Computing diagonal form and Jacobson normal form of a matrix using Gr"obner bases

Computing diagonal form and Jacobson normal form of a matrix using Gr¨ obner bases Viktor Lev ando vskyy , Kristina Sc hindelar Lehrstuhl D f ¨ ur M athematik, R WTH Aac hen, T emplergrab en 6 4, 52062 Aac hen, Germany [Viktor.Lev andovskyy,Kr istina.Schindelar]@math.rwth-a achen.de Septem b er 26, 2018 Abstract In this paper we present tw o algorithms for the computation of a di- agonal form of a matrix ove r non- comm utative Euclidean d omain ove r a field with the help of Gr¨ obner bases . This can b e view ed as the pre- processing for the computation of J acobson normal form and also used for the computation of S mith normal form in th e comm utative case. W e prop ose a general f ramew ork for hand ling, among other, op erator alge- bras with rational co efficients. W e employ special ”p olynomial” strategy in Ore localizations of non-commutativ e G -algebras and sho w its merits. In particular, for a given matrix M w e provide an algorithm to compute U, V and D w ith fraction-free entries suc h that U M V = D holds. The p olynomial approac h allow s one to obtain more precise information, than the rational one e. g. about singularities of the system. Our implementa tion of p olynomial strategy show s very impress ive per- formance, compared with metho ds, which directly use fractions. In p ar- ticular, w e exp erience q uite mo derate sw ell of co efficients and obtain un- complicated transformation matrices. This shows that this metho d is w ell suitable for solving nontrivia l practica l problems . W e presen t an i mple- mentatio n of algorithms in Singular:Plural and compare it with oth er a v ailable s ystems. W e leav e questions on the a lgorithmic complexit y of this algorithm op en, but we stress the practical app licabili ty of the p ro- p osed meth o d to a bigger class of non- comm utative algebras. Con ten ts 1 In tro duction 2 2 Algebras, Locali zations and their Prop erties 3 3 Gr¨ obner Bases in the Computation of a Di agonal F orm 8 3.1 Y oga with Gr¨ obner Bases . . . . . . . . . . . . . . . . . . . . . . 8 1 3.2 W orking with Left and Right Mo dules . . . . . . . . . . . . . . . 9 3.3 Polynomial Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Implementation and Examples 18 4.1 Jacobso n F orm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Examples, Applications and Compar is on . . . . . . . . . . . . . . 21 5 Conclusion and F uture W ork 25 1 In tro duction The existence a nd computation of normal forms of matrice s o ver a ring is a fundamen tal mathematical question. The pro of for the existence o f a normal form is mainly constructive and can b e turned in to a n algor ithm. How ever, suc h a direct alg orithm is not very efficient in g eneral. Computer alg ebra fo cuses its attent ion on this kind of pr oblems, sinc e they are o f elementary interest but of high complexity . In tha t sense nearly any computer alg ebra system is able to compute the Smith norma l form for a matrix ov er a commut ative principal ideal domain ( Z or K [ x ] for a field K ). There are man y textb o oks giving a theoretical ba c kground, like fo r instance [10, 29]. W e present a meth o d, whic h is b ased on Gr¨ obner bases. In [19], ther e is a Gr¨ obner basis-ba sed algor ithm for the co mputation of Smith normal form o f a matrix with entries in K [ x ]. Despite the fact that this approa c h seems to be folklore, we were not a ble to find other r eferences. In this pa per w e consider no n-comm utative skew poly nomial rings . Such rings, among other, offer the p ossibility to describ e time v arying systems in Systems and Control theory [32], [17], [1 8]. Many known oper ator algebras can be realized as skew p olynomial rings or so lv able p olynomial rings [22], some of them can be rea lized ev en as muc h easier Ore a lg ebras [9, 8]. Ho wev e r, general solv able p olynomia l ring s are hard to tac kle constructively (say , in a computer algebr a sy stem), while the class of Ore a lgebras of [9, 8] is indeed restrictive. Ba sed on the PBW algebras [5] also k no wn as G -algebras [23, 15], in Section 2 we prop ose a new class of univ ariate skew po lynomial ring s, which are obtained as Ore loca lizations of G -alg ebras. This framework is p ow erful and conv enien t at the s ame time. Mo reov er, it is more gener a l than the clas s of Ore algebras (with defining endomor phism σ b eing an automorphism) and allows algorithmic treatment of mo dules. In Pr o pos ition 2.2 and Theorem 2.6 several nice prope rties o f such algebras (amo ng other , these algebras are Noetheria n domains with PBW basis ) are established. W e stress, that the computations in these alg e bras, especia lly Gr ¨ o bner bases for mo dules, are alg o rithmic and, moreov er, they can be done without using explic it fractions. It is impor ta n t, that such alg ebras and computations in them can b e realized in a n y co mputer algebra s y stem, which ca n handle G -algebras o r poly nomial