Factorizations for difference operators
- Jeffrey Bergen^{1},
- Mark Giesbrecht^{2},
- Pappur N Shivakumar^{3} and
- Yang Zhang^{3}Email author
https://doi.org/10.1186/s13662-015-0402-1
© Bergen et al.; licensee Springer. 2015
Received: 28 October 2014
Accepted: 3 February 2015
Published: 24 February 2015
Abstract
We consider the factorization problems of difference operators in \(\mathbb{C}[x;\sigma]\) for an automorphism σ of finite order. We study the factorization of regular polynomials in \(\mathbb{R}[x]\) in the ring of such difference operators and obtain an analogue of the fundamental theorem of algebra for skew polynomial ring \(\mathsf{K}[x; \sigma]\) over field K.
Keywords
1 Introduction
Linear differential operators (\(\sigma= \mathrm{id}_{\mathsf{R}}\), the identity map on R) and linear difference operators (\(\delta= 0\), the identically zero function) are special cases of above skew polynomials, which have been studied via algebraic methods since [1]. The study of skew polynomials is now an active and important area in algebra with many applications in other areas such as solving differential/difference equations, engineering, coding theory, etc. More recent results and survey can be found in [2–10]. Factorization of skew polynomials is also a recent active area in computer algebra (see, for example, [11–14]). Algorithms for characterization of linear difference and differential equations are of great interest.
One of the great strengths of skew polynomials is their analogy with regular commutative polynomials. A natural question thus arises: Is there an analogue of the fundamental theorem of algebra for skew polynomials? We provide a positive answer for difference operators with finite-order automorphisms over an algebraically closed field.
Throughout this paper, we assume that \(\mathsf{R}[x; \sigma]\) is a skew polynomial ring with automorphism σ as defined above.
2 Some basic results
We next introduce the terminology used throughout the remainder of this paper. We let K denote a field and σ an automorphism of K. Define \(\mathsf{K}^{\sigma}= \{ a \in\mathsf{K} \mid \sigma(a) = a\}\) and, when we examine the special case where \(\sigma^{2} = 1\), an important subset of \(\mathsf{K}^{\sigma}\) is the set of norms defined by \(N(\mathsf{K}, \sigma) = \{t \sigma(t) \mid t \in\mathsf{K}\}\). When dealing with the field ℂ, unless stated otherwise, we always assume that σ represents complex conjugation.
Definition 2.1
Let \(f(x) = a_{n}x^{n} +a_{n-1}x^{n-1} + \cdots+ a_{2}x^{2}+a_{1}x +a_{0} \in\mathsf{K}[x; \sigma]\) and \(t \in\mathsf{K}\). Then we define the σ-evaluation \(f(t; \sigma) \in\mathsf{K}\) as \(f(t; \sigma) = a_{n}(t\sigma(t) \cdots\sigma^{n-1}(t)) + a_{n-1}(t\sigma(t) \cdots\sigma^{n-2}(t)) + \cdots+a_{2}(t\sigma(t)) + a_{1}t + a_{0}\). If \(f(t; \sigma) = 0\), we say that t is a σ-root of \(f(x)\).
Observe that if \(\sigma= \mathrm{id}_{\mathsf{R}}\), and \(\mathsf{K}[x; \sigma ]=\mathsf{K}[x]\) is an ordinary polynomial ring, then a σ-root is the same as an ordinary root. Lemma 2.2 and Theorem 2.3 show us that the role of σ-roots in the skew polynomial case generalizes the role of roots in the ordinary case.
Lemma 2.2
([15])
When \(f(x) = q(x)(x-t)\), instead of simply saying that \(x-t\) is a factor of \(f(x)\), we may sometimes use the more precise terminology that \(x-t\) is a (right) factor of \(f(x)\). We can now see the relationship between (right) factors and σ-roots.
Theorem 2.3
Suppose \(f(x) \in\mathsf{K}[x; \sigma]\). Then \(x-t\) is a right factor of \(f(x)\) if and only if t is a σ-root of \(f(x)\).
Proof
Using Lemma 2.2, there exists a unique \(q(x) \in\mathsf{K}[x; \sigma]\) such that \(f(x) = q(x)(x-t) +f(t; \sigma)\). Therefore, if t is a σ-root of \(f(x)\), then \(f(t; \sigma) = 0\), and we can see that \(x-t\) is certainly a (right) factor of \(f(x)\). Conversely, if \(x-t\) is a (right) factor of \(f(x)\), then \(f(x) = h(x)(x-t)\) for some \(h(x) \in\mathsf{K}[x; \sigma]\). But then, by Lemma 2.2, the quotient \(h(x)\) and remainder (0) are unique, so \(f(t;\sigma)=0\). □
For ordinary polynomials of degree 2, being reducible is equivalent to having a root. In our situation, we now show that it is equivalent to having a σ-root.
