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Abstract
The product formula for evaluating products of skew polynomials is used to construct a class of rings. As an application, we present a method of evaluating quotients of skew polynomials.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
1 Introduction
We present the notion of “skew product”, which is a particular binary operation on sets of functions with values in a fixed skew field (see Definition 2.1). The concept of the skew product is motivated by the “product formula” for evaluating skew polynomials introduced by Lam and Leroy [Citation6, Theorem 2.7]. We show that the skew product gives rise to near-ring structures. Restricting our attention to the class of “skew-convex” functions, we arrive at the notion of skew-convex function rings (see Definition 2.2 and Theorem 2.4). It turns out that skew-convex function rings are closely related to endomorphism rings (see Section 2.2). The method of evaluating skew polynomials has found interesting applications (see, for example, [Citation2, Citation6, Citation8, Citation9]). Using skew-convex function rings, we extend the method of evaluating skew polynomials to a method of evaluating quotients of skew polynomials.
The paper is organized as follows. Section 2.1 introduces the notions of the skew product and skew-convex functions, and gives their basic properties. Section 2.2 deals with some general structural results. In Section 2.3, we study skew-invertible functions, that is, functions that are invertible with respect to the skew product. Section 3 deals with evaluating quotients of skew polynomials.
2 Skew-convex function rings
In [Citation6], the authors presented a method of evaluation of skew polynomials over a skew field using which skew polynomials can naturally be considered as functions on the ground skew field. It turns out that the value of the product of two skew polynomials at a given point may not be equal to the product of the values of the skew polynomials at the same point. The correct formula for evaluating products of skew polynomials, called the product formula, is given in [Citation6, Theorem 2.7]. The product formula can be regarded as a binary operation on functions, which we shall call the skew product (see Formula 2.1). This section introduces the notion of the skew product, and gives some general facts regarding the skew product.
2.1 Skew-convex function rings
Let K be a skew field and X be a (nonempty) set on which the multiplicative group acts (on the left). The action of
on
is denoted by
. We will freely use the standard terminology of Group Theory. In particular, we use the following notions: An invariant subset of X is a set
such that
for all
and
; An orbit is a nonempty invariant subset of X which is minimal with respect to inclusion. We denote the set of all functions
by
. By abuse of notation, a constant function in
, whose value is
, is simply denoted by a. Given functions
, we let f + g denote the pointwise addition of the functions f and g.
Definition 2.1.
The left skew product of functions is a function
defined as follows
(2.1)
(2.1)
The right skew product is defined using the formula
(2.2)
(2.2)
In this paper, we will exclusively work with the left skew product. Therefore, we shall drop the adjective “left”. We leave it to the reader to formulate and prove similar results for the right skew product.
It is easy to see that , for every
. We shall henceforth denote
by af. Note that
turns
into a left K-vector space. In the following lemma, the proof of which is straightforward, we collect some properties of the skew product.
Lemma 2.1.
Let be arbitrary functions. Then:
The constant function 1 is a unit for
, that is,
.
, that is, the right distributive law (with respect to pointwise addition) holds for the skew product.
, that is,
is associative.
It follows from this lemma that the set equipped with pointwise addition and the skew product is a structure known as “right near-ring” in the literature (see [Citation12]). We note that the skew product may not be left distributive with respect to pointwise addition. However, the left distributive law holds for a class of functions described below.
Definition 2.2.
A function is called skew convex if
The set of all skew-convex functions is denoted by
. Any constant function belongs to
since
for all
. In particular, K is a subring of
. More generally, we have the following result. The easy proof is left to the reader.
Proposition 2.2.
Any function that is constant on every orbit in X is skew convex.
The following lemma gives an important property of skew-convex functions.
Lemma 2.3.
Let be given. The condition
holds for all functions
if and only if h is skew convex.
Proof.
The result follows from the identity
□
As a consequence of this lemma, we have the following result.
Theorem 2.4.
Equipped with the left skew product, the additive group is a ring with identity.
Proof.
The result follows from Lemma 2.3 and the general fact that in any right near-ring R, the set
is a ring. □
We call , equipped with pointwise addition and the skew product, the ring of skew-convex functions on X determined by the action of
on X. We now give some examples of skew-convex function rings.
Example 2.1.
If the action of on X is trivial, the ring
is just the familiar ring of all functions
equipped with pointwise addition and pointwise multiplication.
The following example justifies the terminology and explains the link between skew-convex function rings and skew polynomial rings. For an introduction to skew polynomial rings, we refer the reader to [Citation3].
Example 2.2.