Ore algebra s. In [11], applica tions to systems of partial differen tial equations are shown and 2 several c oncrete exa mples are introduced. W e generalize the idea, originating from [1 9], to use Gr¨ o bner bases in computation of normal for ms for ma trices. The crucial improvemen t is introduced in Sectio n 3.3, where we s ho w how to handle the problem in a completely fraction-free p olynomial framework. W e p oin t out adv antages of the poly nomial strateg y and illustrate s ome of them with interesting examples in the Section 4, where we compare our implemen tation with other av aila ble pack ages. In pa rticular, we do comparisons with the implementation of alg orithms, which use fractions directly . Notably , in many exa mples our approach deliv ers muc h mo re compact results with small co efficien ts. The non-commutativ e analogue to the Smith form over a principal ideal do- main is the Jacobso n form [20] ,[10]. How e v er, since the normal form problem is hard in general, w e prop ose the notion of a weak Jacobs on form, that is a diag- onal matr ix, where the units on the diag onal will not b e necessar ily g enerated. Otherwise the adv antage of the p olynomial strateg y is disturb ed. Instead, we prop ose the splitting of the who le pro cess of obtaining a (s tr ong) normal form int o the co mputation of a diago nal for m and the computation of str o nger di- agonal form fr o m a given dia gonal one. The latter, a s w e s how in 4.4, 4.5 and 4.6 dep e nds heavily on the do main one computes in, while the fir st a lgorithm is very gener al. Our implement ation (of weak Jacobson and Smith forms) is realize d as the library jaco bson.lib [30] for the co mputer algebr a system Singular : Plural [16, 1 5], which is freely av aila ble. The libra ry has bee n alre a dy inco rpo rated int o the official distribution of Singular version 3 -1-0. 2 Algebras, Lo calizations and their Prop erties The framework of this pa per is ba s ed o n skew po ly nomial rings that are principal ideal do mains. An imp o rtant sub class of skew p olynomial r ings cons titute so - called p olynomial O re rings. They are non-commutativ e rings p osses sing a n endomorphism σ and a σ -deriv atio n to define the co mmutation r ule of tw o elements, that is giving the extension from co mm utative p olynomial ring to non- commutativ e. This kind of rings is used in a nalyzing the structure o f ana lytic equations, like linear or dinary or partial different ial equa tions or partia l shift or difference equa tions with ratio nal o r p olyno mia l co efficients, see Exa mple 2.3. The na me is inspired by Øystein Ore, who in tro duced a nd studied this kind of rings. These rings were also studied, fo r instance in [9] and [27]. Let K b e a field and A b e a K -alg ebra. F urther let σ : A → A be a ring endomorphism. Then the map δ : A → A is called σ - deriv ation , if δ is K - linear a nd satisfies the skew Leibniz rule δ ( ab ) = σ ( a ) δ ( b ) + δ ( a ) b for all a, b ∈ A. F or a σ -deriv ation δ the ring A [ ∂ ; σ, δ ] consisting of a ll p olynomials in ∂ with co efficien ts in A with the us ual addition and a pr oduct defined by the commu- 3 tation rule ∂ a = σ ( a ) ∂ + δ ( a ) fo r all a ∈ A is called skew p olynom ial ring or a n Ore extensi on of A with ∂ sub ject to σ , δ . It is ea sy to s ee, that any non-z ero element a ∈ A [ ∂ ; σ , δ ] can b e written a s a = a n ∂ n + · · · + a 1 ∂ + a 0 , wher e n ∈ N 0 and a i ∈ A . W e call n the degree of a , sometimes it is also called the order of a . In describing K -alge br as via finite s e ts of genera tors G a nd rela tio ns R , we write A = K h G | R i . It means that A is a fa c tor algebr a of the free a sso ciative algebra, genera ted b y G mo dulo the t wo-sided ideal, generated by R . Hence yet another notation is A = K h G i / h R i . Example 2.1. • Defining σ := id A and δ := 0 we see, that ( K [ x 1 , . . . , x n ])[ ∂ ; σ , δ ] = K [ x 1 , . . . , x n , ∂ ] a nd K ( x 1 , . . . , x n )[ ∂ ; σ , δ ] = K ( x 1 , . . . , x n )[ ∂ ]. • Let A = K [ x ] for a field K of characteristic 0 , σ := id K [ x ] and δ := ∂ ∂ x . W 1 ( K ) := K [ x ][ ∂ ; id K [ x ] , ∂ ∂ x ] = K h x, ∂ | ∂ x = x∂ + 1 i is c alled the first p olynom i al W eyl algebra . Prop osition 2.2. [5] Let A be a division ring, σ : A → A be an endomorphism and R = A [ ∂ ; σ , δ ] b e an Ore extension with a σ -deriv ation δ . If σ is injectiv e (resp ectiv ely bijective), then • (PID) R is a left (resp. r igh t) principal ideal domain. • (Bezout’s Theorem) for any non- zero a, b ∈ R there exists the rig h t (res p. left) greatest common divisor g r (resp. g ℓ ) of a, b and there exist s, t ∈ R , such that g r = sa + tb (r esp. s ′ , t ′ , such that g ℓ = as ′ + b t ′ ). • (ED) R is a left (resp. r igh t) Euclidean domain. Hence, when σ is bijectiv e, there ar e left and right Euclidea n division a lgo- rithms. In the next example w e enlist s o me interesting sk ew polyno mial r ings (whic h are Ore algebras indeed, see [9]). These rings a re of g reat interest in applications, all of them can b e a ddresses with o ur implementation, see Section 4. Example 2.3. Let A = K ( x ), where K is a field of characteris tic 0 . • Let σ := id K ( x ) and δ := ∂ ∂ x . Then B 1 ( K ) := A [ ∂ ; id K ( x ) , ∂ ∂ x ] = K ( x ) h ∂ | ∂ x = x∂ + 1 i is c alled the first rational W eyl algebra . 