NBThis article discusses a recently developed Mathematica tool–
–a collection of functions for manipulating, evaluating and factoring quaternionic polynomials. 
relies on the package 
, which is available for download at w3.math.uminho.pt/QuaternionAnalysis.
Introduction
Some years ago, the first two authors of this article extended the standard Mathematica package implementing Hamilton’s quaternion algebra—the package 
—endowing it with the ability, among other things, to perform numerical and symbolic operations on quaternion-valued functions [1]. Later on, the same authors, in response to the need for including new functions providing basic mathematical tools necessary for dealing with quaternionic-valued functions, wrote a full new package, 
. Since 2014, the package and complete support files have been available for download at the Wolfram Library Archive (see also [2] for updated versions).
Over time, this package has become an important tool, especially in the work that has been developed by the authors in the area of quaternionic polynomials ([3–5]). While this work progressed, new Mathematica functions were written to appropriately deal with problems in the ring of quaternionic polynomials. The main purpose of the present article is to describe these Mathematica functions. There are two parts.
In this first part, we discuss the 
tool, containing several functions for treating the usual problems in the ring of quaternionic polynomials: evaluation, Euclidean division, greatest common divisor and so on. A first version of 
was already introduced in [4], having in mind the user’s point of view. Here, we take another perspective, giving some implementation details and describing some of the experiments performed.
The second part of the article (forthcoming) is entirely dedicated to root-finding methods.
The QuaternionAnalysis Package and the Algebra of Real Quaternions
In 1843, the Irish mathematician William Rowan Hamilton introduced the quaternions, which are numbers of the form
where the imaginary units 
, 
and 
satisfy the multiplication rules
This noncommutative product generates the well-known algebra of real quaternions, usually denoted by 
.
Definition 1
,
;
,
;
,
;
,
.The standard package 
adds rules to 
, 
, 
, 
and the fundamental 
. Among others, the following quaternion functions are included: 
, 
, 
, 
, 
, 
, 
and 
. In 
, a quaternion is an object of the form 
and must have real numeric valued entries; that is, applying the function 
to an argument gives 
.
The extended version 
allows the use of symbolic entries, assuming that all symbols represent real numbers. The 
package adds functionality to the following functions: 
, 
, 
, 
, 
, 
, 
, 
, 
, 
and 
. We briefly illustrate some of the quaternion functions needed in the sequel. In what follows, we assume that the package 
has been installed.
These are the imaginary units.
These are the multiplication rules.
Here are two quaternions with symbolic entries and their product.
The product is noncommutative.
Here are some basic functions.
The function 
, which was extended in 
through the use of de Moivre’s formula for quaternions, works quite well for quaternions with numeric entries.
contains a different implementation of the power function, 
, which we recommend whenever a quaternion has symbolic entries.
We refer the reader to the package documentation for more details on the new functions included in the package.
Manipulating Quaternionic Polynomials
We focus now on the polynomial 
in one formal variable 
of the form
 |
(1) |
where the coefficients 
are to the left of the powers. Denote by 
the set of polynomials of the form (1), defining addition and multiplication as in the commutative case and assuming the variable 
commutes with the coefficients. This is a ring, referred to as the ring of left one-sided (or unilateral) polynomials.
When working with the functions contained in 
, a polynomial 
in 
is an object defined through the use of the function 
, which returns the simplest form of 
, taking into account the following rules.
The function 
tests if an argument is a scalar in the sense that it is not a complex number, a quaternion number or a polynomial.
For polynomials in 
, the rules 
, 
, 
and 
have to be defined.
■ Addition
■ Product by a scalar
■ Multiplication
■ Power
Example 1
The polynomials 
and 
can be defined using their coefficients in 
in descending order.
.
We now define three particularly important polynomials, the first two associated with a given polynomial 
and the last one associated with a given quaternion 
.
Definition 2
a polynomial as in equation (
a quaternion, define:
;
;
.The first two polynomials are constructed with the functions 
and 
.
The built-in function 
now accepts a quaternion argument.
