J Bergen

J von zur Gathen

JC McConnell

Jeffrey Bergen

M Giesbrecht

Mark Giesbrecht

N Jacobson

O Ore

Pappur N Shivakumar

PM Cohn

PN Shivakumar

R Churchill

TY Lam

W Eberly

Yang Zhang

Advances in Difference Equations

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Bergen et al. Advances in Diﬀerence Equations  (2015) 2015:57 DOI 10.1186/s13662-015-0402-1RESEARCH Open AccessFactorizations for difference operatorsJeﬀrey Bergen1, Mark Giesbrecht2, Pappur N Shivakumar3 and Yang Zhang3**Correspondence:zhang39@cc.umanitoba.ca3Department of Mathematics,University of Manitoba, Winnipeg,MB, CanadaFull list of author information isavailable at the end of the articleAbstractWe consider the factorization problems of diﬀerence operators in C[x;σ ] for anautomorphism σ of ﬁnite order. We study the factorization of regular polynomials inR[x] in the ring of such diﬀerence operators and obtain an analogue of thefundamental theorem of algebra for skew polynomial ring K[x;σ ] over ﬁeld K.Keywords: diﬀerence operators; factorizations1 IntroductionLet R be a ring, σ an endomorphism of R, and δ a left σ -derivation of R. That is, δ is anadditive mapping from R to R and for any a,b ∈ R,δ(ab) = σ (a)δ(b) + δ(a)b.The skew polynomial ring R[x;σ , δ] is the set of all polynomialsa + ax + · · · + anxn,where a, . . . ,an ∈ R, addition is deﬁned as usual, and multiplication is deﬁned byxa = σ (a)x + δ(a) for all a ∈ R.Linear diﬀerential operators (σ = idR, the identity map on R) and linear diﬀerence op-erators (δ = , the identically zero function) are special cases of above skew polynomials,which have been studied via algebraic methods since []. The study of skew polynomials isnow an active and important area in algebra with many applications in other areas such assolving diﬀerential/diﬀerence equations, engineering, coding theory, etc.More recent re-sults and survey can be found in [–]. Factorization of skew polynomials is also a recentactive area in computer algebra (see, for example, [–]). Algorithms for characteriza-tion of linear diﬀerence and diﬀerential equations are of great interest.One of the great strengths of skew polynomials is their analogy with regular commuta-tive polynomials. A natural question thus arises: Is there an analogue of the fundamentaltheorem of algebra for skew polynomials?We provide a positive answer for diﬀerence op-erators with ﬁnite-order automorphisms over an algebraically closed ﬁeld.Throughout this paper, we assume that R[x;σ ] is a skew polynomial ring with automor-phism σ as deﬁned above.© 2015 Bergen et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons At-tribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly credited.Bergen et al. Advances in Diﬀerence Equations  (2015) 2015:57 Page 2 of 62 Some basic resultsIt is clear that the polynomials x –  and x +  in R[x] factor uniquely into monic, linearpolynomials in C[x]. However, the situation is quite diﬀerent when we look at R[x] as asubring of the skew polynomial ring C[x;σ ]. Observe that if t ∈ C such that |t| = , andσ (t) is the complex conjugate of t, thenx –  =(x + σ (t))(x – t)inC[x;σ ]. Thus x – can be factored in an inﬁnite number of ways inC[x;σ ]. On the otherhand, x +  remains irreducible in C[x;σ ]. In this paper we contrast how polynomialsfactor in skew polynomial rings with how they factor in ordinary polynomial rings.We next introduce the terminology used throughout the remainder of this paper. Welet K denote a ﬁeld and σ an automorphism of K. Deﬁne