{r}u_{k}b_{-k}\right)+s^{2}b_{-1}\right)=b_{i}+c_{i}\widetilde{b}_{-1}\in\mathfrak{M}\qquad
i=-r,\ldots,-2\text{.}$

By choice of $c_{i}$ we have $\mathfrak{A}\leqslant\mathfrak{M}$. Hence
$(b_{k}-\xi_{k}b_{1})\in\mathfrak{M}$ for $k=2$,$\ldots$,$r$. Thus
$b_{1}\in\mathfrak{M}$. However, by previous step $b_{1}$ and $\mathfrak{A}$
generate a unit ideal. This is a contradiction.

It remains to refer to Lemma 5.4, which implies that
$H(\mu(u,s,v)^{\reflectbox{t}})\in EO(2r,R)^{\leqslant 7r-3}$.

Step 4. Make the row $(b_{1},b_{-r}\ldots,b_{-3})$ unimodular in $A[S^{-1}]$
by $r-1$ transvections.

Since $A[S^{-1}]$ satisfies $\mathrm{AFSR}_{r}$ and the row
$(b_{1},b_{-r}\ldots,b_{-2})$ is unimodular in $A[S^{-1}]$, it follows from
Lemma 3.3 that there exist $c_{1}$,$c_{-r}$,$\ldots$,$c_{-3}\in A$ such that
the row $(b_{1}+c_{2}b_{-2},b_{-r}+c_{-r}b_{-2},\ldots,b_{-3}+c_{-3}b_{-2})$
is unimodular in $A[S^{-1}]$. Thus by applying the transvections
$T_{i,-2}(c_{i})$ for $i=1$,$-r$,$\ldots$,$-3$, we make the row
$(b_{1},b_{-r}\ldots,b_{-3})$ unimodular in $A[S^{-1}]$.

Step 5. Make $b_{-2}$ lexicographically monic by $r-1$ transvections..

Since the row $(b_{1},b_{-r}\ldots,b_{-3})$ is unimodular in $A[S^{-1}]$, it
follows that the ideal generated by $(b_{1},b_{-r}\ldots,b_{-3})$ in $A$
contains a lexicographically monic polynomial. So for some
$f_{1}$,$f_{-r}$,$\ldots$,$f_{-3}\in A$, the polynomial

$\sum_{i\in\\{1,-r,\ldots,-3\\}}f_{i}b_{i}$

is lexicographically monic. Multiplying polynomials $f_{i}$ by a large enough
power of $x_{1}$, we may assume that the polynomial

$b_{-2}+\sum_{i\in\\{1,-r,\ldots,-3\\}}f_{i}b_{i}$

is also lexicographically monic.

Now applying the transvections $T_{-2,i}(f_{i})$ for $i=1$,$-r$,$\ldots$,$-3$,
we obtain a lexicographically monic polynomial in position $-2$.

### Odd case

Take $b\in\operatorname{IUm}_{2r+1}A$. We need to obtain a lexicographically
monic polynomial by $12r-8$ elementary orthogonal transvections. The proof is
similar to the one given for the even case. However, here we need an
additional step at the begining

Step 0. Make the row $(b_{1},\ldots,b_{r},b_{-r},\ldots,b_{-1})$ unimodular in
$A[S^{-1}]$ by $r-1$ transvections.

Let $\mathfrak{A}=\langle b_{1},b_{-r},\ldots,b_{-1}\rangle\unlhd A[S^{-1}]$.
Since $A[S^{-1}]/\mathfrak{A}$ satisfies $\mathrm{AFSR}_{r}$ and the row
$(b_{2},\ldots,b_{r},2b_{0})$ is unimodular in $A[S^{-1}]/\mathfrak{A}$, it
follows from Lemma 3.3 that there exist $c_{2}$,$\ldots$,$c_{r}\in A$ such
that the row $(b_{2}+2c_{2}b_{0},\ldots,b_{r}+2c_{r}b_{0})$ is unimodular in
$A[S^{-1}]/\mathfrak{A}$. Thus by applying the transvections $T_{i,0}(c_{i})$
for $i=2$,$\ldots$,$r$, we make the row $(b_{2},\ldots,b_{r})$ unimodular in
$A[S^{-1}]/\mathfrak{A}$ without changing the ideal $\mathfrak{A}$. Thus the
row $(b_{1},\ldots,b_{r},b_{-r},\ldots,b_{-1})$ becomes unimodular in
$A[S^{-1}]$.

All the remaining steps in the odd case use only long root transvections and
are identical to those in the even case.

######  Remark.

One can notice that the proof above basically repeats the proof of stability
theorem for $K_{1}$-functor given in [34].

##  6\. Eliminating a variable

In this section, we give the proof of Proposition 4.3. First we need some
preparation.

For a commutative ring $R$, we denote by $\Theta(r,R)$ be the following set

$\Theta(r,R)=\\{M\in M^{r\times r}(R)\colon M^{\reflectbox{t}}=-M\text{, and
}M\text{ has zeroes at antidiagonal positions}\\}\text{.}$

The set $\Theta(r,R)$ forms a group under the addition operation, and the map

$U\colon\Theta(r,R)\to O(2r,R)\text{,}\qquad M\mapsto\begin{pmatrix}1_{r}&0\\\
M&1r\end{pmatrix}$

is a group homomorphism.

In the odd case, we denote by $U$ the composition

$\textstyle{\Theta(r,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{O(2r,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{O(2r+1,R)\text{.}}$

######  Lemma 6.1.

Any matrix from the image of the map $U$ can be expressed as a product of
$r(r-1)/2$ elementary orthogonal transvections.

###### Proof.

Suppose that

$M=\sum_{1\leqslant j<i\leqslant r}\xi_{i,j}(e_{r+1-i,j}-e_{r+1-j,i})\text{.}$

Then

$U(M)=e+\sum_{1\leqslant i<j\leqslant
r}\xi_{i,j}(e_{-i,j}-e_{-j,i})=\prod_{1\leqslant i<j\leqslant
r}T_{-i,j}(\xi_{i,j})\text{.}$

∎

######  Remark.

From the Chevalley groups point of view, in the even case the image of $U$ is
a unipotent radical of a certain parabolic subgroup, and $r(r-1)/2$ is a
number of roots in the special part of the corresponding parabolic set of
roots.

######  Lemma 6.2.

Let $r\geqslant 2$. Let $b^{+}=(b_{1},\ldots,b_{r})^{t}\in R^{r}$ be a
unimodular column. Let $b^{-}=(b_{-r},\ldots,b_{-1})^{t}\in R^{r}$ be such
that $\sum_{i=1}^{r}b_{i}b_{-i}=0$. Then there exists a matrix
$M\in\Theta(r,R)$ such that $b^{-}=Mb^{+}$.

###### Proof.

For a system of linear equations over a ring, existence of a solution is a
local property, see, for example, Proposition 1 in [14]. Hence it is enough to
consider the case where $R$ is a local ring.

In the local case, since $b^{+}$ is unimodular, there exists a matrix
$g\in\mathop{\mathrm{GL}}\nolimits(r,R)$ such that $gb^{+}=e_{1}$. Set
$\widetilde{b}^{-}=(\widetilde{b}_{-r},\ldots,\widetilde{b}_{-1})^{t}=(g^{\reflectbox{t}})^{-1}b^{-}$.
Note that the first column of $g^{-1}$ is $b^{+}$. Hence the last row of
$(g^{\reflectbox{t}})^{-1}$ is transposed $b^{+}$ written backwards. Hence
$\widetilde{b}_{-1}=0$.

Further set

$\widetilde{M}=\begin{pmatrix}\widetilde{b}_{-r}&&&\\\ \vdots&&&\\\
\widetilde{b}_{-2}&&&\\\
0&-\widetilde{b}_{-2}&\ldots&-\widetilde{b}_{-r}\end{pmatrix}\text{.}$

Then we have $\widetilde{M}e_{1}=\widetilde{b}^{-}$. It remains to set
$M=g^{\reflectbox{t}}\widetilde{M}g$. ∎

######  Lemma 6.3.

For any $r\geqslant 2$, there exists $\widetilde{k}\in\mathbb{N}$ such that
for any ring $R$, for any columns $b^{+}=(b_{1},\ldots,b_{r})^{t}\in R^{r}$
and $b^{-}=(b_{-r},\ldots,b_{-1})^{t}\in R^{r}$ such that
$\sum_{i=1}^{r}b_{i}b_{-i}=0$, and for any $s\in\langle
b_{1},\ldots,b_{r}\rangle\unlhd R$, there exists a matrix $M\in\Theta(r,R)$
such that $s^{\widetilde{k}}b^{-}=Mb^{+}$.

###### Proof.

First we prove that such a number $\widetilde{k}$ exists for any given $R$,
$b^{+}$, $b^{-}$ and $s$. Since the column $b^{+}$ becomes unimodular in the
localisation $R[s^{-1}]$, it follows from Lemma 6.2 that there exists a matrix
$M_{1}\in\Theta(r,R[s^{-1}])$ such that $b^{-}=M_{1}b^{+}$. Let
$l_{1}\in\mathbb{N}$ be such that $M_{1}=M_{2}s^{-l_{1}}$ for some
$M_{2}\in\Theta(r,R)$. Then the equality $s^{l_{1}}b^{-}=M_{2}b^{+}$ holds in
$R[s^{-1}]$. Hence for a large enough $l_{2}$, we have
$s^{l_{1}+l_{2}}b^{-}=s^{l_{2}}M_{2}b^{+}$ in $R$. Set
$\widetilde{k}=l_{1}+l_{2}$ and $M=s^{l_{2}}M_{2}$.

Now we show that $\widetilde{k}$, actually, can depend only on $r$. Consider
the universal situation where

$R=\mathbb{Z}[b_{1},\ldots,b_{-1},a_{1},\ldots,a_{r}]/\langle
b_{1}b_{-1}+\ldots+b_{r}b_{-r}\rangle\text{,}\qquad\text{and}\qquad
s=a_{1}b_{1}+\ldots+a_{r}b_{r}\text{.}$

Since this case can be specialised to any other, it follows that the number
$\widetilde{k}$ that works in this case should work in any other case as well.
∎

######  Lemma 6.4.

Let $R$ be a commutative Noetherian ring, $s\in R$ and $r\geqslant 2$. Then
there exists $k\in\mathbb{N}$ such that for any column
$b^{+}=(b_{1},\ldots,b_{r})^{t}\in R^{r}$ such that $s\in\langle
b_{1},\ldots,b_{r}\rangle$, and for any column
$b^{-}=(b_{-r},\ldots,b_{-1})^{t}\in R^{r}$ such that
$\sum_{i=1}^{r}b_{i}b_{-i}=0$ and all the entries $b_{-i}$ are divisible by
$s^{k}$, there exists a matrix $M\in\Theta(r,R)$ such that $b^{-}=Mb^{+}$.