4 • The firs t rational diffe rence algebra is defined by S 1 := A [∆; σ , δ ] = K ( x ) h ∆ | ∆ x = x ∆ + ∆ + 1 i , where σ ( p ( x )) = p ( x + 1) and δ ( p ) = σ ( p ) − p for all p ∈ K ( x ). Let q 6 = 0 b e a unit (a para meter) in the ground field. • Let σ ( p ( x )) = p ( q x ) and δ := ( ∂ ∂ x ) q , δ ( f ( x )) = f ( q x ) − f ( x ) ( q − 1) x . Then W q 1 ( K ) := A [ ∂ ; σ , ( ∂ ∂ x ) q ] = K ( x ) h ∂ | ∂ x = q · x∂ + 1 i is c alled the first rational q -W eyl algebra . • The firs t rational q -diff e rence algebra is defined by Q := A [ ∂ ; σ , δ ] = K ( x ) h ∂ | q · x∂ + ( q − 1) x i , where σ ( p ) = p ( q x ) and δ ( p ) = p ( q x ) − p ( x ). Indeed, we can work within the more general algebraic fr amew ork a s fo llo ws. Let S b e a multiplicativ ely closed s et (see [27]) in a No etherian integral domain A , such that 0 6∈ S . S is ca lle d an Ore set in A , if for all s 1 ∈ S, a 1 ∈ A there exis t s 2 ∈ S, a 2 ∈ A , such that a 1 s 2 = s 1 a 2 . Then o ne can see, that formally (that is, allowing fra ctional expres s ions) s − 1 1 a 1 = a 2 s − 1 2 holds. Then one defines a ring of fractions or an Ore l o calization o f A with resp ect to S to b e a ring A S (often denoted as S − 1 A ) together with an injectiv e homomorphism φ : A → A S , such that (i) for all s ∈ S , φ ( s ) is a unit in A S , (ii) for all f ∈ A S , f = φ ( s ) − 1 φ ( a ) for some a ∈ A, s ∈ S . The Or e prop erty of S in A g ua rantees, that a n y left-sided fra ction can b e written (non-uniquely!) as a right-sided fra ction. Moreover, given a 1 , . . . , a m ∈ A and s 1 , . . . , s m ∈ S , there exist a ′ 1 , . . . , a ′ m ∈ A and s ′ ∈ S , such that a i s ′ = s i a ′ i holds for eac h i . Thus there exist common righ t and common left m ultiples. Remark 2.4. Wh y such loca lizations are important? Among man y motiv ating connections let us state the following. Given an A -mo dule homomorphism ϕ : M → N , where M , N are finitely generated. Then, if S − 1 A exists, o ne has an induced homomorphism of S − 1 A -mo dules S − 1 ϕ : S − 1 M → S − 1 N . How ever, if o ne finds an appropriate multiplicativ ely closed Ore set ˜ S in A and proves that ˜ S − 1 ϕ : ˜ S − 1 M → ˜ S − 1 N is not an isomorphism, it implies tha t M 6 ∼ = N as A -mo dules. This g ives an impo rtant to ol to chec k the isomorphy o f mo dules. In contrast with common lo caliza tions o f co mm utativ e ring a t c o mplemen ts of prime idea ls, we do no t know a pr iori for which S we ar e lo oking for and how many differ en t S should we examine. Note, that the question, whether tw o mo dules ar e isomorphic, is one of the fundamen tal questions in a lgebra. I t is known to b e not algorithmic in general, hence any partial algorithmic ans wer to this que s tion is of big imp ortance . 5 Definition 2. 5. Let A b e a quotien t of the free ass ocia tiv e algebr a K h x 1 , . . . , x n i by the t wo-sided ide a l I , generated b y the finite set { x j x i − c ij x i x j − d ij } for all 1 ≤ i < j ≤ n , where c ij ∈ K ∗ and d ij are p olynomia ls in x 1 , . . . , x n . Without loss o f gener alit y [23] we ca n ass ume that d ij are giv en in terms of standard monomials x a 1 1 . . . x a n n . A is c alled a G –algebr a [24, 2 3], if • for all 1 ≤ i < j < k ≤ n the expression c ik c j k · d ij x k − x k d ij + c j k · x j d ik − c ij · d ik x j + d j k x i − c ij c ik · x i d j k reduces to zer o modulo I a nd • there exists a mono mial o rdering ≺ on K [ x 1 , . . . , x n ], such that for each i < j , such tha t d ij 6 = 0, lm( d ij ) ≺ x i x j . Her e, lm sta nds for the clas s ical notion of leading mono mial of a p o lynomial from K [ x 1 , . . . , x n ]. W e call an ordering o n a G -alg ebra admi ssible , if it satisfies second c o ndi- tion of the definition. A G -algebra A is Noe therian integral domain [24], hence there ex is ts its to ta l tw o -sided ring of fra ctions Quot( A ) = A A \{ 0 } , which is a division ring (skew field). Assume that A is gener ated b y x 1 , . . . , x n +1 and sup- po se that the set Λ n ( A ) = { λ = { i 1 , . . . , i n } | i 1 < . . . < i n , K h x i 1 , . . . , x i n | I λ i is a G -alg ebra } is not empty , where I λ = { x j x i − c ij x i x j − d ij | i, j ∈ λ, i < j } . F or any λ = { i 1 , . . . , i n } ∈ Λ n , let us define B λ to b e a G -a lgebra, genera ted by { x i 1 , . . . , x i n } . Theorem 2.6. Le t A be a G -a lgebra in v aria ble s x 1 , . . . , x n , ∂ and assume that λ = { x 1 , . . . , x n } ∈ Λ n . Mor eov e r , let B := B λ and B ∗ = B \ { 0 } . Supp ose, there exists an a dmissible mono mial or dering ≺ on A , satisfying x k ≺ ∂ for all 1 ≤ k ≤ n . Then the following holds • B ∗ is multiplicatively closed Or e set in