Kσ = {a ∈ K | σ (a) = a} and, whenwe examine the special case where σ  = , an important subset of Kσ is the set of normsdeﬁned byN(K,σ ) = {tσ (t) | t ∈ K}.When dealing with the ﬁeldC, unless stated otherwise,we always assume that σ represents complex conjugation.For f (x) = anxn + · · · + ax + a ∈ K[x;σ ] and t ∈ K, the (standard) evaluation of f (x) atx = t is deﬁned as f (t) = antn + · · · + at + a. When we study polynomials over ﬁelds, oneof the ﬁrst facts we prove is that linear factors correspond to roots. Note that this is notthe case in skew polynomial rings: if σ denotes complex conjugation, thenf (x) = x –  = (x – i)(x – i)in C[x;σ ], yet f (i) = . To better understand the role of linear factors in skew polynomialrings, we deﬁne σ -evaluation as follows.Deﬁnition . Let f (x) = anxn+an–xn– + · · ·+ax +ax+a ∈ K[x;σ ] and t ∈ K. Thenwedeﬁne the σ -evaluation f (t;σ ) ∈ K as f (t;σ ) = an(tσ (t) · · ·σ n–(t))+an–(tσ (t) · · ·σ n–(t))+· · · + a(tσ (t)) + at + a. If f (t;σ ) = , we say that t is a σ -root of f (x).Observe that if σ = idR, and K[x;σ ] = K[x] is an ordinary polynomial ring, then a σ -rootis the same as an ordinary root. Lemma . and Theorem . show us that the role ofσ -roots in the skew polynomial case generalizes the role of roots in the ordinary case.Lemma . ([]) If f (x) = anxn + · · · + ax + a ∈ K[x;σ ] and t ∈ K, then there exists aunique q(x) ∈ K[x;σ ] such thatf (x) = q(x)(x – t) + f (t;σ ).When f (x) = q(x)(x – t), instead of simply saying that x – t is a factor of f (x), we maysometimes use the more precise terminology that x – t is a (right) factor of f (x). We cannow see the relationship between (right) factors and σ -roots.Theorem . Suppose f (x) ∈ K[x;σ ]. Then x – t is a right factor of f (x) if and only if t is aσ -root of f (x).Proof Using Lemma ., there exists a unique q(x) ∈ K[x;σ ] such that f (x) = q(x)(x – t) +f (t;σ ). Therefore, if t is a σ -root of f (x), then f (t;σ ) = , andwe can see that x–t is certainlyBergen et al. Advances in Diﬀerence Equations  (2015) 2015:57 Page 3 of 6a (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) ∈ K[x;σ ]. But then, by Lemma ., the quotient h(x) and remainder () areunique, so f (t;σ ) = . For ordinary polynomials of degree , being reducible is equivalent to having a root. Inour situation, we now show that it is equivalent to having a σ -root.Corollary . Suppose f (x) ∈ K[x;σ ] has degree ; then f (x) is reducible in K[x;σ ] if andonly if f (x) has a σ -root in K.Proof One direction follows immediately from Theorem ., for if f (x) has a σ -root in K,then it has a right factor of degree one in K[x;σ ] and is therefore reducible. In the otherdirection, if f (x) is reducible and has degree , then it is easy to see that it must have amonic, linear right factor. Theorem . now asserts that f (x) has a σ -root in K. One of the examples that motivated this paper was the fact that x +  remains irre-ducible in C[x;σ ]. Thus the situation of quadratic polynomials where σ  =  is of particu-lar interest. We can now determine precisely that irreducible quadratics in Kσ [x] remainirreducible in K[x;σ ].Corollary . Let f (x) = x + ax + b ∈ Kσ [x] and suppose σ  = . Then f (x) is reducible inK[x;σ ] if and only if either (i) f (x) is reducible in Kσ [x], or (ii) a =  and –b ∈ N(K,σ ).Proof If f (x) is reducible in Kσ [x], then it is certainly reducible in K[x;σ ]. Furthermore, ifa =  and –b ∈ N(K,σ ), then –b = tσ (t) for some t ∈ K andf (x) = x – (–b) = x –(tσ (t))=(x + σ (t))(x – t).This