###### Proof.

The annihilators of the elements $s^{i}$, $l\in\mathbb{N}$ form a ascending
chain

$\operatorname{Ann}s\leqslant\operatorname{Ann}s^{2}\leqslant\ldots\text{.}$

Since the ring $R$ is Noetherian, it follows that for some $l\in\mathbb{N}$,
we have $\operatorname{Ann}s^{l+m}=\operatorname{Ann}s^{l}$ for any
$m\in\mathbb{N}$.

Take $\widetilde{k}$ from Lemma 6.3, and set $k=\widetilde{k}+l$. Now let
$b^{+}=(b_{1},\ldots,b_{r})^{t}\in R^{r}$ be such that $s\in\langle
b_{1},\ldots,b_{r}\rangle$, and $b^{-}=(b_{-r},\ldots,b_{-1})^{t}\in R^{r}$
such that $\sum_{i=1}^{r}b_{i}b_{-i}=0$ and $b^{-}=s^{k}(b^{-})^{\prime}$ for
some $(b^{-})^{\prime}\in R^{k}$.

Since $\sum_{i=1}^{r}b_{i}b_{-i}=0$, it follows that
$\sum_{i=1}^{r}b_{i}b_{-i}^{\prime}\in\operatorname{Ann}s^{k}=\operatorname{Ann}s^{l}$.
Hence $\sum_{i=1}^{r}b_{i}(s^{l}b_{-i}^{\prime})=0$. By the choice of
$\widetilde{k}$ there exists $M\in\Theta(r,R)$ such that
$b^{-}=s^{\widetilde{k}}(s^{l}(b^{-})^{\prime})=Mb^{+}$. ∎

######  Lemma 6.5.

Let $B$ be a commutative Noetherian ring, $A=B[y]$, $r\geqslant 3$,
$b=b(y)=(b_{1},\ldots,b_{-1})^{t}\in\operatorname{IUm}_{2r}A$ resp.
$\operatorname{IUm}_{2r+1}A$, $s\in B\cap\langle b_{1},\ldots,b_{r-1}\rangle$.
Then there exists $m\in\mathbb{N}$ such that

$b(y+s^{m}z)\in EO(2r,A[z])^{\leqslant
r(r-1)/2+8r-4}b(y)\;\text{resp.}\;b(y+s^{m}z)\in EO(2r+1,A[z])^{\leqslant
r(r-1)/2+9r-5}b(y)\text{.}$

###### Proof.

First we consider the even case.

Take $k$ from Lemma 6.4 (for $R=A[z]$) and set $m=k+2$. Let
$b=(\begin{smallmatrix}b^{+}\\\ b^{-}\end{smallmatrix})$, where
$b^{+}=(b_{1},\ldots,b_{r})^{t}\in A^{r}$, and
$b^{-}=(b_{-r},\ldots,b_{-1})^{t}\in A^{r}$. Since $s\in B\cap\langle
b_{1},\ldots,b_{r-1}\rangle$, by Corollary 2.4 of [44] there exists a matrix
$g(z)\in E(r,A[z])^{\leqslant 8r-4}$ such that $g(z)b^{+}(y)=b^{+}(y+s^{2}z)$.

Moreover, it follows from the proof that $g$ is congruent to the identity
matrix modulo $z$. Indeed, in Vaserstein’s proof he reduces the column
$b(y+s^{2}z)$ to $b(y)$ in two steps. At the first step he makes the first
$r-1$ components of the column equal to those of $b(y)$ using matrix
$\mu(u,s,v)$ (see Definition 5.3 of the present paper) with all the entries of
the column $v$ being divisible by $z$; hence this matrix is congruent to the
identity matrix modulo $z$. It follows that the difference between $b_{r}(y)$
and the last entry of the column obtained on the first step is divisible by
$z$. At the second step Vaserstein uses addition operations to make the last
entry equal to $b_{r}(y)$. In a view of the above, the coefficients of these
additions must be divisible by $z$. Thus the matrix of the whole
transformation is congruent to the identity matrix modulo $z$.

Therefore, for some $\widetilde{b}^{-}\in A[z]^{r}$ we have

$b^{\prime}=\begin{pmatrix}b^{+}(y+s^{m}z)\\\
\widetilde{b}^{-}\end{pmatrix}=H(g(s^{k}z))b\in EO(2r,A[z])^{\leqslant
8r-4}b\text{.}$

In addition, $\widetilde{b}^{-}$ is congruent to $b^{-}(y)$ modulo $s^{k}$.
Since both $b(y+s^{m}z)$ and $b^{\prime}$ are isotropic, and
$b^{-}(y+s^{m}z)-\widetilde{b}^{-}$ is divisible by $s^{k}$, it follows by
Lemma 6.4 that there exists a matrix $M\in\Theta(r,A[z])$ such that
$b^{-}(y+s^{m}z)-\widetilde{b}^{-}=Mb^{+}(y+s^{m}z)$. Here we used that $s\in
B$, so the shift of the variable does not change the fact that $s\in\langle
b_{1},\ldots,b_{r-1}\rangle$. Hence by Lemma 6.1 we have

$b(y+s^{m}z)=U(M)b^{\prime}\in EO(2r,A[z])^{\leqslant r(r-1)/2+8r-4}b$

In the odd case, where we have

$b=\begin{pmatrix}b^{+}\\\ b_{0}\\\ b^{-}\end{pmatrix}\text{,}$

the proof is the similar. However, after creating $b^{+}(y+s^{m}z)$ at the top
and before creating $b^{-}(y+s^{m}z)$ at the bottom, we have to create
$b_{0}(y+s^{m}z)$ in the middle by the transvections $T_{-i,0}(\xi_{i})$,
$i=1,\ldots,r-1$ with suitable $\xi_{i}$. Here we again use that $s\in\langle
b_{1},\ldots,b_{r-1}\rangle$. ∎

######  Lemma 6.6.

Let $B$ be a commutative ring, $P_{1}$,$\ldots$,$P_{m}$ be distinct maximal
ideals in $B$, $A=B[y]$, $r\geqslant 3$,
$b=b(y)=(b_{1},\ldots,b_{-1})^{t}\in\operatorname{IUm}_{2r}A$ resp.
$\operatorname{IUm}_{2r+1}A$ such that $b_{-2}$ is monic. Then there exists a
column

$b^{(1)}\in EO(2r,A)^{\leqslant\lceil{r-1\over
2}\rceil+2}b\qquad\text{resp.}\qquad EO(2r+1,A)^{\leqslant\lceil{r-1\over
2}\rceil+3}b$

such that $b^{(1)}_{-2}$ is monic and

$\left(\langle b^{(1)}_{1},\ldots b^{(1)}_{r-1}\rangle\cap
B\right)\setminus\bigcup_{i=1}^{m}P_{i}\neq\varnothing\text{.}$

###### Proof.

Set

$R=B/\left(\bigcap_{i=1}^{m}P_{i}\right)=\prod_{i=1}^{m}B/P_{i}\text{.}$

First we show that the last condition on $b^{(1)}$ holds if $b^{(1)}_{1}$
monic and the row $(b^{(1)}_{1},\ldots,b^{(1)}_{r-1})$ becomes unimodular in
$R[y]$.

Let $c_{1}$,$\ldots$,$c_{r-1}\in A$ be such that
$c_{1}b^{(1)}_{1}+\ldots+c_{r-1}b^{(1)}_{r-1}\equiv 1\mod P_{i}$ for every
$i$.

Set $f=c_{2}b^{(1)}_{2}+\ldots+c_{r-1}b^{(1)}_{r-1}$. Then $b^{(1)}_{1}$ and
$f$ are coprime in $B/P_{i}[y]$ for every $i$.

Since $b^{(1)}_{1}$ is monic, it follows that the resultant
$\operatorname{res}(b^{(1)}_{1},f)$ modulo $P_{i}$ is equal to the resultant
of $b^{(1)}$ taken modulo $P_{i}$ and $f$ taken modulo $P_{i}$ (even if $f$
modulo $P_{i}$ has smaller degree).

Therefore, we have

$\operatorname{res}(b^{(1)}_{1},f)\in\left(\langle b^{(1)}_{1},\ldots
b^{(1)}_{r-1}\rangle\cap B\right)\setminus\bigcup_{i=1}^{m}P_{i}\text{.}$

Thus it remains to prove that a given column $b\in\operatorname{IUm}_{2r}A$
resp. $\operatorname{IUm}_{2r+1}A$, with $b_{-2}$ being monic, can be
transformed by $\lceil{r-1\over 2}\rceil+2$ resp. $\lceil{r-1\over 2}\rceil+3$
transvections so that $b_{1}$ becomes monic, $b_{-2}$ remains monic, and the
row $(b_{1},\ldots,b_{r-1})$ becomes unimodular in $R[y]$.

First we consider the even case. Here we perform the following steps.

Step 1. Make the polynomial $b_{1}$ monic and the row $(b_{1},\ldots,b_{-2})$
unimodular in $R[y]$ by the transvection $T_{1,-2}(\xi)$.

Since $R$ is a product of fields and $b_{-2}$ is monic, it follows that the
ring $R[y]/\langle b_{1},b_{3},\ldots,b_{-2}\rangle$ is semilocal; hence this
ring satisfies $\mathrm{SR}_{2}$. Then there exists $\widetilde{\xi}$ such
that the row $(b_{1},b_{2}+\widetilde{\xi}b_{-1},b_{3},\ldots,b_{-2})$ is
unimodular in $R[y]$. Therefore, if we take

$\xi=\widetilde{\xi}+y^{K}b_{-2}$

for some $K\in\mathbb{N}$, then we guarantee that the row
$(b_{1},\ldots,b_{-2})$ becomes unimodular in $R[y]$. It remains to notice
that if $K$ is large enough, then we also make $b_{1}$ monic.

Step 2. If $r$ is even, make the row $(b_{1},\ldots,b_{-3})$ unimodular in
$R[y]$ by the transvection $T_{-3,-2}(\xi)$. If $r$ is odd, then skip this
step.

Since $b_{1}$ is now monic, it follows that the ring $R[y]/\langle
b_{1},\ldots,b_{-4}\rangle$ is semilocal; hence this ring satisfies
$\mathrm{SR}_{2}$. Therefore, there exists an element $\xi$ with the required
property.

Step 3. Make the row $(b_{1},\ldots,b_{r})$ unimodular in $R[y]$ by
$\lfloor{r-1\over 2}\rfloor$ transvections.

It is enough to show that if the row $(b_{1},\ldots,b_{-i+2})$ is unimodular
in $R[y]$, then we can make the row $(b_{1},\ldots,b_{-i})$ unimodular in
$R[y]$ by the transvection $T_{i-1,-i+2}(\xi)$, where $4\leqslant i\leqslant
r+1$. Here if $i=r+1$, then by $(b_{1},\ldots,b_{-i})$ we mean
$(b_{1},\ldots,b_{r})$.