Let be an endomorphism and
be a σ-derivation. Let
denote the ring of skew polynomials determined by σ and δ. Every nonzero element of
can uniquely be written as
where
with
. The identity
, where
, holds in
. Following [Citation6], we consider the
-action of
on K, that is,
(2.3)
(2.3)
The ring of skew-convex functions determined by the -action is denoted by
. One can verify that there exists a unique ring homomorphism
that sends each
to the constant function a, and T to the identity function
. In particular, every skew polynomial
can, under this homomorphism, be considered as a skew-convex function
. The reader can verify that the value P(a) of P at
coincides with the evaluation map introduced in [Citation6], that is, P(a) is the unique element of K for which we have
Let us remark that the product formula for evaluating products of skew polynomials is an important consequence of the existence of the above ring homomorphism.
Example 2.3.
Consider the left regular action of on
, i.e.,
. The reader can easily verify that a function
is skew-convex (with respect to the left regular action) iff there exists a group homomorphism
such that
for all
. It is straightforward to check that the assignment
establishes an isomorphism between the ring of skew-convex functions on
and the endomorphism ring
. For a more general result, see Proposition 2.7.
2.2 Structure of skew-convex function rings
We begin with a result regarding homomorphisms between skew-convex function rings for which we need some preliminaries. Let K be a skew field, and X, Y be (nonempty) sets on which acts (on the left). For a map
, let
denote the pullback map
. Recall that a map
is called action-preserving if
for all
and
.
Proposition 2.5.
Assume that is action-preserving. Then, for any function
, we have
. Moreover, the map
is a homomorphism of rings.
Proof.
The fact that , for any
, follows from the identity
The rest of the proof is straightforward. □
The following result reduces the problem of classifying skew-convex function rings to the case of transitive actions, i.e., actions with a single orbit.
Proposition 2.6.
Let be the family of all orbits in X. Then, the ring
is isomorphic to the direct product of the rings
.
Proof.
For any function , let fi
denote the restriction of f to the set Xi
. It is easy to check that the assignment
gives an isomorphism between
and the direct product
. □
Assume now that acts transitively on X. Fix an element
. Then, the map
, sending
, is onto. It is easy to see that the set
is a subgroup of
, called the stabilizer subgroup with respect to x. Furthermore, X can be identified with the set of cosets
. In this way, the action on X becomes isomorphic with the so-called (left) regular action of
on
, that is,
Therefore, it suffices to treat the case of regular actions. We also need the following definition. A map is called right G-linear if
and
, for all
and
. The following proposition gives a characterization of skew-convex function rings for the case of transitive actions.
Proposition 2.7.
Let G be a subgroup of and consider the left regular action of
on
. Then:
A function
is skew convex iff there exists a (unique) right G-linear map
such that
for all
.
The assignment
establishes an isomorphism between
and the endomorphism ring
of right G-linear operators on K.
Proof.
Given a function , we define
as follows
It is straightforward to check that f is skew-convex iff is right G-linear. The easy proof of (2) is left to the reader. □
We end this section with a remark.
Remark 2.1.
Keeping the notations as in Example 2.2, let be fixed. The
-conjugacy class of a, that is, the set
is an invariant subset of K, and the -action is transitive on
. One can show that the set
is a skew subfield of K (see Lemma 3.2 in [Citation6]). An application of Proposition 2.7 reveals that the ring of skew-convex functions on
is isomorphic to the ring of right
-linear operators on K. This isomorphism has implicitly been used in the proof of [Citation6, Proposition 3.16]. In particular, we obtain a ring homomorphism
This ring homomorphism coincides with the homomorphism Λa
introduced and studied in [Citation11] (see Corollary 1.13 in the reference). We also note that where
is the so-called λ-transform defined in Definition 4.10 of [Citation8].
2.3 Skew-invertible functions
As before, let K be a skew field and X be a set on which acts. A function
is called skew invertible if it is invertible with respect to the skew product, in which case, the inverse of f with respect to the skew product is called its skew inverse and denoted by
.
Lemma 2.8.
If is skew invertible, then the skew inverse of f belongs to
.
Proof.
By Lemma 2.3, we have
The result follows from this identification and the following general fact whose proof is left to the reader: Let R be a right near-ring and consider the ring
If is invertible in R, then its inverse belongs to
. □
Next, we give a characterization of skew-invertible functions.
Lemma 2.9.
Let
. There exists
such that
if and only if for every
, there exists some
such that
.
Let
. There exists
such that
if and only if
, and the map
is 1-1.
A function
is skew invertible if and only if
and the assignment
establishes a bijection from X onto X.
Proof.