proves one half of the result.To see the converse, Corollary . tells us that if f (x) is reducible in K[x;σ ], then it has aσ -root t ∈ K. If t is a σ -root, then it satisﬁes the equationtσ (t) + at + b = .Since σ  = , we know that tσ (t) ∈ Kσ . However, b ∈ Kσ , therefore it follows that at ∈ Kσ .If a = , then the fact that a ∈ Kσ implies that t ∈ Kσ . Since t is an ordinary root of f (x) inKσ , we see that f (x) is reducible in Kσ [x]. On the other hand, suppose a = . Our equationabove now tells us that –b = tσ ∈ N(K,σ ), thereby concluding the proof. The next corollary completely describes the situation for the examples x – ,x +  ∈R[x]⊆C[x;σ ] that were mentioned at the start of this paper.Corollary . Let f (x) = x + ax + b ∈R[x].(i) f (x) is reducible in C[x;σ ] if and only if f (x) is reducible in R[x].(ii) If f (x) is reducible, then the factorization of f (x) in C[x;σ ] into monic, linear factorsis unique when a = , whereas f (x) factors an inﬁnite number of ways into monic,linear factors when a = .Bergen et al. Advances in Diﬀerence Equations  (2015) 2015:57 Page 4 of 6Proof In light of Corollary ., in order to prove part (i), it suﬃces to show that if a =  and–b ∈ N(C,σ ), then f (x) is reducible in R[x]. However, in this situation, N(C,σ ) is equal tothe set of positive real numbers. Therefore, if –b ∈ N(C,σ ), then –b = c for some c ∈ R.Thusf (x) = x – (–b) = (x + c)(x – c),as required.For the proof of part (ii), let cis(θ ) = cos θ + i sin θ for every θ ∈R. Then, when we are inthe case where a = , we can rewrite the equation above asf (x) = x – (–b) = (x + c)(x – c) =(x + c(cis(–θ )))(x – c(cis(θ ))).By letting θ range between  and π , we can see that f (x) does factor an inﬁnite numberof ways in C[x;σ ].Finally, suppose a = ; the argument in the proof of Corollary . indicates that anyfactorization of f (x) actually takes place in R[x]. Since there is only one way to factor f (x)in R[x], it follows that the factorization of f (x) in C[x;σ ] is also unique. When we look back at Corollaries . and ., it becomes natural to look for an examplewhere reducibility in K[x;σ ] is not equivalent to reducibility in Kσ [x].Example . Let K =Q(i) and once again let σ denote complex conjugation. In this case,x –  and x –  are irreducible in Kσ [x] =Q[x]. However, since ,  ∈ N(Q,σ ), these poly-nomials become reducible in Q(i)[x;σ ] as we havex –  =(x + ( – i))(x – ( + i))and x –  =(x + ( – i))(x – ( + i)).On the other hand, since –,  /∈ N(Q(i),σ ), the polynomials x +  and x –  are irre-ducible in Q[x] and remain irreducible in Q(i)[x;σ ].3 Main theoremsIn order to complete our description of the factoring of polynomials in C[x;σ ], we ﬁrstneed to prove a result that holds in any algebraically closed ﬁeld.Theorem . Let K be an algebraically closed ﬁeld and let σ be an automorphism of K oforder n. Then every non-constant reducible skew polynomial in K[x;σ ] can be written as aproduct of irreducible skew polynomials of degree less than or equal to n.Proof Since σ has order n, K has dimension n over the subﬁeld Kσ . It is well known that thecentre of K[x;σ ] is the polynomial ring Kσ [xn]. Now suppose that f (x) is a non-constantelement of K[x;σ ]. Since K[x;σ ] is a ﬁnite module over the Noetherian ring Kσ [xn], K[x;σ ]contains no inﬁnite direct sums of Kσ [xn]-submodules. As a result, the sumf (x)Kσ[xn]+ f (x)Kσ[xn]+ f (x)Kσ[xn]+ · · ·cannot be direct. Therefore, there exist a(x), . . . ,am(x) ∈ Kσ [xn], with at least two nonzeroai(x), such thatf (x)a(x) + f (x)a(x) + · · · + f (x)mam(x) = .Bergen et al. Advances in Diﬀerence Equations  (2015) 2015:57 Page 5 of 6By cancelling any common factors of f (x) on the left, we obtainaj(x) + f (x)aj+(x) + · · · + f (x)m–jam(x) = .Observe that aj(x) is a nonzero element of Kσ [xn] ∩ f (x)K[x;σ ]. Therefore, there existsg(x) ∈ K[x;σ ] such that f (x)g(x) is a nonzero element of Kσ [xn].Since K is algebraically closed and n-dimensional over Kσ , non-constant polynomials inKσ [y] can all be factored into a product of polynomials of degree at most n. If we let y = xn,thenf (x)g(x) ∈ Kσ [xn] = Kσ [y].Therefore, there exist non-constant h(y), . . . ,hm(y) ∈ Kσ [y], all of degree at most n, suchthatf (x)g(x) = h(y) · · ·hm(y).For each i, the polynomial hi(y) = hi(xn) ∈ Kσ [x] can also be factored in Kσ [x] as a productof non-constant polynomials of degree at most n. As a result, we have actually factoredf (x)g(x) in K[x,σ ] as a product of polynomials of degree at most n.By Theorem  of Chapter II in [], the degrees of factors of f (x)g(x) are at most n.Thus f (x) has been factored in K[x;σ ] as a product of polynomials of degree at most n.Using Theorem . and Corollary ., we now have the following.Corollary . Every non-constant polynomial in C[x;σ ] is a product of linear and irre-ducible polynomials. Furthermore, the quadratic f (x) = x+ax+a is irreducible inC[x;σ ]if and only if there does not exist some t ∈C such that tσ (t) + at + a = .Proof Theorem . asserts that every non-constant polynomial in C[x;σ ] is a product oflinear and irreducible polynomials. In addition, Corollary . tells us that f (x) = x +ax+a is irreducible in C[x;σ ] if and only if f (x) does not have a σ -root in C. However, whenwe look at the deﬁnition of σ -roots in the quadratic case, it is clear that f (x) does not havea σ -root if and only if there does not exist t ∈ C such that tσ (t) + at + a = . Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.Author details1Department of Mathematical Sciences, DePaul University, Chicago, IL 60614, USA. 2David R. Cheriton School ofComputer Science, University of Waterloo, Waterloo, ON, Canada. 3Department of Mathematics, University of Manitoba,Winnipeg, MB, Canada.AcknowledgementsWe would like to thank the anonymous referees for their careful reading and valuable comments that have improved thereadability of this article. The research of the ﬁrst author was supported by a grant from the University Research Council atDePaul University.Received: 28 October 2014 Accepted: 3 February 2015Bergen et al. Advances in Diﬀerence Equations  (2015) 2015:57 Page 6 of 6References1. Ore, O: Theory of non-commutative polynomials. Ann. Math. 34(22), 480-508 (1933)2. Bergen, J: Derivation algebras of rings related to Heisenberg algebras. J. Algebra Appl. 3(2), 181-191 (2004)3. Bergen, J, Grzeszczuk, P: Jacobson radicals of ring extensions. J. Pure Appl. Algebra 216(12), 2601-2607 (2012)4. Bergen, J, Grzeszczuk, P: Skew power series rings of derivation type. J. Algebra Appl. 10(6), 1383-1399 (2011)5. 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Giesbrecht, M, Zhang, Y: Factoring and decomposing Ore polynomials over Fq(t). In: Proceedings of the ACM 28thInternational Symposium on Symbolic and Algebraic Computation (ISSAC ’03), pp. 127-134. ACM, New York (2003)15. Lam, TY, Leroy, A: Vandermonde and Wronskian matrices over division rings. J. Algebra 119(2), 308-336 (1988)

Factorizations for difference operators

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