Since $b_{1}$ is now monic, it follows that the ring $R[y]/\langle
b_{1},\ldots,b_{i-3},b_{i}\ldots,b_{-i}\rangle$ is semilocal. Let
$\mathfrak{M}_{1}$,$\ldots$, $\mathfrak{M}_{l}$ be its maximal ideals. Then it
is enough to choose $\xi$ to be contained in those and only those
$\mathfrak{M}_{j}$ that do not contain the image of the ideal $\langle
b_{i-2},b_{i-1}\rangle$.

Step 4. Make the row $(b_{1},\ldots,b_{r-1})$ unimodular in $R[y]$ by the
transvection $T_{r-1,r}(\xi)$.

This step is similar to Step 2.

It remains to note that neither of those steps change $b_{-2}$. Hence it
remains unimodular.

The proof in the odd case is similar. However, here we need an additional step
at the begining

Step 0. Make the row $(b_{1},\ldots,b_{r},b_{-r},\ldots,b_{-1})$ unimodular in
$R[y]$ by the transvection $T_{1,0}(\xi)$.

This step is similar to Steps 2 and 4. ∎

Now we are ready to prove Proposition 4.3.

For simplicity, we will write

$\textstyle{v\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{1}}$$\scriptstyle{N_{2}}$$\textstyle{w}$

instead of

$w\in EO(2r,R)^{\leqslant N_{1}}v\qquad\text{resp.}\qquad w\in
EO(2r+1,R)^{\leqslant N_{2}}v\text{,}$

where $v$ and $w$ are columns.

Applying Lemmas 2.1 and 6.6 $\delta$ times, we obtain elements
$s_{1}$,$\ldots$,$s_{\delta}\in B$ and columns
$b=b^{(0)}$,$b^{(1)}$,$\ldots$,$b^{(\delta)}\in\operatorname{IUm}_{2r}A$ resp.
$\operatorname{IUm}_{2r+1}A$ such that, firstly,

$\textstyle{b^{(i)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lceil{r-1\over
2}\rceil+2}$$\scriptstyle{\lceil{r-1\over
2}\rceil+3}$$\textstyle{b^{(i+1)}}$$\textstyle{i=0,\ldots,\delta-1\text{,}}$

secondly, $s_{i}\in\langle b^{(i)}_{1},\ldots,b^{(i)}_{r-1}\rangle$ for
$i=1,\ldots,\delta$, and thirdly, $\operatorname{BSdim}B/(s_{1},\ldots
s_{i+1})<\operatorname{BSdim}B/(s_{1},\ldots,s_{i})$ for
$i=0,\ldots,\delta-1$. In particular, the elements
$s_{1}$,$\ldots$,$s_{\delta}$ generate the unit ideal.

By Lemma 6.5 we have

$\textstyle{b^{(i)}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r(r-1)/2+8r-4}$$\scriptstyle{r(r-1)/2+9r-5}$$\textstyle{b^{(i)}(y+s_{i}^{m_{i}}z)}$

in $A[z]$.

Therefore, we have the following chain of transformations in
$A[z_{1},\ldots,z_{\delta}]$:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.05574pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b=b^{(0)}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
48.05574pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
48.05574pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{b^{(1)}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
102.54234pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.54234pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{b^{(1)}(y+s_{1}^{m_{1}}z_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
184.62889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
184.62889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{b^{(2)}(y+s_{1}^{m_{1}}z_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
266.71544pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
266.71544pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ldots}$}}}}}}}\ignorespaces}}}}\ignorespaces\\\
\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
53.57106pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{b^{(\delta)}(y+s_{1}^{m_{1}}z_{1}+\ldots+s_{\delta-1}^{m_{\delta-1}}z_{\delta-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
77.57106pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
77.57106pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{b^{(\delta)}(y+s_{1}^{m_{1}}z_{1}+\ldots+s_{\delta}^{m_{\delta}}z_{\delta})\text{.}}$}}}}}}}\ignorespaces}}}}\ignorespaces$

Thus we have

$\textstyle{b(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\delta\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+8r-2\right)}$$\scriptstyle{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\delta\left(\tfrac{r(r-1)}{2}+\lceil\tfrac{r-1}{2}\rceil+9r-2\right)}$$\textstyle{b^{(\delta)}(y+s_{1}^{m_{1}}z_{1}+\ldots+s_{\delta}^{m_{\delta}}z_{\delta})\text{.}}$

Since the elements $s_{1}$,$\ldots$,$s_{\delta}$ generate the unit ideal, it
follows that so do the elements
$s_{1}^{m_{1}}$,$\ldots$,$s_{\delta}^{m_{\delta}}$. Specializing the
indeterminates $z_{i}$ to elements in $yB$, we make
$y+s_{1}^{m_{1}}z_{1}+\ldots+s_{\delta}^{m_{\delta}}z_{\delta}$ equal to zero;
this concludes the proof of Proposition 4.3.

## Acknowledgment

The author is a participant of a scientific group that won "Leader" grant by
"BASIS" foundation in 2020, grant #20-7-1-27-4. Research is also supported by
«Native towns», a social investment program of PJSC ’’Gazprom Neft’’.

## References

  * [1] _Abe E._ Whitehead groups of Chevalley groups over polynomial rings // _Comm. Algebra_. —  1983\. —  Vol. 11, no. 12. —  p. 1271–1307.
  * [2] _Bak A._ Nonabelian K-theory: The nilpotent class of $\mathrm{K}_{1}$ and general stability // _K-Theory_. —  1991\. —  Vol. 4. —  p. 363–397.
  * [3] _Bass H._ K-theory and stable algebra // _Publ. Math. Inst. Hautes Études Sci_. —  1964\. —  Vol. 22. —  p. 5–60.
  * [4] _Carter D., Keller G. E._ Bounded elementary generation of $\operatorname{SL}_{n}(\mathcal{O})$ // _Amer. J. Math._ —  1983\. —  Vol. 105. —  p. 673–687.
  * [5] _Carter D., Keller G. E._ Elementary expressions for unimodular matrices // _Comm. Algebra_. —  1984\. —  Vol. 12. —  p. 379–389.
  * [6] Commutator width in Chevalley groups / R. Hazrat, A. V. Stepanov, N. A. Vavilov, Z. Zhang // _Note Mat._ —  2013\. —  Vol. 33, no. 1. —  p. 139–170. http://arxiv.org/abs/1206.2128.
  * [7] _Erovenko I. V., Rapinchuk A. S._ Bounded generation of some $S$-arithmetic orthogonal groups // _C. R. Acad. Sci., Paris, Sér. A_. —  2001\. —  Vol. 333, no. 5. —  p. 395–398.
  * [8] _Erovenko I. V., Rapinchuk A. S._ Bounded generation of $S$-arithmetic subgroups of isotropic orthogonal groups over number fields // _J. Number Theory_. —  2006\. —  Vol. 119, no. 1. —  p. 28–48.
  * [9] _Ershov M., Jaikin-Zapirain A._ Property $(T)$ for noncommutative universal lattices // _Invent. Math._ —  2010\. —  Vol. 179, no. 2. —  p. 303–347.
  * [10] _Ershov M., Jaikin-Zapirain A., Kassabov M._ Property $(T)$ for groups graded by root systems. —  Providence, RI: American Mathematical Society (AMS), 2017. —  Vol. 1186. —  p. vi + 140.
  * [11] _Estes D., Ohm J._ Stable range in commutative rings // _J. Algebra_. —  1967\. —  Vol. 7. —  p. 343–362.
  * [12] _Grunewald F., Mennicke J., Vaserstein L._ On symplectic groups over polynomial rings // _Math. Z._ —  1991\. —  Vol. 206. —  p. 35–56.
  * [13] _Gvozdevsky P.B._ Improved ${K}_{1}$-stability for the embedding ${D}_{5}$ into ${E}_{6}$ // _Communications in Algebra_. —  2020\. —  Vol. 48, no. 11. —  p. 4922–4931.
  * [14] _Hermida J. A, Sanchez-Giralda T._ Linear Equations over Commutative Rings and Determinantal Ideals // _J. Algebra_. —  1986\. —  Vol. 99. —  p. 72–79.
  * [15] _Humphreys J. E._ Introduction to Lie algebras and representation theory. —  Springer, 1973.
  * [16] _Kassabov M._ Kazhdan constants for SL${}_{n}(\mathbb{Z})$ // _Int. J. Algebra Comput._ —  2005\. —  Vol. 15, no. 5-6. —  p. 971–995.
  * [17] _Kopeiko V. I._ The stabilization of symplectic groups over a polynomial ring // _Math. of the USSR – Sbornik_. —  1978\. —  Vol. 34, no. 5. —  p. 655–669.
  * [18] _Kopeiko V. I._ On the structure of the symplectic group of polynomial rings over regular rings of dimension $\leqslant 1$ // _Russian Math. Surveys_. —  1992\. —  Vol. 47, no. 4. —  P. 210.
  * [19] _Kopeiko V. I._ On the structure of the special linear groups over Laurent polynomial rings // _Fundanientalnaya i prikladnaya matematika_. —  1995\. —  Т. 1. —  С. 1111–1114.
  * [20] _Kopeiko V. I._ Symplectic groups over Laurent polynomial rings and patching diagrams // _Fundanientalnaya i prikladnaya matematika_. —  1999\. —  Т. 5, № 3. —  С. 943–945.
  * [21] _Lubotzky A._ Subgroup growth and congruence subgroups // _Invent. Math._ —  1995\. —  Vol. 119. —  p. 267–295.
  * [22] _Magurn B. A., Wan dar Kallen, Vaserstein L. N._ Absolute stable rank and Witt cancellation for noncommutative rings // _Invent. Math._ —  1988\. —  Vol. 91. —  p. 525–542.
  * [23] _Morris D. W._ Bounded generation of $\mathop{\mathrm{SL}}\nolimits(n,A)$ (after D. Carter, G. Keller, and E. Paige) // _N. Y. J. Math_. —  2007\. —  Vol. 13. —  p. 383–421.
  * [24] _Platonov V.P., Rapinchuk A.S._ Abstract properties of S-arithmetic groups and the congruence subgroup problem // _Izv. Ross. Akad. Nauk. Ser. Mat_. —  1992\. —  Vol. 56. —  p. 483–508.
  * [25] _Plotkin E. B._ Stability theorems for ${K}_{1}$ -functors for Chevalley groups // Proceedings of the Conference on Nonassociative Algebras and Related Topics, Hiroshima. —  1991\.
  * [26] _Plotkin E. B._ Surjective stabilization for ${K}_{1}$ -functor for some exceptional Chevalley groups // _J. Soviet Math._ —  1993\. —  Vol. 64. —  p. 751–767.
  * [27] _Plotkin E. B._ On the Stability of the ${K}_{1}$ -Functor for Chevalley Groups of Type ${E}_{7}$ // _J. Algebra_. —  1998\. —  Vol. 210. —  p. 67–85.
  * [28] _Shalom Y._ Bounded generation and Kazhdan’s property (T) // _Publ. Math. Inst. Hautes Études Sci_. —  1999\. —  Vol. 90. —  p. 145–168.
  * [29] _Shalom Y._ The algebraization of Kazhdan’s property (T) // Proceedings of the International Congress of Mathematicians, Madrid, Spain. —  2006\. —  p. 1283–1310.
  * [30] _Shalom Y., Willis G._ Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity // _Geom. Funct. Anal_. —  2013\. —  Vol. 23, no. 5. —  p. 1631–1683.
  * [31] _Sivatski A., Stepanov A._ On the word length of commutators in $\mathop{\mathrm{GL}}\nolimits_{n}(R)$ // _K-Theory_. —  1999\. —  Vol. 17. —  p. 295–302.
  * [32] _Stavrova A. K._ Homotopy invariance of non-stable $K_{1}$-functors // _J. K-Theory_. —  2014\. —  Vol. 13. —  p. 199–248.
  * [33] _Stavrova A. K._ Chevalley groups of polynomial rings over Dedekind domains // _Journal of Group Theory_. —  2020\. —  Vol. 23, no. 1. —  p. 121–132. https://doi.org/10.1515/jgth-2019-0100.
  * [34] _Stein M. R._ Stability theorems for $K_{1}$, $K_{2}$ and related functors modeled on Chevalley groups // _Japan J. Math_. —  1978\. —  Vol. 4. —  p. 77–108.
  * [35] _Stein M. R._ Matsumoto’s solution of the congruence subgroup problem and stability theorems in algebraic K-theory // Proceedings of the 19th Meeting Algebra Section Math. Soc. Japan. —  1983\. —  p. 32–44.
  * [36] _Stepanov A., Vavilov N._ Length of commutators in Chevalley groups // _Israel J. Math._ —  2011\. —  Vol. 185. —  p. 253–276.
  * [37] _Stepanov A. V., Vavilov N. A._ Decomposition of transvections: Theme with variations // _K-Theory_. —  2000\. —  Vol. 19, no. 2. —  p. 109–153.
  * [38] _Suslin A. A._ On the Structure of the Special Linear Group over Polynomial Rings // _Math. USSR. Izvestija_. —  1977\. —  Vol. 11. —  p. 221–238.
  * [39] _Suslin A. A., Kopeiko V. I._ Quadratic Modules and Orthogonal Group over Polynomial Rings // _J. Soviet Math._ —  1982\. —  Vol. 20, no. 6. —  p. 2665–2691.
  * [40] _Tavgen O. I._ Bounded generation of Chevalley groups over rings of algebraic $S$-integers // _Math. USSR. Izvestija_. —  1991\. —  Vol. 36, no. 1. —  p. 101–128.
  * [41] _Tavgen O. I._ Bounded generation of normal and twisted Chevalley groups over the rings of $S$-integers // _Contemp. Math._ —  1992\. —  Vol. 131, no. 1. —  p. 409–421.
  * [42] _V. Morgan A., S. Rapinchuk A., Sury B._ Bounded generation of $\mathop{\mathrm{SL}}\nolimits_{2}$ over rings of S-integers with infinitely many units // _Algebra and Number Theory_. —  2017\. —  Vol. 12.
  * [43] _Vaserstein L. N._ On the stabilization of the general linear group over a ring // _Sb. Math._ —  1969\. —  Vol. 8, no. 3. —  p. 383–400.
  * [44] _Vaserstein L. N._ Bounded reduction of invertible matrices over polynomial rings by addition operations // _Unpublished_. —  2006\. http://www.personal.psu.edu/lxv1/pm2.pdf.
  * [45] _Vaserstein L. N., Suslin A. A._ Serre’s problem on projective modules over polynomial rings, and algebraic K-theory // _Math. USSR. Izvestija_. —  1976\. —  Vol. 10, no. 5. —  p. 937–1001.