First, let f satisfy the property that for every
, there exists some
such that
. Given any
, we can choose
such that
. Define
by
. It is easy to see that
, that is, g is a right skew-inverse of f. The proof of the other direction is straightforward.
Assume that
, and the map
is 1-1. We define a map
as follows: If
for some
, we set
. Otherwise, we set
where
is a fixed element. It is easy to see that f is well-defined and
. The proof of the reverse direction is straightforward.
Assume that f is skew invertible and let g be its skew inverse. By (2), f is nonzero on X and
is 1–1. The fact that
is onto follows from the following identity
Conversely, assume that f satisfies the stated properties. By Part (2), there exists such that
. We need only show that
. Given an arbitrary element
, we can choose
such that
. We have
Therefore, we have from which it follows that
Since was arbitrary, we conclude that
. □
Remark 2.2.
The map has also been used in the context of W-polynomials (see the definition of the
-transform in [Citation8, Definition 4.5]).
It turns out that it is somewhat easier to decide whether a skew-convex function is skew invertible. More precisely, we have the following characterization.
Proposition 2.10.
A skew-convex function is skew invertible if and only if
and for any
, there exists some
such that
.
Proof.
The “only if” direction follows from Part 1 and Part 2 of Lemma 2.9. To prove the reverse direction, we first show that if , then x = y. Let
for some
. Then y and x are in the same orbit, implying that there exists
such that
. So,
It follows that for some
satisfying
. Note that
because f is skew-convex, and therefore,
We can write
It follows that b = a, since . Thus
. This concludes the proof of the injectivity of the map
. It now follows from Part 1 and Part 2 of Lemma 2.9 that f is both left and right skew invertible. Therefore, f is skew invertible, and we are done. □
3 Evaluation of skew rational functions
The method of evaluating skew polynomials introduced in [Citation6] is a natural generalization of evaluation of polynomials in the commutative setting. Therefore, it is of importance to extend the method to skew rational functions, that is, quotients of skew polynomials. In this section, using the material developed in the previous section, we present a method of evaluating skew rational functions. Let us fix some notations. Throughout this section, K is a skew field, is an endomorphism and
is a σ-derivation. We will work with the
-action (see Example 2.2). The
-conjugacy class of an element
is denoted by
. If
are
-conjugate, we write
. The notation
means
.
3.1 Basic definitions
We begin with some general facts. For more details, see [Citation3]. It is known that is a left PID, and therefore, a left Ore domain. Its field of fractions (called the skew rational function field) is denoted by
. Every skew rational function can be represented as a left quotient
for some skew polynomials
, where P(T) is called the denominator and Q(T) is called the numerator of the quotient. Although such a representation is not unique for a given
, there exists a unique representation
of f(T) such that P(T) is monic and has the least possible degree among all representations of f(T). This representation will be called the minimal representation of f(T).
After the above preliminaries, we shall describe a method of evaluating skew rational functions. Since in the commutative setting, a rational function is not defined at the roots of its denominator, some care is needed in evaluating skew rational functions. Therefore, we introduce the following definition.
Definition 3.1.
A skew rational function is said to be defined at
if the denominator of the minimal representation of f(T) is skew-invertible as a function on the
-conjugacy class
of a.
We now define the evaluation of a skew rational function at elements of K. Recall that every skew polynomial P(T) can be regarded as a skew-convex function on any invariant set (see Example 2.2). By abuse of notation, the skew-convex function associated to P(T) is denoted by
.
Definition 3.2.
Let have the minimal representation
. Assume that f(T) is defined at
. The value of f(T) at a (denoted by f(a)) is defined to be
where
is the skew-inverse of
.
It is convenient to introduce one more definition.
Definition 3.3.
Let . The set of all
, at which f(T) is defined, is denoted by dom(f), and called the domain of f(T). The function sending
to f(a) is denoted by
.
It follows from the definitions that if is defined at some
, then it is also defined at all
. Therefore, dom(f) is a
-invariant subset of K, for all
.
In general, it is difficult to find the domain of an arbitrary skew rational function. Here, we present a partial result. Recall that a skew polynomial is called semi-invariant if
. For a detailed account of semi-invariant skew polynomials, we refer the reader to [Citation10].
Proposition 3.1.
Assume that is an automorphism. Let
have the minimal representation
such that P(T) is semi-invariant and of degree n. Then, for any
, f(T) is defined at a iff
. Furthermore, we have
Proof.