A . • ( B ∗ ) − 1 A (O re lo calization of A with res pect to B ∗ ) c a n b e presented as an Ore extension of Q uot( B ) by the v aria ble ∂ . Pr o of. Since B is a G -algebra itself, it is an integral domain, hence B ∗ is mul- tiplicatively closed and do es not contain zero . Since A and B are G -algebra s and ≺ is a n a dmissible or dering, for a re la tion ∂ x j = c j x j ∂ + d j with c j ∈ K ∗ and a p olynomial d j ∈ A holds d j = 0 or lm( d j ) ≺ x j ∂ . Sin ce x j ≺ ∂ , then x j ∂ ≺ ∂ 2 , hence d j is at mos t linear in ∂ . W r iting d j = a j · ∂ + b j for a j , b j ∈ B , we define c ′ j = c j x j + a j and thus we obtain a relation ∂ x j = c ′ j ∂ + b j , where x j , c ′ j , b j ∈ B . Then, by defining σ ( x j ) = c j x j + a j and δ ( x j ) = b j for all 1 ≤ j ≤ n , we s ee, that σ is a n automorphism of Quo t( B ). Th us an Ore extens ion Quot( B )[ ∂ ; σ, δ ] is indeed another presentation o f ( B ∗ ) − 1 A as so on as B ∗ is a n Ore set in A . Since lm( d j ) = lm ( a j ∂ + b j ) ≺ x j ∂ , b oth lm( a j ) ≺ x j and lm ( b j ) ≺ x j ∂ hold. The latter implies, that there exist po sitiv e weigh ts ω a nd w 1 , . . . , w n for v aria bles { ∂ , x 1 , . . . , x n } , such that for lm( a j ) = x α and lm ( b j ) = x β one has P i w i α i ≤ w j and P i w i β i ≤ w j + ω . In par ticular, this can b e a c hieved by setting ω big enough. Then w e follow the recip e from [5] a nd construct a blo ck ordering from this setting. Consider an ordering ≺ ∂ on A , which is a blo ck ordering for blo c ks of v ar iables { ∂ } , { x 1 , . . . , x n } . It means that ∂ ≫ x j for all j , that is the v aria ble ∂ is bigger than any p ow er of x j . The s econd block is an 6 ordering ≺ B on B , for which lm( a j ) ≺ B x j holds. F or ins tance, one ca n take ≺ B to b e the restr iction o f ≺ to B . Then lm( d j ) = max ≺ ∂ ( a j ∂ , b j ) ≺ ∂ x j ∂ holds, hence ≺ ∂ is admissible ordering on A . F rom the Pro positio n 28 of [13] (whic h holds fo r muc h mo re g eneral situation), the exis tence of such a blo ck ordering a s ≺ ∂ implies, tha t the set B ∗ is a n Ore set in A . Remark 2.7. Note, that by constructio n A B ∗ := ( B ∗ ) − 1 A is a Euclidean (prin- cipal ideal) doma in by the Pr opos ition 2.2. In par ticular, all but one v a riables are inv ertible (w e call them also r ational v a riables). W e call non-inv ertible v ar i- ables p olynomial . In a more gener al setting, we like to pres e n t lo calizations of the type A B ∗ , where B is a sub- G -a lgebra o f A , as a r ing of solv a ble t yp e [22] or, equiv a len tly , as a P BW r ing [5]. In the case of several polyno mial v ariables, the a na logue to the Theorem 2.6 seem to be muc h more involv ed. Example 2 . 8. T o illustrate the Theor em 2.6, consider the difference algebra S 1 := K h x, ∆ | ∆ x = x ∆ + ∆ + 1 i . Since ∆ ≺ x ∆ is a consequence of 1 ≺ x (w e assume we ar e dealing with well-orderings only), S 1 can b e loc a lized at bo th K [ x ] ∗ and K [∆] ∗ . How ever, the algebr a, asso ciated with the o pera tor of partial int egratio n I 1 := K h x, I | I x = xI − I 2 i ca n be lo calized only at K [ I ] ∗ but not at K [ x ] ∗ , since I 2 ≺ xI is a co nsequence o f I ≺ x and any or dering, satisfying x ≺ I is no t admissible for I 1 . F or ma n y problems in mo dule theory a nd in applications we would like to analyze complicated problems via lo calizing at big subalgebr a s. In the situation as ab ov e , w e obtain non-c o mm utative Euc lidea n domain as the r esult, hence we are interested in computing Ja cobson form in this setting. One of the complica- tions, which ar ise in co nstructiv e handling of o b jects ov er suc h a lgebras, is quite hard arithmetics in the skew field. Several fundamental questions lik e the tr a ns- formation of a left frac tio n into the rig h t one (which is p o ssible, since the Ore condition is s atisfied), simplifica tion of a one-s ided fraction etc. re q uire quite nontrivial and c omplex algor ithms (like computation of syz y gy modules and so on) to b e used, see for instance [1]. Ev en in the comm utative ca se the com- putations (even with o ne v ariable) over a transcendental extensio n by se veral generator s are still nontrivial and reso urce-consuming for mo st computer alge- bra sy stems. Hence saying “ring R is a (non-commutativ e) Euclidea n domain” do es not automatically mean “computations in R are easy”. Remark 2.9. Let us come bac k to the justification o f terminology . Usually , sp eaking on “op erator algebra with p olynomial co efficien ts”, o ne means that one w orks with the s et of op erators ∂ 1 , . . . , ∂ m ov er a c omm utative po lynomial ring, say , K [ x 1 , . . . , x n ]. By saying “oper ator algebra with rational co efficients” one addresses an Ore extension o f K ( x 1 , . . . , x n ) by the ope r ators ∂ i . It is imp ortant to men tion, that K ( x 1 , . . . , x n ) is a lo calization of K [ x 1 , . . . , x n ] with resp ect to m ultiplicatively closed se t K [ x 1 , . . . , x n ] \ { 0 } . Thus it is enough to define an algebr a with p olynomial co efficien ts and then sp eak on different lo calizations of it. Therefor e the notion of Or e lo calization r e v eals the origin of 7 v ario us “rationa l” coe fficie nts a nd a llows to treat different lo calizations (among them e.g . pa ssage to the to rus K [ x ± 1 1 , . . . , x ± 1 n ] ⊂ K ( x 1 , . . . , x n )) uniformly . 