# Narayana’s cows numbers which are concatenations of three repdigits in base
$\rho$

Pagdame Tiebekabe and Kouèssi Norbert Adédji

###### Abstract

Narayana’s sequence is a ternary recurrent sequence defined by the recurrence
relation $\mathcal{N}_{n}=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}$ with initial
terms $\mathcal{N}_{0}=0$ and
$\mathcal{N}_{1}=\mathcal{N}_{2}=\mathcal{N}_{3}=1$. Let $\rho\geqslant 2$ be
a positive integer. In this study, it is proved that the $n$th Narayana’s
number $\mathcal{N}_{n}$ which is concatenations of three repdigits in base
$\rho$ satisfies $n<5.6\cdot 10^{48}\cdot\log^{7}\rho$. Moreover, it is shown
that the largest Narayana’s number which is concatenations of three repdigits
in base $\rho$ with $1\leqslant\rho\leqslant 10$ is
$58425=\mathcal{N}_{31}=\overline{3332200}_{5}=\overline{332223}_{7}.$

Keywords: Narayana’s numbers, base $\rho,$ concatenations, repdigits, linear
forms in logarithms, reduction method.
2020 Mathematics Subject Classification: 11B39, 11J86, 11D61.

##  1 Introduction

The power of linear forms in logarithms in solving Diophantine exponential
equations in general and Diophantine equations in linear recurrent sequences
in particular is well established. Since the introduction of this powerful
tool by the British mathematician Alan Baker in $1966$ by proving a landmark
result about linear forms in the logarithms of algebraic numbers, which helped
to earn him the Fields Medal in $1970$, several exponential Diophantine
equations have been solved. Since few years now, researchers in number theory
have been interested in the resolution of Diophantine equations in linear
recurrent sequences. Variants of these equations have been intensely studied.
The determination of the terms of linear recurrent sequences which are product
or concatenations of two or three repdigits have been widely studied. For
details of recent work related to this problem, we refer the reader to [4, 13,
2, 1, 5, 7, 8, 9, 10].

Our work aims to be more general than previous work. In this article,
following what was done by Mahadi Ddamulira et al [4], we study the problem of
writing all Narayana numbers which are concatenations of three repdigits in
base $\rho$ with $\rho\geq 2$. To be precise, we find all solutions of the
Diophantine equation

$\mathcal{N}_{n}=\overline{\underbrace{d_{1}\ldots
d_{1}}_{\ell\leavevmode\nobreak\ times}}\overline{\underbrace{d_{2}\ldots
d_{2}}_{m\leavevmode\nobreak\ times}}\overline{\underbrace{d_{3}\ldots
d_{3}}_{k\leavevmode\nobreak\ times}}$ (1)

in non-negative integers $n,d_{1},d_{2},d_{3},\ell,m,k$ with $n\geqslant
0,1\leqslant k\leqslant m\leqslant\ell$ and
$d_{1},d_{2},d_{3}\in\\{0,1,2,3,\ldots,9\\},d_{1}>0$. Contrary to most
previous works on this question done in base $10$, we determined an upper
bound of $n$ and $\ell+m+k$ in terms of $\rho$ with $\rho\geqslant 2$. For
application, we found in section $4$ all Narayana’s cows numbers that are
concatenations of three repdigits in the cases $2\leqslant\rho\leqslant 10$.
The main results are as follows :

######  Theorem 1.1.

Let $\rho\geqslant 2$ be an integer. All solutions to the Diophantine equation
(1) satisfy

$\ell+m+k<n<5.6\cdot 10^{48}\cdot\log^{7}\rho.$

Moreover, the above result implies.

######  Corollary 1.1.

The Diophantine equation (1) has only finitely many solutions in positive
integers $n,d_{1},d_{2},d_{3},\ell,m$ and $k.$

By considering the cases $2\leqslant\rho\leqslant 10,$ we get the following
result.

######  Theorem 1.2.

The only Nayarana numbers which are concatenations of three repdigits in base
$\rho$ with $2\leq\rho\leq 10$ are

$4,6,9,13,19,28,41,60,88,129,189,277,406,595,872,1278,1873,2745,4023,18560,58425.$

More precisely, we have: $4=\mathcal{N}_{6}=\overline{100}_{2},$
$6=\mathcal{N}_{7}=\overline{110}_{2},$
$13=\mathcal{N}_{9}=\overline{111}_{3}=\overline{1101}_{2},$
$19=\mathcal{N}_{10}=\overline{10011}_{2}=\overline{201}_{3}=\overline{103}_{4},$

$\begin{array}[]{ccccl}28&=&\mathcal{N}_{11}&=&\overline{1001}_{3}=\overline{130}_{4}=\overline{103}_{5}=\overline{11100}_{2},\\\
41&=&\mathcal{N}_{12}&=&\overline{1112}_{3}=\overline{221}_{4}=\overline{131}_{5},\\\
60&=&\mathcal{N}_{13}&=&\overline{111100}_{2}=\overline{330}_{4}=\overline{220}_{5}=\overline{140}_{6}=\overline{114}_{7},\\\
88&=&\mathcal{N}_{14}&=&\overline{323}_{5}=\overline{224}_{6}=\overline{154}_{7}=\overline{130}_{8}=\overline{1120}_{4}=\overline{107}_{9},\\\
129&=&\mathcal{N}_{15}&=&\overline{333}_{6}=\overline{243}_{7}=\overline{201}_{8}=\overline{1004}_{5}=\overline{2001}_{4}=\overline{153}_{9}=\overline{10000001}_{2}=\overline{129}_{10},\\\
189&=&\mathcal{N}_{16}&=&\overline{2331}_{4}=\overline{21000}_{3}=\overline{360}_{7}=\overline{230}_{9}=\overline{1224}_{5}=\overline{513}_{6}=\overline{275}_{8}=\overline{189}_{10},\\\
277&=&\mathcal{N}_{17}&=&\overline{1141}_{6}=\overline{544}_{7}=\overline{425}_{8}=\overline{337}_{9}=\overline{10111}_{4},\\\
406&=&\mathcal{N}_{18}&=&\overline{3111}_{5}=\overline{626}_{8}=\overline{1120}_{7}=\overline{501}_{9}=\overline{406}_{10},\\\
595&=&\mathcal{N}_{19}&=&\overline{1123}_{8}=\overline{731}_{9}=\overline{595}_{10},\\\
872&=&\mathcal{N}_{20}&=&\overline{11442}_{5}=\overline{1168}_{9}=\overline{1550}_{8}=\overline{872}_{10},\\\
1278&=&\mathcal{N}_{21}&=&\overline{5530}_{6},\\\
1873&=&\mathcal{N}_{22}&=&\overline{24443}_{5}=\overline{2511}_{9},\\\
2745&=&\mathcal{N}_{23}&=&\overline{11001}_{7},\\\
4023&=&\mathcal{N}_{24}&=&\overline{7667}_{8},\\\
18560&=&\mathcal{N}_{28}&=&\overline{44200}_{8},\\\
58425&=&\mathcal{N}_{31}&=&\overline{3332200}_{5}=\overline{332223}_{7}.\end{array}$

For the proof of Theorem 1.1, we use $n>560$ by writing Diophantine (11) in
three different ways to get three linear forms in logarithms. Next we apply
lower bound for linear forms in logarithms to obtain an upper bound on $n$. To
do this we need some preliminary results which are discussed in the following
section.