For every and
, we have
(see [Citation10, Lemma 2.2]). Evaluating at any
, we obtain
. It is clear that if f is defined at a, then
. To prove the reverse direction, we use Proposition 2.10 to show that P is skew invertible on the conjugacy class Δ of a. By the assumption
and the identity
, we see that P is nowhere vanishing on Δ. Since σ is an automorphism, we can find
such that
. It follows from the identity
that
. By Proposition 2.10, P is skew invertible on Δ, and therefore, f is defined at a. Moreover, the value of
at a is
, from which the value of f at a can be derived. □
Let us give an example illustrating the proposition.
Example 3.1.
Let σ be an involution and δ = 0. Let belong to the center of K. Then, the polynomial
is semi-invariant (see also Example 2.5.(a) in [Citation10]). By Proposition 3.1,
is defined at
iff
. Moreover, we have
We end this part with some remarks.
Remark 3.1.
Our method of evaluation of skew rational functions has some features not present in the commutative setting. One such feature is that it may happen that dom(f) is empty for every skew rational function not in . There exist examples of
in which every irreducible skew polynomial is of degree one, and
for all
. Examples are universal differential fields discovered by Kolchin [Citation5]. For such examples, we have
, for every skew rational function f not in
.
The following remarks deal with some equivalent formulations of the above definitions.
Remark 3.2.
Working with the notations of Corollary 1.13 in [Citation11], we can see that is defined at
iff
is a bijection. Moreover, the value of f(T) at
is
This approach has the advantage that one can define evaluation of skew rational functions over arbitrary rings.
Remark 3.3.
Using the notion of λ-transform (see Definition 4.10 in [Citation8]), we can see that is defined at
iff
is a bijection. The referee pointed out that the map
coincides with the map
in the preceding remark. In fact, for all
, we have
3.2 Evaluation of the skew rational function ![](//:0)
![](//:0)
This part deals with evaluation of skew rational functions of the form , where
.
Proposition 3.2.
Let . Then,
is defined at a iff
and the
-metro equation
has a solution
, for all
.
If
is defined at a, then the value of
at a is the (unique) solution x of the
-metro equation
Proof.
Using Lemma 2.9 and Proposition 2.10, we see that T – b is skew invertible as an element of
if and only if (1) T – b does not have a root in
, and (2) for any
, there exists
such that
. The first condition is equivalent to
. The second condition is equivalent to saying that for any
, there exists
satisfying the equation
, or equivalently, the
-metro equation
This is a direct consequence of (1) and the definition of the skew product.
□
In [Citation8], the notion of a -metro equation is studied in the context of Wedderburn polynomials (also called W-polynomials). Recall that a monic polynomial
is called a Wedderburn polynomial if it is the “defining” polynomial of an algebraic subset V of K, i.e., P(T) is the monic polynomial of least degree which vanishes on V. Using the results of [Citation8], we obtain the following criterion.
Corollary 3.3.
Let . Then,
is defined at a iff
and
is a W-polynomial for every
. In particular, if
is an algebraic
-conjugacy class, then
is defined at a, for all
.
Proof.
The first statement follows from Proposition 3.2 and [Citation8, Theorem 6.6]. The second statement follows from [Citation8, Corollary 6.7]. □
The following result sheds light on evaluating skew rational polynomials of the type .
Proposition 3.4.
Let satisfy
and
. If
is defined at a, then
is defined at c.
Proof.
The result follows from the definition and the identity
where
satisfies
. □
We now give some examples.
Example 3.2.
Let be a skew polynomial ring of endomorphism type, that is, δ = 0. It follows from Proposition 3.2 that the skew rational function
is defined at
if and only if
and
. Furthermore, the value of
at such an element a is
. In particular, if σ is an automorphism, then
.
Example 3.3.
Consider the skew polynomial ring where
is the complex conjugate map and δ = 0. Using Proposition 3.2, one can show that the domain of
, where
, is the set
Furthermore, the value of
at
is
We remark that similar results hold more generally for the case when is an involution of a commutative field K.
Example 3.4.
Consider the ring of polynomials over the skew field of quaternions in a central indeterminate T. Using Proposition 3.2, one can show that the domain of
, where
, is the set of all
not conjugate to q0. Moreover, the value of
at
is
This function plays a central role in the version of “quaternionic analysis” introduced in [Citation1]. It has also been used as a Cauchy kernel, see [Citation4].
3.3 Evaluation of skew rational functions over centrally finite skew fields
For , we set
Note that is a skew subfield of K (see [Citation6, Lemma 3.1]).
Proposition 3.5.
Let such that K is finite-dimensional as a right
-vector space. Then, a skew rational function with minimal representation
is defined at a iff
for all
.
Proof.
The “only if” direction is trivial. To prove the other direction, one can use Proposition 2.7 (see also Remark 2.1) and the fact that K is finite-dimensional as a right -vector space. Alternatively, one can use Part (a) of Theorem 6.2 in [Citation8]. □
Remark 3.4.