Observe that 
is a polynomial with real coefficients. For simplicity, in this context and in what follows, we assume that a quaternion with vector part zero is real.
Example 2
Consider the polynomial 
of Example 1 and the quaternion 
.
Evaluating Quaternionic Polynomials
The evaluation map at a given quaternion 
, defined for the polynomial 
given by (1), is
 |
(2) |
It is not an algebra homomorphism, as
does not lead, in general, to 
, as the next theorem remarks.
and 
be two polynomials in 
and consider the polynomial 
and 
. Then:
.
, then 
.
, then 
, where 
.
is a real polynomial, then 
.
, then 
.As usual, we say that 
is a zero (or root) of 
if 
An immediate consequence of Theorem 1 is that if 
, then 
is a zero of 
if and only if 
is a zero of 
.
A straightforward implementation of equation (2) can be obtained through 
.
As in the classical (real or complex) case, the evaluation of a polynomial can also be obtained by the use of Horner’s rule [3]. The nested form of equation (2) is
and the quaternionic version of Horner’s rule can be implemented as 
.
Example 3
Consider again the polynomial 
. The problem of evaluating 
at 
can be solved through one of the following (formally) equivalent expressions.
Example 4
We now illustrate some of the conclusions of Theorem 1 by considering the polynomials 
, 
and 
and the quaternion 
.
The Euclidean Algorithm
For the theoretical background of this section, we refer the reader to [6] (see also [7] where basic division algorithms in 
are presented). Since 
is a principal ideal domain, left and right division algorithms can be defined. The following theorem gives more details.
If 
and 
are polynomials in 
(with 
), then there exist unique 
, 
, 
and 
such that
 |
(3) |
 |
(4) |
and 
.If in equation (3), 
, then 
is called a right divisor of 
, and if in equation (4), 
, 
is called a left divisor of 
. This article only presents right versions of the division functions; in 
both the left and right versions are implemented. The function 
performs the right division of two quaternionic polynomials, returning a list with the quotient and remainder of the division.
Example 5
Consider the polynomials 
and 
.
, 
is a right divisor of 
and 
. On the other hand, 
does not right-divide 
(but it is a left divisor).
implements this procedure for the case of the greatest common right divisor.
is defined as follows.
Example 5 (continued)
and 
.
The Zero Structure in 
Before describing the zero set 
of a quaternionic polynomial 
, we need to introduce more concepts.
Definition 3
is congruent (or similar) to a quaternion 
(and write 
) if there exists a nonzero quaternion 
such that 
.This is an equivalence relation in 
that partitions 
into congruence classes. The congruence class containing a given quaternion 
is denoted by 
. It can be shown (see, e.g. [8]) that
This result gives a simple way to test if two or more quaternions are similar, implemented with the function 
.
For zero or equality testing, we use the 
test function.
It follows that 
if and only if 
. The congruence class of a nonreal quaternion 
can be identified with the three-dimensional sphere in the hyperplane 
with center 
and radius 
.
Definition 4
of 
is called an isolated zero of 
if 
contains no other zeros of 
. Otherwise, 
is called a spherical zero of 
and 
is referred to as a sphere of zeros.It can be proved that if 
is a zero that is not isolated, then all quaternions in 
are in fact zeros of 
(see Theorem 4); therefore the choice of the term spherical to designate this type of zero is natural. According to the definition, real zeros are always isolated zeros. Identifying zeros can be done, taking into account the following results.
and 
. The following conditions are equivalent:
such that 
.
, 
, is a divisor of the companion polynomial 
of 
.
is a root of 
.A nonreal zero 
is a spherical zero of 
if and only if any of the following equivalent conditions hold:
and 
are both zeros of 
.
.
of 
is a right divisor of 
; that is, there exists a polynomial 
such that 
.Example 6
We are going to show that the polynomial
has a spherical zero: 
and an isolated one: 
.
We first observe that both 
and 
are zeros of 
.
Now we use Theorem 4-1 to conclude that the zero 
is spherical, while the zero 
is isolated.