##  2 Preliminary results

###  2.1 Some properties of Narayana’s cows sequence

Narayana’s cows sequence comes from a problem with cows proposed by the Indian
mathematician Narayana in the $14$th century. In this problem, we assume that
there is a cow at the beginning and each cow produces a calf every year from
the $4$ years old. Narayana’s cow problem counts the number of calves produced
each year [3]. This problem seems to be similar to the Fibonacci rabbit
problem. So also the answers, known as Narayana sequence and Fibonacci
sequence.

Narayana’s cows sequence (A000930 in OEIS) satisfies a third-order recurrence
relation :

$\mathcal{N}_{n}=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}\,\text{for}\,n\geqslant
3.$

This has the initial values $\mathcal{N}_{0}$ and
$\mathcal{N}_{1}=\mathcal{N}_{2}=\mathcal{N}_{3}=1$ [3]. Explicitly, the
characteristic polynomial of $\mathcal{N}_{n}$ is :

$\varphi(x)=x^{3}-x^{2}-1,$

and the characteristic roots are:

$\displaystyle\alpha=$
$\displaystyle\dfrac{1}{3}\left(\sqrt[3]{\dfrac{1}{2}(29-3\sqrt{93})}+\sqrt[3]{\dfrac{1}{2}(3\sqrt{93}+29)}+1\right),$
(2) $\displaystyle\beta=$
$\displaystyle\dfrac{1}{3}-\dfrac{1}{6}\left(1-i\sqrt{3}\right)\sqrt[3]{\dfrac{1}{2}(29-3\sqrt{93})}-\dfrac{1}{6}\left(1+i\sqrt{3}\right)\sqrt[3]{\dfrac{1}{2}(3\sqrt{93}+29)},$
(3) $\displaystyle\gamma=$
$\displaystyle\dfrac{1}{3}-\dfrac{1}{6}\left(1+i\sqrt{3}\right)\sqrt[3]{\dfrac{1}{2}(29-3\sqrt{93})}-\dfrac{1}{6}\left(1-i\sqrt{3}\right)\sqrt[3]{\dfrac{1}{2}(3\sqrt{93}+29)}.$
(4)

Then, the Narayana sequence can be obtained by Binet’s formula:

$\mathcal{N}_{n}=a\alpha^{n}+b\beta^{n}+c\gamma^{n}.$ (5)

For $n\in\mathbb{Z}_{\geqslant 0}$ the generating function of the Narayana’s
cows sequence is

$g(x)=\dfrac{1}{1-x-x^{3}}=\sum_{n=0}^{\infty}\mathcal{N}_{n+1}x^{n}.$

With the Vieta theorem, we have:

$\left\\{\begin{array}[]{lllll}\alpha+\beta+\gamma=1,\\\
\alpha\beta+\beta\gamma+\alpha\gamma=0,\\\
\alpha\beta\gamma=1.\end{array}\right.$ (6)

From formula (5), we otain

$\displaystyle\mathcal{N}_{0}$ $\displaystyle=a+b+c=0,$
$\displaystyle\mathcal{N}_{1}$ $\displaystyle=a\alpha+b\beta+c\gamma=1,$
$\displaystyle\mathcal{N}_{2}$
$\displaystyle=a\alpha^{2}+b\beta^{2}+c\gamma^{2}=1,$

which implies

$a=\dfrac{1-\beta-\gamma}{(\alpha-\beta)(\alpha-\gamma)},\quad
b=\dfrac{1-\alpha-\gamma}{(\beta-\alpha)(\beta-\gamma)},\quad\text{and}\quad
c=\dfrac{1-\alpha-\beta}{(\gamma-\beta)(\gamma-\alpha)}.$

Also, from formula (6), we can simplify $a,b$ and $c$ and obtain

$a=\dfrac{\alpha}{\alpha^{2}-\alpha\beta-\alpha\gamma+\beta\gamma}=\dfrac{\alpha}{\alpha^{2}+2\beta\gamma}=\dfrac{\alpha^{2}}{\alpha^{3}+2}$
$b=\dfrac{\beta^{2}}{\beta^{3}+2}\quad\text{and}\quad
c=\dfrac{\gamma^{2}}{\gamma^{3}+2}$ (7)

and the minimal polynomial of $a$ over integers is $31x^{3}-3x-1$. Setting
$\Pi(n)=\mathcal{N}_{n}-a\alpha^{n}=b\beta^{n}+c\gamma^{n}$, we notice that
$|\Pi(n)|<\dfrac{1}{\alpha^{n/2}}$ for all $n>1$.

The Narayana’s sequence was originally defined at positive indices. Actually,
it can be extended to negative indices by defining

$\mathcal{N}_{-n}=\dfrac{a}{\alpha^{n}}+\dfrac{b}{\beta^{n}}+\dfrac{c}{\gamma^{n}}.$
(8)

So, the following recurrence relation holds for all integral indices

$\mathcal{N}_{n}=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}\quad\text{for}\quad
n\in\mathbb{Z}.$ (9)

Through a simple computation, the first few terms of $\mathcal{N}_{n}$ at
negative indices can be obtained from formulas (7) and (8), so that
$\mathcal{N}_{-1}=0,\,\mathcal{N}_{-2}=1,\,\mathcal{N}_{-3}=0,\,\mathcal{N}_{-4}=-1$,
which also satisfies relation (9).

The characteristic polynomial for $(\mathcal{N}_{n})_{n\geqslant 0}$ is
irreducible in $\mathbb{Q}[x]$. We note that it has a real zero $\alpha(>1)$
and two conjugate complex zeros $\beta$ and $\gamma$ with
$|\beta|=|\gamma|<1$. In fact, $\alpha\approx 1.46557$. We also have the
following properties of $(\mathcal{N}_{n})_{n\geqslant 0}$.

######  Lemma 2.1.

For the sequence $(\mathcal{N}_{n})_{n\geqslant 0}$, we have

$\alpha^{n-2}\leqslant\mathcal{N}_{n}\leqslant\alpha^{n-1},\quad\text{for}\quad
n\geqslant 1.$

###### Proof.

Using induction, one can easily prove Lemma 2.1. ∎

Let $\mathbb{K}:=\mathbb{Q}(\alpha,\beta)$ be the splitting field of the
characteristic polynomial $\varphi$ over $\mathbb{Q}$. Then
$[\mathbb{K}:\mathbb{Q}]=6$ and $[\mathbb{Q}(\alpha):\mathbb{Q}]=3.$ The
Galois group of $\mathbb{K}/\mathbb{Q}$ is given by

$\mathcal{G}:=\text{Gal}(\mathbb{K}/\mathbb{Q})\cong\\{(1),(\alpha\beta),(\alpha\gamma),(\beta\gamma),(\alpha\beta\gamma)\\}\cong
S_{3}.$

We identify the automorphisms of $\mathcal{G}$ with the permutation group of
the zeroes of $\varphi$. We highlight the permutation $(\alpha\beta)$,
corresponding to the automorphism
$\sigma:\alpha\mapsto\beta,\beta\mapsto\alpha,\gamma\mapsto\gamma$, which we
use later to obtain a contradiction on the size of the absolute value of
certain bound.

###  2.2 Linear forms in Logarithms

We begin this section with a few reminders about logarithmic height of an
algebraic number. Let $\eta$ be an algebraic number of degree $d,$ let
$a_{0}>0$ be the leading coefficient of its minimal polynomial over
$\mathbb{Z}$ and let $\eta=\eta^{(1)},\ldots,\eta^{(d)}$ denote its
conjugates. The quantity defined by

$h(\eta)=\frac{1}{d}\left(\log|a_{0}|+\sum_{j=1}^{d}\log\max\left(1,\left|\eta^{(j)}\right|\right)\right)$

is called the logarithmic height of $\eta.$ Some properties of height are as
follows. For $\eta_{1},\eta_{2}$ algebraic numbers and $m\in\mathbb{Z},$ we
have

$\displaystyle h(\eta_{1}\pm\eta_{2})$ $\displaystyle\leq
h(\eta_{1})+h(\eta_{2})+\log 2,$ $\displaystyle h(\eta_{1}\eta_{2}^{\pm 1})$
$\displaystyle\leq h(\eta_{1})+h(\eta_{2}),$ $\displaystyle h(\eta_{1}^{m})$
$\displaystyle=|m|h(\eta_{1}).$

If $\eta=\dfrac{p}{q}\in\mathbb{Q}$ is a rational number in reduced form with
$q>0,$ then the above definition reduces to $h(\eta)=\log(\max\\{|p|,q\\}).$
We can now present the famous Matveev result used in this study. Thus, let
$\mathbb{L}$ be a real number field of degree $d_{\mathbb{L}}$,
$\eta_{1},\ldots,\eta_{s}\in\mathbb{L}$ and
$b_{1},\ldots,b_{s}\in\mathbb{Z}\setminus\\{0\\}.$ Let
$B\geq\max\\{|b_{1}|,\ldots,|b_{s}|\\}$ and

$\Lambda=\eta_{1}^{b_{1}}\cdots\eta_{s}^{b_{s}}-1.$

Let $A_{1},\ldots,A_{s}$ be real numbers with

$A_{i}\geq\max\\{d_{\mathbb{L}}h(\eta_{i}),|\log\eta_{i}|,0.16\\},\quad
i=1,2,\ldots,s.$

With the above notations, Matveev proved the following result [12].

######  Theorem 2.1.

Assume that $\Lambda\neq 0.$ Then

$\log|\Lambda|>-1.4\cdot 30^{s+3}\cdot s^{4.5}\cdot
d_{\mathbb{L}}^{2}\cdot(1+\log d_{\mathbb{L}})\cdot(1+\log B)\cdot A_{1}\cdots
A_{s}.$

We also need the following result of [11].

######  Lemma 2.2.

Let $r\geqslant 1$ and $H>0$ be such that $H>(4r^{2})^{r}$ and $H>L/(\log
L)^{r}$. Then

$L<2^{r}H(\log H)^{r}.$

###  2.3 Reduction method

The bounds on the variables obtained via Baker’s theory are too large for any
computational purposes. To reduce the bounds, we use reduction method due to
Dujella and Pethő ([6], Lemma 5a). For a real number $X$, we write
$\left\lVert X\right\rVert:=\min\\{\mid X-n\mid:n\in\mathbb{Z}\\}$ for the
distance of $X$ to the nearest integer.