3 Gr¨ obner Bases in the Computation of a Diag- onal F orm 3.1 Y oga with Gr¨ obner Bases Let us give a short in tro duction to non-co mm utative Gr¨ obner basis theory , whic h has b een studied b y e. g. [7, 22, 2 3]. Supp ose, that there is a G - algebra R ∗ ov er a field K , which is genera ted by x 1 , . . . , x n , ∂ , such that R ∗ = A ∗ [ ∂ ; σ , δ ] is an Ore extension of a G -a lgebra A ∗ , gener ated by { x i } . By using the low er index ∗ , w e p oin t out that we deal with structures, ob jects in which always have a po lynomial presentation. A nice pro p erty of a G -algebr a is that as a K - v ector space it is g enerated by mono m ials o f R ∗ : Mon( R ∗ ) = { x α 1 1 · . . . · x α n n ∂ k | α ∈ N n , k ∈ N } = { x α ∂ k | x α ∈ Mon( A ∗ ) , k ∈ N } . Based on a mo dule ordering we define leading coe fficien t (lc), leading monomial (lm), lea ding term (lt) a nd le ading p osition (lp os) notions a s usual. Let e i := (0 , . . . , 1 , . . . , 0 ) b e the i -th unit vector. In this pap er w e will compute Gr¨ obner ba s is of modules ov er R ∗ with respect to an monomial mo dule ordering P OT (pos ition-ov e r-term), defined as follows. F or r , s ∈ Mon( R ∗ ), re i < se j ⇔ i < j or if i = j then r < s, (1) and r < s with resp ect to an admissible well-ordering on R ∗ , eliminating ∂ , that is satisfying ∂ ≫ x n > · · · > x 1 on R ∗ . In R , a Gr ¨ obner bas is is co mputed with res pect to the induced POT or de r ing, which takes only degree of ∂ into account since Mon( R ) = { ∂ k | k ∈ N } . W e call a ∈ R ∗ a s tri ct le ft (resp. right) divis or of b ∈ R ∗ if a nd only if ∃ f ∈ R ∗ such that af = b (resp. f a = b ). Extending this notation to R p ∗ requires that both elemen ts a, b ∈ R p ∗ hav e the same leading position. Moreover, a is said to be a prop er strict divisor of b , if either b = af o r b = f a holds, where f is not a n unit in R ∗ . F or t wo monomials m 1 , m 2 ∈ R ∗ we wr ite m 1 ≤ m 2 for the comparison with the fixed monomial ordering . W e say that m 1 divides m 2 , if each exp onent o f m 1 is no t gre a ter than the corres p onding exp onent of m 2 . Definition 3 .1. Let M b e a left submodule of R p ∗ and < b e a monomial module ordering on R p ∗ . A finite subs e t G ⊂ M is called a Gr¨ obner Basis of M with resp ect to < , if for every f ∈ M \ { 0 } there ex is ts a g ∈ G , s o that lm( g ) divides lm( f ). A Gr¨ obner ba sis G is called reduced if and only if for any pair of po lynomials h 6 = f ∈ G , the leading monomial lm( h ) do es not div ide a n y monomial of f . It can b e shown, tha t a normalized (that is with leading co efficients 1) re duce d 8 Gr¨ obner ba sis is unique for a fixed o rdering. W e recall the common prop erty of a Gr¨ o bner basis to b e, in particula r, a gener a ting set. Remark 3.2. Let M ⊆ R p ∗ with a Gr¨ obner bas is G and f ∈ M . Define the submo dule S of M to be genera ted b y a ll s ∈ G such that lm( s ) ≤ lm( f ). Then f ∈ S . 3.2 W orking with Left and Right Mo dules Opp osite algebra . In or der to work with left and right mo dules ov er an a s- so ciative K -alge br a A , o ne ha s to use b oth A and its opp osite alge br a A op in general. Recall, that A op is the same vector space as A , endow ed with the op- po site multiplication: ∀ a, b ∈ A op , a ⋆ A op b = b · a . A natural opp osing ma p makes from a right (r e sp. left) A -module a left (resp. right) A op -mo dule. There is an alg orithmic pro cedure to s et up an opp osite algebr a to a given G -alg ebra, see [23]. In v ol utiv e anti-automorphism . Alternatively , for “ sw apping sides” one can employ a n ant i-automor phism θ of A , that is a K -linear map, which ob eys θ ( ab ) = θ ( b ) θ ( a ) for all a , b ∈ A , which is inv olutive, that is θ 2 = id A . Often such an an ti-automor phism is called in v o lution . In clas sical o per a tor alge- bras, particular ly simple involutions a re known [8]. Moreov er , it is p ossible to determine linear ly pr esen ted in volution of a G -algebr a v ia an alg orithm (Le v an- dovskyy et al., unpublished, see Singular library i nvolut. lib [2] for an imple- men tation). A constructive adv ant age o f using in volution versus using opp osite algebra lies in the fa c t, tha t one do es no t need to crea te o pposite a lgebra and make to an o b ject its o ppos ite. Instead, we apply an in v olution to an ob ject and remain in the same ring. One application of inv olution mea ns that the ob ject we dea l with change its side fr o m left to right or vice versa. An inv olution can b e defined on matrices as follows. Let θ : A → A b e an inv o lution a s ab ov e. W e define the map e θ : A p × q → A q × p , M 7→ ( θ ( M )) T , where M T is the tra nspos e d matrix of M and θ ( M ) = [ θ ( M ij )] for 1 ≤ i ≤ p and 1 ≤ j ≤ q . One can e asily s ho w that ( θ ( B · C )) T = ( θ ( C )) T · ( θ ( B )) T for B ∈ A p × q , C ∈ A q × k . Applied t wice, we g e t B · C back. Diagonalization . Let R b e a K -alg ebra and a non-comm utative Euclidean PID. Recall, that a matrix U ∈ R p × p is called unimo dul ar if and only if there exists U − 1 ∈ R p × p such that U U − 1 = U − 1 U = id p × p . Let M ∈ R p × q and assume, without loss of genera lit y , that p > q . Then one can show, that ther e 9 exist unimo dular matrices U ∈ R p × p and V ∈ R q × q such that U M V =      m 1 0 . . . 