We can reach the same conclusion from Theorem 4-3.
Taking all this into account, the verification of the nature of a zero can be done using the function 
.
Consider the same polynomial and quaternions again.
We now list other results needed in the next section.
Let 
and 
. Then 
is a zero of 
if and only if there exists 
such that 
.
Any nonconstant polynomial in 
always has a zero in 
.
Factoring
In this section, we address the problem of factoring a polynomial 
. We mostly follow [4]. As in the classical case, it is always possible to write a quaternionic polynomial as a product of linear factors; however the link between these factors and the corresponding zeros is not straightforward. As an immediate consequence of Theorems 5 and 6, one has the following theorem.
Any monic polynomial 
of degree 
in 
factors into linear factors; that is, there exist 
such that
 |
(5) |
Definition 5
of the form (5), the quaternions 
are called factor terms of 
and the 
-tuple 
is called a factor terms chain associated with 
or simply a chain of 
.If 
and 
are chains associated with the same polynomial 
, then we say that the chains are similar and write 
.
The function 
constructs a polynomial with a given chain, and the function 
checks if two given chains are similar.
The repeated use of the next result allows the constructions of similar chains, if any.
be a chain of a polynomial 
. If 
, then
Theorem 8 can be implemented using the function 
.
Example 7
This constructs chains similar to the chain 
.
Observe that 
, 
and 
are similar chains.
We emphasize that there are polynomials with just one chain. This issue is addressed in Theorem 12. For the moment, we just give an example of such a polynomial.
These computations lead us to the conclusion that the polynomial 
factors uniquely as 
.
The next fundamental results shed light on the relation between factor terms and zeros of a quaternionic polynomial.
Let 
be a chain of the polynomial 
. Then every zero of 
is similar to some factor term 
in the chain and conversely, every factor term 
is similar to some zero of 
.
Consider a chain 
of the polynomial 
. If the similarity classes 
are distinct, then 
has exactly 
zeros 
, which are given by:
 |
(6) |
The function 
determines the zeros of a polynomial with a prescribed chain in the case where no two factors in the chain are similar quaternions, giving a warning if this condition does not hold.
Example 8
Consider the polynomial 
. One of its chains is 
, and it follows at once that the similarity classes of the factor terms are all distinct. Therefore, we conclude from Theorem 10 that 
has four distinct isolated roots, which can be obtained with the following code.
On the other hand, the polynomial 
has 
as one of its chains. Since 
, one cannot apply Theorem 10 to find the roots of 
.
Observe that this does not mean that the roots of 
are spherical.
This issue will be resumed later in connection with the notion of the multiplicity of a zero. The following theorem indicates how, under certain conditions, one can construct a polynomial having prescribed zeros.
If 
are quaternions such that the similarity classes 
are distinct, then there is a unique polynomial 
of degree 
with zeros 
that can be constructed from the chain 
, where
The function 
implements the procedure described in Theorem 11.
Example 9
Consider the problem of constructing a polynomial having the isolated roots 
. We first determine one chain associated with these zeros.
Now we determine the polynomial associated with this chain.
Check the solution.
Let 
be a quaternionic polynomial of degree 
. Then 
is the unique zero of 
if and only if 
admits a unique chain 
with the property
 |
(7) |
.
associated with a polynomial 
has property (7), 
is a polynomial of degree 
such that 
is its unique zero and 
, then the polynomial 
(of degree 
) has only two zeros, namely 
and 
.We can now introduce the concept of the multiplicity of a zero and a new kind of zero. In this context, we have to note that several notions of multiplicity are available in the literature (see [9], [15–17]).
Definition 6
of 
is defined as the maximum degree of the right factors of 
with 
as their unique zero and is denoted by 
. The multiplicity of a sphere of zeros 
of 
, denoted by 
, is the largest 
for which 
divides 
.
of 
is called a mixed zero if 
and 
for all 
.Example 10
The polynomial 
has an isolated root 
with multiplicity 
and an isolated root 
with multiplicity 
.