It is known that K is finite-dimensional as a right -vector space iff the conjugacy class of a is
-algebraic (see Proposition 4.2 in [Citation10]).
As special cases of this proposition, we give the following results.
Corollary 3.6.
Assume that every conjugacy class of K is -algebraic. Then, for any irreducible element
of degree > 1, the skew rational function
is defined everywhere, i.e.,
.
Corollary 3.7.
Let K be a centrally finite skew field. Assume that σ is the identity homomorphism and δ = 0. Then, a skew rational function in is defined at
iff the denominator of its minimal representation does not have a root in the conjugacy class
.
Let us now give some examples.
Example 3.5.
The skew field of quaternions is centrally finite. A classical result of Niven states that
is left algebraically closed. Therefore, every skew rational function f(T) over
has a minimal representation of the form
The domain of f(T) consists of all which are not conjugate to any of the qi
’s. Note that every conjugacy class in
is either a singleton or a 2-dimensional sphere.
More generally, we have the following example.
Example 3.6.
Assume that every -conjugacy class in K is algebraic. In the light of Remark 3.4, we see that for every skew rational function f, the complement of dom(f) in K is a finite union of conjugacy classes.
In the light of Proposition 3.5, the following example completes the discussion of evaluating skew rational functions in .
Example 3.7.
Consider the skew polynomial ring where – is the complex conjugate map and δ = 0. It is known that every irreducible monic element of
is either of the form T – a or the form
, where
for all
(see Example 1.15.4 in [Citation11]). The case T – b was treated in Example 3.3. For an irreducible element
, the skew rational function
is defined everywhere, and moreover, the value of
at
is equal to
3.4 The product formula for skew rational functions
For , we let
denote the set of all skew rational functions that are defined at a. Note that any
is defined at all points of the conjugacy class
. Therefore, the assignment
gives rise to the evaluation map
where
is the ring of skew-convex functions on the conjugacy class of a. By abuse of notaion, we denote the map
by eva
. This part deals with Def(a) and eva
, and their properties.
Fix . Using the observation that the union of any family of left Ore subsets of
is a left Ore set, we see that there exists a unique left Ore subset S(a) of
which is maximal with respect to inclusion among all left Ore subsets S of
satisfying the property
Here, the word “factor” means that for some skew polynomials
. It is clear that the Ore localization
is a subset of Def(a). We denote
by
.
Proposition 3.8.
The evaluation map is a ring homomorphism.
Proof.
This follows from the above discussion and the universal property of the Ore localization. □
As an application of this proposition, we present the product formula for skew rational functions.
Corollary 3.9.
Let and set
. Then
In general, the problem of determining S(a) seems to be difficult. Here, we present two partial results. The first result concerns skew polynomials of degree 1.
Proposition 3.10.
A polynomial T – b belongs to S(a) iff .
Proof.
The “only if” direction is trivial. To prove the other direction, let . Consider the set S of all skew polynomials T – c where c is a conjugate of b. It is enough to show that S is a left Ore set since, by Proposition 3.4,
, for all
. The fact that S is a left Ore set is a consequence of the fact that
for any
with
. □
Our second result solves the problem in the case of algebraic conjugacy classes.
Theorem 3.11.
Assume that is
-algebraic. Then, a skew polynomial P(T) belongs to S(a) iff P(T) has no (right) roots in the conjugacy class of a. In particular, we have
.
Proof.
The “only if” direction is proved in Proposition 3.5 (note that if a factor of P(T) has a root in , then P(T) has a root in
, as proved in [Citation8, Corollary 6.3]). To prove the reverse direction, let S be the set of all skew polynomials P(T) with (right) roots in the conjugacy class of a. It is enough to show that S is a left Ore set. We need to show that the set
is nonempty for every
and
. Without loss of generality, we may assume that P(T) is irreducible. If
, the proof is trivial. So, assume that
. Let L(T) be the least left common multiple of P(T) and Q(T). Then,
for some
. Since
is a UFD, we see that
and P(T) must be similar, i.e., the
-modules
and
are isomorphic. It follows from Lemma 6.4 in [Citation7] that every right root of
is conjugate to a right root of P(T). Therefore,
has no right roots in
, and consequently
. This completes the proof. □
We conclude the paper with the following example as an application of the above theorem.
Example 3.8.
Consider the ring of polynomials over the skew field of quaternions in a central indeterminate T. For any
is a ring consisting of all
such that
when q is conjugate to q0.
Acknowledgments
I would like to thank the referee for his/her comments which led to an improvement of the paper.
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