0 m q 0 p − q      . There a re several ways to prove this sta temen t, a ll based on the E uclidean (and thus PID) prop erty of the underly ing ring. F r om now o n, w e assume that R is a lo calization of a G -algebra as in Remark 2.7. W e present algo rithms to o btain diag onal form together with unimo dular tr ansformation ma trices v ia Gr¨ obner bas es. The main idea ab out the computation is the s equen tial alter- nation b etw ee n the computation of a reduced Gr¨ obner basis of the submo dule, generated b y , say , the rows of a matrix a nd acting by the inv olution e θ o n a s ub- mo dule. In the PhD thesis [19] this idea w as applied to K [ x ] (of course, without using an involution θ , which is sup erfluous in that case) in o rder to co mpute a Smith nor mal form. In the following, by R M we denote the left R - module genera ted b y the r ows of a matrix M . F urther on, by G ( R M ) w e denote the reduced left Gr¨ obner basis of the submo dule, gener ated by R M with resp ect to the mo dule ordering (1 ). F or the i -th row of a matrix M we write M i and M ij stands, as usual, for the entry in the i -th row and j -th column. With r e spect to the context we identify G ( R M ) = { g 1 , . . . , g m } with the matrix [ g t 1 , . . . , g t m ] t . Define the degree of an element 0 6 = m ∈ R 1 × q to b e the degree of the corr esponding leading mono mial, tha t is, deg( m ) := deg(lm( m )). Since R is a P ID, this degree measur e s the highest expo nen t in the v ar iable ∂ . F ollowing the standard conv en tion, deg(0) = −∞ . Note that the elements o f G ( R M ) hav e pairwise distinct leading monomials, since they form a reduced Gr¨ o bner basis. In a reduced Gr¨ obner bas is lm ( G ( R M ) i ) | lm( G ( R M ) j ) if and only if G ( R M ) i = G ( R M ) j . Lemma 3.3. Order a reduced Gr¨ o bner basis in suc h a w ay , that lm( G ( R M ) 1 ) < · · · < lm ( G ( R M ) m ). Then    G ( R M ) 1 . . . G ( R M ) m    is a low e r triangular matr ix. Pr o of. Suppose the claim do es no t hold. Then ther e exists G ( R M ) i and G ( R M ) j with lp os( G ( R M ) i ) = lpo s( G ( R M ) j ) for i < j . Th us lm( G ( R M ) i ) = ∂ α i e k and lm( G ( R M ) j ) = ∂ α j e k such that α i < α j . But then evidently lm( G ( R M ) i ) divides lm( G ( R M ) j ), which is a contradiction to G ( R M ) b eing re duced. Due to the pr e v ious lemma, we may assume witho ut loss o f genera lit y , that the matrix G ( R M ) is lo wer triangular. Since R is an in tegral domain, we define 10 the r ank of a matrix M to b e the rank o f M ov er the field o f fractio ns of R . Now, let us assume that p = q and M is of full ra nk , that is r o w a nd column ranks o f M are equal to p . The no n-square cas e will b e discussed in Remark 3.7. Lemma 3. 4. Let I deno te the left ideal genera ted by the elements in the last column of e θ ( G ( R M )), that is, b y θ ( G ( R M ) p 1 ) , . . . , θ ( G ( R M ) pp ). Then I = R h G ( R e θ ( G ( R M ) ) ) pp i . Pr o of. Note, that due to Lemma 3.3    ∗ . . . . . . G ( R M ) p 1 · · · G ( R M ) pp    | {z } G ( R M ) e θ    θ ( G ( R M ) p 1 ) . . . . . . ∗ · · · θ ( G ( R M ) pp )    G    ∗ . . . . . . ∗ · · · G ( R e θ ( G ( R M ) ) ) pp    . According to the definition of G the left ideal generated by G ( R e θ ( G ( R M ) ) ) pp coincides with R h θ ( G ( R M ) p 1 ) , . . . , θ ( G ( R M ) pp ) i . Now we ca n formulate the alg orithm that yields the des ired diagona l fo r m. Algorithm 3.5 ( Di agonali zation with Gr¨ obner B ases ) . Input: M ∈ R g × g of full rank, e θ inv olution as ab ov e. Output: Matrices U , V , D ∈ R g × g , such that U, V are unimo dular a nd U · M · V = Diag( r 1 , . . . , r g ) = D. M (0) ← M , U ← id g × g , V ← id g × g i ← 0 while ( M ( i ) is not a dia gonal matrix or i ≡ 2 1) do i ← i + 1 Compute U ( i ) such that U ( i ) · M ( i − 1) = G ( R M ( i − 1) ) M ( i ) ← e θ ( G ( R M ( i − 1) )) if ( i ≡ 2 0) then V ← V · e θ ( U ( i ) ) else U ← U ( i ) · U end if end whil e return ( U, V , M ( i ) ) Theorem 3.6. The Algorithm 3.5 terminates and it is c o rrect. That is, for M ∈ R g × g , let M ( i ) denote the matrix we get after the i -th execution of the wh i le lo op. Then ther e exists an element k ∈ N such tha t M ( k ) is a diagonal ma tr ix. If k is o dd, then the whil e lo op is re peated just one mo r e time (define l := k + ( k mo d 2 ) in this case). The ma trices U, V obtained in the la s t lo op are unimodula r and satisfy U M V = Diag( m 1 , . . . , m g ). 