The polynomial 
has an isolated root 
with multiplicity 
and a sphere of zeros 
with multiplicity 
.
The polynomial 
has a mixed root 
with multiplicity 
and 
.
Finally, one can construct a polynomial with assigned zeros by the repeated use of the following result.
A polynomial with 
and 
as its isolated zeros with multiplicities 
and 
, respectively, and a sphere of zeros 
with multiplicity 
can be constructed through the chain
and 
.An alternative syntax for the function 
addresses the problem of constructing a polynomial (in fact it constructs a chain) once one knows the nature and multiplicity of its roots.
Example 11
We reconsider here Example 6 of [4]. An example of a polynomial 
that has 
as a zero of multiplicity three, 
as a zero of multiplicity two and 
as a sphere of zeros with multiplicity two is
, 
and 
; that is,
Of course this solution is not unique. For example, the polynomial
We confirm this using the function 
with the new syntax.
Here are two spherical roots corresponding to the same sphere.
Observe that the result is, of course, the same as this one.
Recall that a real root is always an isolated root, and two roots in the same congruence class cannot be isolated.
Conclusion
This article has discussed implementation issues related to the manipulation, evaluation and factorization of quaternionic polynomials. We recommend that interested readers download the support file 
to get complete access to all the implemented functions. The increasing interest in the use of quaternions in areas such as number theory, robotics, virtual reality and image processing [18] makes us believe that developing a computational tool for operating in the quaternions framework will be useful for other researchers, especially taking into account the power of Mathematica as a symbolic language.
In the ring of quaternionic polynomials, new problems arise mainly because the structure of zero sets, as we have described, is very different from the complex case. In this article, we did not discuss the problem of computing the roots or the factor terms of a polynomial; all the results we have presented assumed that either the zeros or the factor terms of a given polynomial are known. Methods for computing the roots or factor terms of a quaternionic polynomial are considered in Part II.
Acknowledgments
Research at the Centre of Mathematics at the University of Minho was financed by Portuguese Funds through FCT – Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013. Research at the Economics Politics Research Unit was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (UID/ECO/03182/2013), with the FCT/MEC’s (Fundação para a Ciência e a Tecnologia, I.P.) financial support through national funding and by the European Regional Development Fund through the Operational Programme on “Competitiveness and Internationalization – COMPETE 2020” under the PT2020 Partnership Agreement.
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Additional Material
.Available at: w3.math.uminho.pt/QuaternionAnalysis
.Available at: content.wolfram.com/sites/19/2018/05/QPolynomial.m
About the Authors
M. Irene Falcão is an associate professor in the Department of Mathematics and Applications of the University of Minho. Her research interests are numerical analysis, hypercomplex analysis and scientific software.
Fernando Miranda is an assistant professor in the Department of Mathematics and Applications of the University of Minho. His research interests are differential equations, quaternions and related algebras and scientific software.
Ricardo Severino is an assistant professor in the Department of Mathematics and Applications of the University of Minho. His research interests are dynamical systems, quaternions and related algebras and scientific software.
M. Joana Soares is an associate professor in the Department of Mathematics and Applications of the University of Minho. Her research interests are numerical analysis, wavelets mainly in applications to economics, and quaternions and related algebras.
M. Irene Falcão
CMAT – Centre of Mathematics
DMA – Department of Mathematics and Applications
University of Minho
Campus de Gualtar, 4710-057 Braga
Portugal
mif@math.uminho.pt
Fernando Miranda
CMAT – Centre of Mathematics
DMA – Department of Mathematics and Applications
University of Minho
Campus de Gualtar, 4710-057 Braga
Portugal
fmiranda@math.uminho.pt
Ricardo Severino
DMA – Department of Mathematics and Applications
University of Minho
Campus de Gualtar, 4710-057 Braga
Portugal
ricardo@math.uminho.pt
M. Joana Soares
NIPE – Economics Politics Research Unit
DMA – Department of Mathematics and Applications
University of Minho
Campus de Gualtar, 4710-057 Braga
Portugal
jsoares@math.uminho.pt