######  Lemma 2.3 (Dujella and Pethő, [6]).

Let $M$ be a positive integer, $p/q$ be a convergent of the continued fraction
expansion of the irrational number $\tau$ such that $q>6M,$ and $A,B,\mu$ be
some real numbers with $A>0$ and $B>1.$ Furthermore, let

$\varepsilon:=\parallel\mu q\parallel-M\cdot\parallel\tau q\parallel.$

If $\varepsilon>0,$ then there is no solution to the inequality

$0<|u\tau-v+\mu|<AB^{-w}$ (10)

in positive integers $u,v$ and $w$ with

$u\leqslant M\;\mbox{and}\;w\geqslant\frac{\log(Aq/\varepsilon)}{\log B}.$

The following results will also be used in our proof.

######  Lemma 2.4.

For any non-zero real number $x$, if $|e^{x}-1|<\dfrac{1}{2}$, then
$|x|<2|e^{x}-1|$.

##  3 Proof of Theorem 1.1

For the proof of Theorem 1.1 we can assume that $n>560.$

###  3.1 The initial bound on $n$.

To begin with, we consider the Diophantine equation (1), and rewrite it as

$\displaystyle\mathcal{N}_{n}=$
$\displaystyle\overline{\underbrace{d_{1}\ldots
d_{1}}_{\ell\leavevmode\nobreak\ times}}\overline{\underbrace{d_{2}\ldots
d_{2}}_{m\leavevmode\nobreak\ times}}\overline{\underbrace{d_{3}\ldots
d_{3}}_{k\leavevmode\nobreak\ times}}$ $\displaystyle=$
$\displaystyle\overline{\underbrace{d_{1}\ldots
d_{1}}_{\ell\leavevmode\nobreak\ times}}\cdot
10^{m+k}+\overline{\underbrace{d_{2}\ldots d_{2}}_{m\leavevmode\nobreak\
times}}\cdot 10^{k}+\overline{\underbrace{d_{3}\ldots
d_{3}}_{k\leavevmode\nobreak\ times}}$ $\displaystyle=$ $\displaystyle
d_{1}\left(\dfrac{\rho^{\ell}-1}{\rho-1}\right)\cdot\rho^{m+k}+d_{2}\left(\dfrac{\rho^{m}-1}{\rho-1}\right)\cdot\rho^{k}+d_{3}\left(\dfrac{\rho^{k}-1}{\rho-1}\right).$

Therefore, we have

$\mathcal{N}_{n}=\dfrac{1}{\rho-1}\left(d_{1}\rho^{\ell+m+k}-(d_{1}-d_{2})\rho^{m+k}-(d_{2}-d_{3})\rho^{k}-d_{3}\right).$
(11)

We state and prove the following lemma which gives a relation between $n$ and
$l+m+k$ of (11).

######  Lemma 3.1.

All solutions to the Diophantine equation (1) satisfy

$(\ell+m+k-1)\log\rho+\log\alpha<n\log\alpha<(\ell+m+k)\log\rho+1.$

###### Proof.

From (1) and Lemma 2.1, we get

$\alpha^{n-2}\leqslant\mathcal{N}_{n}<\rho^{\ell+m+k}.$

Taking logarithm on both sides, we have

$(n-2)\log\alpha<(\ell+m+k)\log\rho$

which leads to

$n\log\alpha<(\ell+m+k)\log\rho+2\log\alpha<(\ell+m+k)\log\rho+1.$ (12)

On the other hand, for the lower bound, (11) implies that

$\rho^{\ell+m+k-1}<\mathcal{N}_{n}\leqslant\alpha^{n-1}.$

Taking logarithm on both sides, we get

$(\ell+m+k-1)\log\rho<(n-1)\log\alpha,$

which leads to

$(\ell+m+k-1)\log\rho+\log\alpha<n\log\alpha.$ (13)

Comparing (12) and (13) completes the proof of Lemma 3.1. ∎

####  3.1.1 Upper bound for $\ell\log\rho$ in terms of $n$

######  Lemma 3.2.

All solutions to (1) satisfy

$\ell\log\rho<3.74\cdot 10^{13}\cdot\log^{2}\rho\cdot(1+\log n).$

###### Proof.

Using (5) and (11), we have

$a\alpha^{n}+b\beta^{n}+c\gamma^{n}=\dfrac{1}{\rho-1}\left(d_{1}\cdot\rho^{\ell+m+k}-(d_{1}-d_{2})\cdot\rho^{m+k}-(d_{2}-d_{3})\cdot\rho^{k}-d_{3}\right).$

Equivalently

$(\rho-1)a\alpha^{n}-d_{1}\cdot\rho^{\ell+m+k}=-(\rho-1)\Pi(n)-(d_{1}-d_{2})\cdot\rho^{m+k}-(d_{2}-d_{3})\cdot\rho^{k}-d_{3}.$

Thus, we have that

$\displaystyle\left|(\rho-1)a\alpha^{n}-d_{1}\cdot\rho^{\ell+m+k}\right|$
$\displaystyle=\left|-(\rho-1)\Pi(n)-(d_{1}-d_{2})\cdot\rho^{m+k}-(d_{2}-d_{3})\cdot\rho^{k}-d_{3}\right|$
$\displaystyle\leqslant(\rho-1)\cdot\alpha^{-n/2}+(\rho-1)\cdot\rho^{m+k}+(\rho-1)\cdot\rho^{k}+(\rho-1)$
$\displaystyle<3(\rho-1)\cdot\rho^{m+k},$

where we used the fact that $n>560$.

Dividing both sides of the inequality by $d_{1}\cdot\rho^{\ell+m+k}$ gives

$\left|\left(\dfrac{(\rho-1)a}{d_{1}}\right)\cdot\alpha^{n}\cdot\rho^{-(\ell+m+k)}-1\right|<\dfrac{3(\rho-1)\cdot\rho^{m+k}}{d_{1}\cdot\rho^{\ell+m+k}}<\dfrac{3}{\rho^{\ell-1}}.$
(14)

Let

$\Lambda_{1}:=\left(\dfrac{(\rho-1)a}{d_{1}}\right)\cdot\alpha^{n}\cdot\rho^{-(\ell+m+k)}-1.$
(15)

We then proceed to apply Theorem 2.1 on (15). We have first to observe that
$\Lambda_{1}\neq 0$. Indeed, if it were zero, we would then get that

$a\alpha^{n}=\dfrac{d_{1}}{\rho-1}\cdot\rho^{\ell+m+k}.$

In this case therefore, applying the automorphism $\sigma$ of the Galois group
$\mathcal{G}$ on both sides of the preceeding equation and taking absolute
values, we obtain

$\left|\left(\dfrac{d_{1}}{\rho-1}\cdot\rho^{\ell+m+k}\right)\right|=|\sigma(a\alpha^{n})|=|c\gamma^{n}|<1,$

which is false. Thus, we have $\Lambda_{1}\neq 0$. Theorem 2.1 is then applied
on (15) with the following parameters :

$\eta_{1}:=\dfrac{(\rho-1)a}{d_{1}},\,\eta_{2}:=\alpha,\,\eta_{3}:=\rho,\,b_{1}:=1,\,b_{2}:=n,\,b_{3}:=-\ell-
m-k,\,s:=3.$

From Lemma 3.1, we have $\ell+m+k<n$. Consequently, we choose $B:=n$. Notice
that $\mathbb{K}:=\mathbb{Q}(\eta_{1},\eta_{2},\eta_{3})=\mathbb{Q}(\alpha)$,
since $a=\dfrac{\alpha^{2}}{\alpha^{3}+2}$. Moreover its minimal polynomial
over integers is $31x^{3}-3x-1.$ Therefore,
$d_{\mathbb{K}}:=[\mathbb{K}:\mathbb{Q}]=3$.

Using the properties of the logarithmic height, we estimate $h(\eta_{1})$ as
follows :

$\displaystyle h(\eta_{1})$
$\displaystyle=h\left(\dfrac{(\rho-1)a}{d_{1}}\right)\leqslant
h(\rho-1)+h(a)+h(d_{1})$ $\displaystyle\leqslant 2\log\rho+\dfrac{1}{3}\log
31\leqslant 4\log\rho,$

which holds for $\rho\geqslant 2.$ Similarly, we have
$h(\eta_{2})=h(\alpha)=\dfrac{\log\alpha}{3}$ and
$h(\eta_{3})=h(\rho)=\log\rho$. Therefore, we choose

$A_{1}:=12\log\rho,\,A_{2}:=\log\alpha,\,\text{and}\,A_{3}:=3\log\rho.$

By Theorem 2.1, we get

$\displaystyle\log|\Lambda_{1}|$ $\displaystyle>-1.4\cdot 30^{6}\cdot
3^{4.5}\cdot 3^{2}\cdot(1+\log 3)\cdot(1+\log n)\cdot
12\log\rho\cdot\log\alpha\cdot 3\log\rho$ $\displaystyle>-3.73\cdot
10^{13}\cdot\log^{2}\rho\cdot(1+\log n)$

which when compared with (14) gives

$(\ell-1)\log\rho-\log 3<3.73\cdot 10^{13}\cdot\log^{2}\rho\cdot(1+\log n),$

leading to

$\ell\log\rho<3.74\cdot 10^{13}\cdot\log^{2}\rho\cdot(1+\log n).$ (16)

∎

####  3.1.2 Upper bound for $m\log\rho$ in terms of $n$

######  Lemma 3.3.

All solutions to (1) satisfy

$m\log\rho<3.5\cdot 10^{26}\cdot\log^{3}\rho\cdot(1+\log n)^{2}.$

###### Proof.

Rewriting (5), we obtain

$(\rho-1)a\alpha^{n}-(d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2}))\rho^{m+k}=-(\rho-1)\Pi(n)-(d_{2}-d_{3})\cdot\rho^{k}-d_{3},$

which shows that

$\displaystyle\left|(\rho-1)a\alpha^{n}-(d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2}))\rho^{m+k}\right|$
$\displaystyle=\left|-(\rho-1)\Pi(n)-(d_{2}-d_{3})\cdot\rho^{k}-d_{3}\right|$
$\displaystyle\leqslant(\rho-1)\cdot\alpha^{-n/2}+(\rho-1)\cdot\rho^{k}+\rho-1$
$\displaystyle<2\cdot\rho^{k+1}.$

Dividing both sides of the inequality by
$(d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2}))\rho^{m+k}$, we have that

$\left|\left(\dfrac{(\rho-1)a}{d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2})}\right)\cdot\alpha^{n}\cdot\rho^{-m-k}-1\right|<\dfrac{2\cdot\rho^{k+1}}{(d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2}))\rho^{m+k}}<\dfrac{2}{\rho^{m-1}}.$
(17)

Let

$\Lambda_{2}:=\left(\dfrac{(\rho-1)a}{d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2})}\right)\cdot\alpha^{n}\cdot\rho^{-m-k}-1.$

Using similar argument as in $\Lambda_{1}$, we apply Theorem 2.1 on
$\Lambda_{2}$. We notice that $\Lambda_{2}\neq 0$. If it were, then we would
have that