11 Pr o of. W e prov e the cla im by induction on g , the size of the square matrix M . F or g = 1 there is no thing to show. Using Lemma 3.4, th e eq ualit y R h θ (( M ( i +1) ) gg ) i = R h ( M ( i ) ) 1 g , . . . , ( M ( i ) ) gg i holds . Hence we get R h ( M ( i ) ) gg i ⊆ R h θ (( M ( i +1) ) gg ) i for all i. Note that θ preserves the degree. Then the previous inclusion implies by degree arguments that R h ( M ( r ) ) gg i = R h ( M ( r +1) ) gg i for so me r . Using Lemma 3 .4 and ( M ( r ) ) gg 6 = 0 (since M is of full ra nk), we obtain tha t ( M ( r ) ) gg is a strict left divisor of ( M ( r ) ) ig for ea ch 1 ≤ i ≤ g − 1. Then the de finitio n of G y ields that M ( r +1) =      0 M ′ . . . 0 0 . . . 0 ( M ( r +1) ) gg      . , or, in a different notation, M ( r +1) = M ′ ⊕ ( M ( r +1) ) gg , that is M ( r +1) is a blo c k matrix. The ( g − 1) × ( g − 1) matrix M ′ can b e transfor med to a diago na l ma tr ix via unimo dular op erations b y induction. It remains to consider the transformatio n matrices U and V . F o r each i ∈ N , after exec uting the whil e lo op i times, we obtain ( M ( i ) = U ( i − 1) · U ( i − 3) · · · U (1) · M · e θ ( U (2) ) · e θ ( U (4) ) · · · e θ ( U ( i ) ) , if i is even M ( i ) = U ( i − 1) · U ( i − 3) · · · U (1) · e θ ( M ) · e θ ( U (2) ) · e θ ( U (4) ) · · · e θ ( U ( i ) ) , if i is o dd, which co mpletes the pro of. Remark 3.7. In orde r to extend Theorem 3.6 and Algorithm 3.5 to non-square and non- full ra nk matrices, we need to add suitable syzygies to U resp ectively V and zero rows respectively columns to the diag onal matrix, in order to maintain the initial s ize of M . F or a computational so lution it is sufficien t to extend Algorithm 3.5 in the fo llowing wa y . Let M i ∈ R s × t where either s = p, t = q or s = q , t = p in the i - th while lo op. Instead of co mputing U i , sa tisfying U i · M i − 1 = G ( R M i − 1 ), w e compute G ( R ˜ M ) for the extended matrix ˜ M := [id s × s M i − 1 ]. Suc h ˜ M is ob viously a full row rank ma tr ix. Defining U i := [ G ( R ˜ M ) T 1 , . . . , G ( R ˜ M ) T s ] T and M i := [ G ( R ˜ M ) T s +1 , . . . , G ( R ˜ M ) T t ] T , it is e asy to see that U i M i − 1 = M i . The matrix M i consists of the rows of G ( R M i − 1 ) and additional zer o rows, such that M i ∈ R s × t . 3.3 P olynomial Str ategy W e are given a matr ix M ov er a non-co mm utative Euclidean domain R . In this section, we s ho w our main a pproach of this chapter. W e intro duce a method that allows to execute the Algorithm 3 .5 in a completely po lynomial (that is, fraction-free) fr a mew ork. The idea comes fro m the commutativ e case and was app eared e. g. in [1 4]. 12 Let A ∗ be a G -algebr a a nd A = Quot( A ∗ ). Moreover, let R = A [ ∂ ; σ , δ ], such that R ∗ = A ∗ [ ∂ ; σ , δ ] is a G -alge bra. Evidently R ∗ ⊆ R , since A ∗ ⊆ A . Without los s of gener alit y , w e supp ose tha t M do es not co n tain a ze ro row. W e define the de gree of an elemen t in R ∗ (or R 1 × g ∗ ) to b e the weigh ted deg ree function with weigh t 0 to any gener a tor of A ∗ and weigh t 1 to ∂ . Thus this weigh ted deg ree o f f ∈ R ∗ coincides with the degr ee of f in R . Such degree is clearly inv ariant under the multip lication of elements in A ∗ . Lemma 3.8 . Le t M ∈ R g × q . Then there exists an algo rithm to co mpute a R -unimo dular matrix T ∈ R g × g ∗ such that T M ∈ R ∗ g × q . Pr o of. If M ∈ R ∗ g × g , there is nothing to do . Supp ose that M contains elements with fractio ns. At first, w e show how to bring t w o frac tio nal elements a − 1 b, c − 1 d for a, c ∈ A ∗ , b , d ∈ R ∗ to a common left denominator, cf. [1]. F or any h 1 , h 2 ∈ A ∗ , such that h 1 a = h 2 c , it is e a sy to s e e that ( h 1 a ) − 1 ( h 1 b ) = a − 1 h − 1 1 h 1 b = a − 1 b and ( h 1 a ) − 1 ( h 2 d ) = ( h 2 c ) − 1 ( h 2 d ) = c − 1 d, hence ( h 1 a ) − 1 = a − 1 h − 1 1 = ( h 2 c ) − 1 is a common le ft denominator . Analogo us ly we can compute a common left denominator for any finite s et of fractions. Let T ii be a common left denominator o f non-zero elements from the i -th row of M , then T M contains no fractions. Moreov er, T is a diagonal matrix with non-ze ro po lynomial entries, so it is R -unimo dular. Remark 3.9 . Note that the computation o f compatible factors h i for a 1 , a 2 ∈ A ∗ can b e achiev ed by computing syzyg ies, since { ( h 1 , h 2 ) ∈ A 2 ∗ | h 1 a 1 = h 2 a 2 } is exa ctly the mo dule S y z ( a 1 , − a 2 ) ⊂ A 2 ∗ . The factors h i for mo re a i ’s ca n b e obtained as well. Define M ∗ := T M ∈ R p × q ∗ using the notatio n of Lemma 3.8. Then the relations R ∗ M ∗ ⊆ R M and R M ∗ = R M hold obviously . Thus whenever we sp eak a bout a finitely genera ted submo dule R M ⊂ R 1 × q , we denote by R M ∗ a presentation of R M with generato r s co n tained in R ∗ . In what follows, we will show how to find R -unimo dula r matrices U ∈ R p × p ∗ and V ∈ R q × q ∗ such that U ( T M ) V =      r 1 . . . r q 0      ∈ R p × q ∗ . Since the equality U ( T M ) V = ( U T ) M V holds and U T is a R -unimo dular ma - trix, our initial aim follows. As in the previo us subsection, by G ( R ∗ M ∗ ) we denote the reduced left Gr¨ o bner basis of the submo dule R ∗ M ∗ with r espect to the mo dule ordering < ∗ on R ∗ , which w as alrea dy defined in (1). Unlike the rational case, the lea ding monomi- als of elements in R 1 × g ∗ are of the fo r m x α 1 1 · · · x α n n ∂ β for α i , β ∈ N . 