Corollary 2.4
Suppose \(f(x) \in\mathsf{K}[x; \sigma]\) has degree 2; then \(f(x)\) is reducible in \(\mathsf{K}[x; \sigma]\) if and only if \(f(x)\) has a σ-root in K.
Proof
One direction follows immediately from Theorem 2.3, for if \(f(x)\) has a σ-root in K, then it has a right factor of degree one in \(\mathsf{K}[x; \sigma]\) and is therefore reducible. In the other direction, if \(f(x)\) is reducible and has degree 2, then it is easy to see that it must have a monic, linear right factor. Theorem 2.3 now asserts that \(f(x)\) has a σ-root in K. □
One of the examples that motivated this paper was the fact that \(x^{2}+1\) remains irreducible in \(\mathbb{C}[x; \sigma]\). Thus the situation of quadratic polynomials where \(\sigma^{2} = 1\) is of particular interest. We can now determine precisely that irreducible quadratics in \(\mathsf{K}^{\sigma}[x]\) remain irreducible in \(\mathsf {K}[x; \sigma]\).
Corollary 2.5
Let \(f(x) = x^{2}+ax+b \in\mathsf{K}^{\sigma}[x]\) and suppose \(\sigma^{2} =1\). Then \(f(x)\) is reducible in \(\mathsf{K}[x; \sigma]\) if and only if either (i) \(f(x)\) is reducible in \(\mathsf{K}^{\sigma}[x]\), or (ii) \(a =0 \) and \(-b \in N(\mathsf{K}, \sigma)\).
Proof
The next corollary completely describes the situation for the examples \(x^{2}-1, x^{2}+1 \in\mathbb{R}[x] \subseteq\mathbb{C}[x; \sigma]\) that were mentioned at the start of this paper.
Corollary 2.6
- (i)
\(f(x)\) is reducible in \(\mathbb{C}[x; \sigma]\) if and only if \(f(x)\) is reducible in \(\mathbb{R}[x]\).
- (ii)
If \(f(x)\) is reducible, then the factorization of \(f(x)\) in \(\mathbb{C}[x; \sigma]\) into monic, linear factors is unique when \(a \ne0\), whereas \(f(x)\) factors an infinite number of ways into monic, linear factors when \(a =0\).
Proof
Finally, suppose \(a \ne0\); the argument in the proof of Corollary 2.5 indicates that any factorization of \(f(x)\) actually takes place in \(\mathbb{R}[x]\). Since there is only one way to factor \(f(x)\) in \(\mathbb{R}[x]\), it follows that the factorization of \(f(x)\) in \(\mathbb{C}[x; \sigma]\) is also unique. □
When we look back at Corollaries 2.5 and 2.6, it becomes natural to look for an example where reducibility in \(\mathsf{K}[x; \sigma]\) is not equivalent to reducibility in \(\mathsf{K}^{\sigma}[x]\).
Example 2.7
3 Main theorems
In order to complete our description of the factoring of polynomials in \(\mathbb{C}[x; \sigma]\), we first need to prove a result that holds in any algebraically closed field.
Theorem 3.1
Let K be an algebraically closed field and let σ be an automorphism of K of order n. Then every non-constant reducible skew polynomial in \(\mathsf{K}[x; \sigma]\) can be written as a product of irreducible skew polynomials of degree less than or equal to n.
Proof
By Theorem 1 of Chapter II in [1], the degrees of factors of \(f(x)g(x)\) are at most n.
Thus \(f(x)\) has been factored in \(\mathsf{K}[x; \sigma]\) as a product of polynomials of degree at most n. □
Using Theorem 3.1 and Corollary 2.4, we now have the following.
Corollary 3.2
Every non-constant polynomial in \(\mathbb{C}[x; \sigma]\) is a product of linear and irreducible polynomials. Furthermore, the quadratic \(f(x) = x^{2}+a_{1}x+a_{0}\) is irreducible in \(\mathbb{C}[x; \sigma]\) if and only if there does not exist some \(t \in\mathbb{C}\) such that \(t\sigma(t) +a_{1}t +a_{0} = 0\).
Proof
Theorem 3.1 asserts that every non-constant polynomial in \(\mathbb{C}[x; \sigma]\) is a product of linear and irreducible polynomials. In addition, Corollary 2.4 tells us that \(f(x) = x^{2}+a_{1}x+a_{0}\) is irreducible in \(\mathbb{C}[x; \sigma]\) if and only if \(f(x)\) does not have a σ-root in ℂ. However, when we look at the definition of σ-roots in the quadratic case, it is clear that \(f(x)\) does not have a σ-root if and only if there does not exist \(t \in\mathbb{C}\) such that \(t\sigma(t)+ a_{1}t + a_{0} = 0\). □
Declarations
Acknowledgements
We would like to thank the anonymous referees for their careful reading and valuable comments that have improved the readability of this article. The research of the first author was supported by a grant from the University Research Council at DePaul University.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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