$a\alpha^{n}=\left(\dfrac{d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2})}{\rho-1}\right)\cdot\rho^{m+k}.$

Applying the automorphism $\sigma$ of the Galois group $\mathcal{G}$ on both
sides, and taking the absolute values, we obtain

$1<\left|\left(\dfrac{d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2})}{\rho-1}\right)\cdot\rho^{m+k}\right|=|\sigma(a\alpha^{n})|=|c\gamma^{n}|<1,$

which is false. Therefore, $\Lambda_{2}\neq 0$. We then proceed to apply
Theorem 2.1with the following parameters:

$\eta_{1}:=\dfrac{(\rho-1)a}{d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2})},\,\eta_{2}:=\alpha,\,\eta_{3}:=\rho,\,b_{1}:=1,\,b_{2}:=n,\,b_{3}:=-m-k,\,s=3.$

Since $m+k<n$, we take $B:=n$. Again taking
$\mathbb{K}:=\mathbb{Q}(\eta_{1},\eta_{2},\eta_{3})=\mathbb{Q}(\alpha)$, we
have that $d_{\mathbb{K}}:=[\mathbb{K}:\mathbb{Q}]=3$. Next, we use the
properties of the logarithmic height to estimate $h(\eta_{1})$ as before, and
obtain

$\displaystyle h(\eta_{1})$
$\displaystyle=h\left(\dfrac{(\rho-1)a}{d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2})}\right)$
$\displaystyle\leqslant h((\rho-1)a)+h(d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2}))$
$\displaystyle\leqslant h(\rho-1)+h(a)+\log
2+h(d_{1}\cdot\rho^{\ell})+h(d_{1}-d_{2})$
$\displaystyle\leqslant\dfrac{1}{3}\log 31+4\log(\rho-1)+\ell\log\rho+2\log 2$
$\displaystyle\leqslant 3.74\cdot 10^{13}\cdot\log^{2}\rho\cdot(1+\log
n)+4\log(\rho-1)+\dfrac{1}{3}\log 31+2\log 2$ $\displaystyle<3.75\cdot
10^{13}\cdot\log^{2}\rho\cdot(1+\log n),$

where we used the fact that $\ell\log\rho<3.74\cdot
10^{13}\cdot\log^{2}\rho\cdot(1+\log n).$ We then take

$A_{1}:=1.125\cdot 10^{14}\cdot\log^{2}\rho\cdot(1+\log
n),\,A_{2}:=\log\alpha,\,\text{and}\,A_{3}:=3\log\rho.$

Theorem 2.1 says that

$\displaystyle\log|\Lambda_{2}|$ $\displaystyle>-1.4\cdot 30^{6}\cdot
3^{4.5}\cdot 3^{2}\cdot(1+\log 3)\cdot(1+\log n)\cdot(1.125\cdot
10^{14}\cdot\log^{2}\rho\cdot(1+\log n))\cdot\log\alpha\cdot 3\log\rho$
$\displaystyle>-3.49\cdot 10^{26}\cdot\log^{3}\rho\cdot(1+\log n)^{2}$

which when compared with (17) gives

$(m-1)\log\rho-\log 2<3.49\cdot 10^{26}\cdot\log^{3}\rho\cdot(1+\log n)^{2},$

which simplifies to

$m\log\rho<3.5\cdot 10^{26}\cdot\log^{3}\rho\cdot(1+\log n)^{2}.$ (18)

∎

####  3.1.3 Upper bound for $\ell+m+k$ and $n$

Rewriting (5), we obtain

$(\rho-1)a\alpha^{n}-\left(d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})\right)\cdot\rho^{k}=-(\rho-1)\Pi(n)-d_{3},$

which shows that

$\displaystyle\left|(\rho-1)a\alpha^{n}-\left(d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})\right)\cdot\rho^{k}\right|$
$\displaystyle=|-(\rho-1)\Pi(n)-d_{3}|$
$\displaystyle\leqslant(\rho-1)\cdot\alpha^{-n/2}+(\rho-1)<2(\rho-1).$

Dividing both sides of the inequality by $(\rho-1)a\alpha^{n}$, we have that

$\left|\left(\dfrac{d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})}{(\rho-1)a}\right)\cdot\alpha^{-n}\cdot\rho^{k}-1\right|<\dfrac{2}{a\alpha^{n}}<\dfrac{5}{\alpha^{n}}.$
(19)

Now, we let

$\Lambda_{3}:=\left(\dfrac{d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})}{(\rho-1)a}\right)\cdot\alpha^{-n}\cdot\rho^{k}-1.$

Using similar arguments as in $\Lambda_{1}$ and $\Lambda_{2}$, we apply
Theorem 2.1 on $\Lambda_{3}$. We notice that $\Lambda_{3}\neq 0.$ If it were,
then we would have

$a\alpha^{n}=\left(\dfrac{d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})}{\rho-1}\right)\cdot\rho^{k}.$

Applying the automorphism $\sigma$ of the Galois group $\mathcal{G}$ on both
sides, and taking the absolute values, we obtain

$1<\left|\left(\dfrac{d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})}{\rho-1}\right)\cdot\rho^{k}\right|=|\sigma(a\alpha^{n})|=|c\gamma^{n}|<1,$

which is false. Therefore, $\Lambda_{3}\neq 0$. We proceed to apply Theorem
2.1 with the following parameters :

$\eta_{1}:=\left(\dfrac{d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})}{(\rho-1)a}\right),\,\eta_{2}:=\alpha,\,\eta_{3}:=\rho,\,b_{1}:=1,\,b_{2}:=-n,\,b_{3}:=k,\,s:=3.$

Since $k<n$, we take $B:=n$. Again, taking
$\mathbb{K}=\mathbb{Q}(\eta_{1},\eta_{2},\eta_{3})=\mathbb{Q}(\alpha)$, we
have that $d_{\mathbb{K}}:=[\mathbb{K}:\mathbb{Q}]=3$. Nest, we use the
properties of the logarithmic height to estimate $h(\eta_{1})$ as before, and
obtain:

$\displaystyle h(\eta_{1})$
$\displaystyle=h\left(\dfrac{d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})}{(\rho-1)a}\right)$
$\displaystyle\leqslant
h(d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3}))+h((\rho-1)a)$
$\displaystyle\leqslant
h(d_{1}\cdot\rho^{\ell+m})+h((d_{1}-d_{2})\cdot\rho^{m})+h(d_{2}-d_{3})+\log
2+h(\rho-1)+h(a)$ $\displaystyle\leqslant
h(d_{1})+(\ell+m)h(\rho)+h(d_{1}-d_{2})+mh(\rho)+h(d_{2}-d_{3})+\log
2+h(\rho-1)+h(a)$ $\displaystyle\leqslant
h(d_{1})+(\ell+m)h(\rho)+h(d_{1})+h(d_{2})+\log
2+mh(\rho)+h(d_{2})+h(d_{3})+\log 2+h(\rho-1)+h(a)$ $\displaystyle\leqslant
2h(d_{1})+2h(d_{2})+h(d_{3})+(2m+\ell)h(\rho)+h(\rho-1)+h(a)+2\log 2$
$\displaystyle<6\log(\rho-1)+2\times 3.5\cdot
10^{26}\cdot\log^{3}\rho\cdot(1+\log n)^{2}+3.74\cdot
10^{13}\cdot\log^{2}\rho\cdot(1+\log n)+\dfrac{1}{3}\log 31+2\log 2$
$\displaystyle<1.4\cdot 10^{27}\cdot\log^{3}\rho\cdot(1+\log n)^{2}.$

Note that in above estimate, we have used the estimates from (16) and (18). We
then take

$A_{1}:=4.2\cdot 10^{27}\cdot\log^{3}\rho\cdot(1+\log
n)^{2},\,A_{2}:=\log\alpha,\,\text{and}\,A_{3}:=3\log\rho.$

Theorem 2.1 says that

$\displaystyle\log|\Lambda_{3}|$ $\displaystyle>-1.4\cdot 30^{6}\cdot
3^{4.5}\cdot 3^{2}\cdot(1+\log 3)\cdot(1+\log n)\cdot(4.2\cdot
10^{27}\cdot\log^{3}\rho\cdot(1+\log n)^{2})\cdot\log\alpha\cdot 3\log\rho$
$\displaystyle>-1.31\cdot 10^{40}\cdot\log^{4}\rho\cdot(1+\log n)^{3}$

and comparison of this inequality with (19) gives

$n\log\alpha-\log 5<1.31\cdot 10^{40}\cdot\log^{4}\rho\cdot(1+\log n)^{3},$

which satisfies to

$n<2.75\cdot 10^{41}\cdot\log^{4}\rho\cdot\log^{3}n,\quad\text{with}\;1+\log
n<2\log n.$

Next, we apply Lemma 2.2 that enable us to find an upper bound of $n$, with
following parameters

$r:=3,\quad L:=n,\quad\text{and}\quad H:=2.75\cdot 10^{41}\cdot\log^{4}\rho.$

Therefore, we have

$n<2^{3}\cdot 2.75\cdot
10^{41}\cdot\log^{4}\rho\cdot(95.42+4\log\log\rho)^{3},$

which leads to

$n<5.6\cdot 10^{48}\cdot\log^{7}\rho.$

The above inequality holds because form $\rho\geqslant 2,$ we have
$95.42+4\log\log\rho<136\log\rho.$ By Lemma 3.1, we have that

$\ell+m+k<n<5.6\cdot 10^{48}\cdot\log^{7}\rho.$

This completes the proof of Theorem 1.1.

##  4 The study of the cases $2\leqslant\rho\leqslant 10$

In this section, we explicitly determine all the Narayana numbers which are
concatenations of three repdigits in base $\rho,$ with $\rho$ between $2$ and
$10.$ So our result in this case is Theorem 1.2.

###### Proof of Theorem 1.2.

Then in this range Theorem 1.1 allows us to deduce that all solutions to
Diophantine equation (1) satisfy

$\ell+m+k<n<2\times 10^{51}.$

The next step is therefore the reduction of the upper bound above in order to
identify the real range in which the possible solutions of (1) are found. For
this, we proceed in three steps as follow.
Step 1: Using (14), let

$\Gamma_{1}:=-\log(\Lambda_{1}+1)=(\ell+m+k)\log\rho-n\log\alpha-\log\left(\dfrac{(\rho-1)a}{d_{1}}\right).$

Notice that (14) is rewritten as

$\left|e^{-\Gamma_{1}}-1\right|<\dfrac{3}{\rho^{\ell-1}}.$

Observe that $-\Gamma_{1}\neq 0$, since $e^{-\Gamma_{1}}-1=\Lambda_{1}\neq 0$.
Assume that $\ell\geqslant 4$, then

$\left|e^{-\Gamma_{1}}-1\right|<\dfrac{3}{\rho^{\ell-1}}<\dfrac{1}{2}.$

Therefore, by Lemma 2.4, we have

$\left|\Gamma_{1}\right|<\dfrac{6}{\rho^{\ell-1}}.$

Substituting $\Gamma_{1}$ in the above inéquality with its value and dividing
through by $\log\alpha$, we obtain

$\left|(\ell+m+k)\left(\dfrac{\log\rho}{\log\alpha}\right)-n+\left(\dfrac{\log(d_{1}/(\rho-1)a)}{\log\alpha}\right)\right|<\dfrac{6/\log\alpha}{\rho^{\ell-1}}.$

Thus to apply Lemma 2.3 we can choose

$\tau:=\dfrac{\log\rho}{\log\alpha},\quad\mu:=\dfrac{\log(d_{1}/(\rho-1)a)}{\log\alpha},\quad
A:=\dfrac{6}{\log\alpha},\quad B:=\rho,\quad\text{and}\quad
w:=\ell-1\quad\text{with}\quad 1\leqslant d_{1}\leqslant\rho-1.$

Because $\ell+m+k<n<2\times 10^{51},$ we can take $M:=2\times 10^{51}$. So,
for the remaining proof, we use Mathematica to apply Lemma 2.3. For the
computations, if the first convergent $q_{t}$ such that $q_{t}>6M$ does not
satisfy the condition $\varepsilon>0,$ then we use the next convergent until
we find the one that satisfies the conditions. Thus, we have that

$\rho$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$q_{t}$ | $q_{115}$ | $q_{93}$ | $q_{107}$ | $q_{108}$ | $q_{86}$ | $q_{101}$ | $q_{107}$ | $q_{95}$ | $q_{93}$
$\varepsilon\geqslant$ | 0.28 | 0.21 | 0.13 | 0.24 | 0.03 | 0.03 | 0.02 | 0.05 | 0.03
$\ell-1\leqslant$ | 183 | 114 | 92 | 78 | 70 | 64 | 61 | 57 | 55

Therefore,

$1\leqslant\ell\leqslant\dfrac{\log((6/\log\alpha)\cdot q_{115}/0.28)}{\log
2}+1\leqslant 184.$

Step 2: For the next step we have to reduce the upper bound on $m.$ To do
this, let us consider

$\Gamma_{2}:=-\log(\Lambda_{2}+1)=(m+k)\log\rho-n\log\alpha+\log\left(\dfrac{d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2})}{(\rho-1)a}\right).$

Thus inequalities (17) become

$\left|e^{-\Gamma_{2}}-1\right|<\dfrac{2}{\rho^{m-1}}<\dfrac{1}{2},$

which holds for $m\geqslant 3.$ It follows from Lemma 2.4 that

$\displaystyle\left|(m+k)\left(\dfrac{\log\rho}{\log\alpha}\right)-n+\left(\dfrac{\log((d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2}))/(\rho-1)a)}{\log\alpha}\right)\right|<\dfrac{4/\log\alpha}{\rho^{m-1}}.$
(20)

Hence, since the conditions of Lemma 2.3 are satisfied, we may now apply it to
inequality (20) with the following data

$\tau:=\dfrac{\log\rho}{\log\alpha},\quad\mu:=\dfrac{\log((d_{1}\cdot\rho^{\ell}-(d_{1}-d_{2}))/(\rho-1)a)}{\log\alpha},\quad
A:=\dfrac{4}{\log\alpha},\quad B:=\rho,\quad\text{and}\quad w:=m-1$

with $1\leqslant d_{1},d_{2}\leqslant\rho-1$ and $1\leqslant\ell\leqslant
184.$ As $m+k<n<2\times 10^{51},$ we can take $M:=2\times 10^{51}.$ With
Mathematica we got the following results

$\rho$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$q_{t}$ | $q_{115}$ | $q_{93}$ | $q_{107}$ | $q_{108}$ | $q_{86}$ | $q_{101}$ | $q_{107}$ | $q_{95}$ | $q_{93}$
$\varepsilon\geqslant$ | 0.0009 | 0.0001 | 0.001 | 0.0002 | 0.0001 | 0.0006 | 0.0002 | $10^{-6}$ | 0.00003
$m-1\leqslant$ | 191 | 120 | 95 | 82 | 73 | 66 | 63 | 61 | 58

In all cases we can conclude that

$1\leqslant m\leqslant\dfrac{\log((4/\log\alpha)\cdot
q_{115}/0.0009)}{\log\rho}+1\leqslant 192.$

Step 3: Finaly, to reduce the bound on $n$ we have to choose

$\Gamma_{3}:=\log(\Lambda_{3}+1)=(m+k)\log\rho-n\log\alpha+\log\left(\dfrac{d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3})}{(\rho-1)a}\right).$

By inequalities (19) we have that

$\left|e^{-\Gamma_{3}}-1\right|<\dfrac{5}{\rho^{n}}<\dfrac{1}{2},$

which is valid for $n\geqslant 3$ and $\rho\geqslant 2.$ It follows from Lemma
2.4 that

$\displaystyle\left|k\left(\dfrac{\log\rho}{\log\alpha}\right)-n+\dfrac{\log((d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3}))/(\rho-1)a)}{\log\alpha}\right|<\dfrac{10/\log\alpha}{\rho^{n}}.$
(21)

Now we have to apply Lemma 2.3 to (21) by taking the following parameters

$\tau:=\dfrac{\log\rho}{\log\alpha},\quad\mu:=\dfrac{\log((d_{1}\cdot\rho^{\ell+m}-(d_{1}-d_{2})\cdot\rho^{m}-(d_{2}-d_{3}))/(\rho-1)a)}{\log\alpha},\quad
A:=\dfrac{10}{\log\alpha},\quad B:=\rho$

and $w:=n$ with $1\leqslant d_{1},d_{2}\leqslant\rho-1,$
$1\leqslant\ell\leqslant 184$ and $1\leqslant m\leqslant 183.$ Using the fact
that $k<n<2\times 10^{51},$ we can take $M:=2\times 10^{51}$

$\rho$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$q_{t}$ | $q_{115}$ | $q_{93}$ | $q_{107}$ | $q_{108}$ | $q_{86}$ | $q_{101}$ | $q_{107}$ | $q_{95}$ | $q_{93}$
$\varepsilon\geq$ | $10^{-6}$ | $10^{-7}$ | $10^{-6}$ | $10^{-7}$ | $10^{-7}$ | $10^{-8}$ | $10^{-7}$ | $10^{-7}$ | $10^{-8}$
$n\leqslant$ | 201 | 126 | 100 | 87 | 76 | 72 | 67 | 63 | 64

It follows from the above table that

$1\leqslant n\leqslant\dfrac{\log((10/\log\alpha)\cdot q_{115}/10^{-6})}{\log
2}\leqslant 201,$

which is valid for all $\rho$ such that $2\leqslant\rho\leqslant 10.$ In the
light of the above results, we need to check equation (1) in the cases
$2\leqslant\rho\leqslant 10$ for $1\leq d_{1},d_{2},d_{3}\leq 9$, $1\leq n\leq
201,$ $1\leq k\leq 201,$ $1\leq\ell\leq 184$ and $1\leq m\leq 192.$ A quick
inspection using Maple reveals that Diophantine equation (1) has only the
solution listed in the statement of Theorem 1.2. This completes the proof of
Theorem 1.2. ∎

## Acknowledgements

The first author is partially supported by Université de Kara (Togo) and the
second author is supported by IMSP, Institut de Mathématiques et de Sciences
Physiques de l’Université d’Abomey-Calavi. This project was initiated when the
first author visited IMSP for Number Theory research school. He thanks the
authorities for the warm hospitality and the working environment.

## References

  * [1] K. N. Adédji, A. Filipin and A. Togbé, _Padovan and Perrin numbers which are products of two repdigits in base $b$_, Results Math. 78, 193 (2021), https://doi.org/10.1007/s00025-021-01502-6.
  * [2] K. N. Adédji, A. Filipin and A. Togbé, _Fibonacci and Lucas numbers as products of three repdigits in base $g$_, Rend. Circ. Mat. Palermo, II. Ser (2023). https://doi.org/10.1007/s12215-023-00878-4.
  * [3] J. P. Allouche and T. Johnson, Narayana’s cows and delayed morphisms. J. d’Inform. Music. 1996, hal-02986050. Available online: https://hal.archives-ouvertes.fr/hal-02986050 (accessed on 22 March 2023).
  * [4] M. Ddamulira, P. Emong, and G. I. Mirumbe, _Members of Narayana’s cow sequence that are concatenations of two repdigits_ , https://arxiv.org/abs/2210.00926v1 (2022).
  * [5] M. Ddamulira, _Tribonacci numbers that are concatenations of two repdigits_ , RACSAM 114, 203, (2020). https://doi.org/10.1007/
  * [6] A. Dujella and A. Pethő, _A generalization of a theorem of Baker and Davenport_ , Quart. J. Math. Oxford Ser. 49 (1998), no. 195, 291-306.
  * [7] F. Erduvan and R. Keskin, _Fibonacci and Lucas numbers as products of two repdigits_ , Turk. J. Math., 43 (2019), 2142–2153.
  * [8] F. Erduvan, R. Keskin and Z. Şiar, _Repdigits as product of two Pell or Pell-Lucas numbers_ , Acta Math. Univ. Comenian. 88, (2019), 247–256.
  * [9] F. Erduvan, R. Keskin and Z. Şiar, _Repdigits base $b$ as products of two Pell numbers or Pell-Lucas numbers_, Bol. Soc. Mat. Mex. 27, 70 (2021).https://doi.org/10.1007/
  * [10] F. Erduvan, R. Keskin and Z. Şiar, _Fibonacci or Lucas Numbers Which are Products of Two Repdigits in Base b_ , Bull Braz Math Soc, New Series https://doi.org/10.1007 s00574-021-00243-y.
  * [11] S. Guzmán and F. Luca, _Linear combinations of factorials and $s$-units in a binary recurrence sequence_, Ann. Math. Qué. 38 (2014), 169–188.
  * [12] E. M. Matveev, _An explicit lower bound for a homogeneous rational linear form in then logarithms of algebraic numbers II_ , Izv. Math. 64 (2000), 1217–1269.
  * [13] P. Tiebekabe and K. N. Adédji, _On Padovan or Perrin numbers as products of three repdigits in base $\delta$_, preprint, (2023).

Université Cheikh Anta Diop de Dakar (UCAD), Laboratoire d’Algèbre, de
Cryptologie, de Géométrie Algébrique et Applications (LACGAA), BP. Fann 5005
Dakar-Senegal.
Universié de Kara, Faculté des Sciences et Techniques (FST), Département de
Mathématiques, Kara-Togo.
<EMAIL_ADDRESS>
Institut de Mathématiques et de Sciences Physiques, Université d’Abomey-
Calavi, Bénin
Email<EMAIL_ADDRESS>