13 Remark 3.10. Using the p olynomial stra tegy , tw o improvemen ts c an b e ob- served. On the one hand, once we hav e mapp ed the ma trix we w ork with from R g × q to R g × q ∗ , the complicated ar ithmetics in the skew field of fractions is not used an ymore. The o ther improvemen t lies in the nature of cons truction of normal for ms for matrices and the cor resp onding transfor mation matrices. The naive appro ach would b e to apply elementary o p eratio ns inclusive division by inv er tibles on the rows and co lumns, that is , op erations fro m the left and from the rig h t. Indeed, there a re metho ds us ing different techniques like, fo r instance, p -adic ar gumen ts to calculate the inv aria n t facto rs of the Smith fo r m ov er in- tegers [26], but this method do es not help in construction of transfor mation matrices. Surely the swap from left to r ight has no influence in the commu- tative fra mew ork. But a lready in the ra tional W ey l alg ebra B 1 (see Example 2.3), 1 x is an unit in B 1 and ∂ 1 x = 1 x ∂ − 1 x 2 . Compar ing the mult iplication w ith the inv erse element, that is, with x , we see that ∂ x = x∂ + 1 holds. Thus a m ultiplication o f any po lynomial containing ∂ with the element 1 x in the field of fractio ns causes an immediate co efficien t swell. Since a normal form o f a matrix is given modulo unimo dular op erations, the previous example illustrates the v ariations of p ossible representations. In section 4, we present nontrivial examples. Especia lly in the Example 4.10, the p olynomial strategy dams up the co efficient increase in a very impr essive wa y . On the other hand, switc hing to the po ly nomial fra mew ork changes the setup. The ring R ∗ is not a P ID anymore, what was the esse ntial prop erty for the existence of a diagonal form over R . In the se quel, we s ho w how that this pro blem can b e r esolved b y intro ducing a suitable sorting condition for the chosen mo dule or de r ing. Referring to the a rgument ation of Remark 3.3 yields the blo ck-diagonal form with the 0 blo ck ab ov e. G ( R ∗ M ∗ ) =                           0 . . . . . . 0 * . . . 0 ∗ * . . . 0 ∗ . . . * ∗ . . . ∗                           . (2) Moreov er, the r o ws with the boxed elemen t ha v e the smalles t leading monomial with resp ect to the chosen ordering in the co rresp onding blo ck. A blo ck denotes all elements of same leading position in G ( R ∗ M ∗ ). In Theorem 3.15 we sho w that these elements indeed generate R M , while in Lemma 3.13 we show that these elements provide us with additional information. How e ver, this re s ult r equires some pre pa rations. 14 Lemma 3.11. Let P b e R or R ∗ . F o r M ∈ P g × q of full rank and every 1 ≤ i ≤ g , define α i := min { deg( a ) | a ∈ P M \ { 0 } and lp os( a ) = i } . Then for all 1 ≤ i ≤ g , there exists h i ∈ G ( P M ) of degree α i with lpo s( h i ) = i . Pr o of. Recall that ∂ ≫ x j for all j . Let f ∈ P M with lpo s( f ) = i and deg( f ) = α i . Supp ose that for all g ∈ G ( P M ) with leading p osition i , deg( g ) > α i holds. Since G ( P M ) is a Gr¨ obner basis, ther e exists g ∈ G ( P M ) such that lm ( g ) divides lm( f ). This happens if a nd only if deg( g ) ≤ deg ( f ) (beca use R ∗ is a G -algebr a and R is an Ore PID), whic h yields a contradiction. The full r ank a ssumption in the lemma g uarantees the existence o f α i for each comp onent 1 ≤ i ≤ g . Note, that ov er R ∗ the cardinality of { deg ( a ) | a ∈ P M \{ 0 } and lp os( a ) = i } is more than one in g e neral, hence ther e might be different s e lection strateg ies. W e pr o pos e to s elect an element accor ding to min < ∗ , see Lemma 3 .13. Corollary 3.12. Lemma 3.1 1 and Le mma 3.3 yield deg( G ( R M ) i ) = min { deg( a ) | a ∈ R M \ { 0 } and lp os( a ) = i } . Lemma 3.13. Let α i be the degree of the b oxed ent ry with leading p osition in the i - th column, tha t is α i := deg( min < ∗ { b | b ∈ G ( R ∗ M ∗ ) and lp os( b ) = i } ) . Then for all h ∈ R M with lp os( h ) = i w e have deg (lm( h )) ≥ α i . Pr o of. No w suppo se the claim do es not hold and there is h ∈ R M with lpo s( h ) = i of degree smaller than α i . Using Lemma 3.8, there exists a ∈ A ∗ such that ah ∈ R ∗ M ∗ . Then deg( ah ) = deg( h ) and lpos( ah ) = i . Due to L e mma 3 .11, deg( f ) ≥ α i for a ll f ∈ R ∗ M ∗ with leading p osition i , hence we obtain a contradiction. Corollary 3.14. Lemma 3.1 3 and Co rollary 3.12 imply , that ∀ 1 ≤ i ≤ g min { deg( a ) | a ∈ R M \{ 0 }∧ lp os( a ) = i } = min { deg ( a ) | a ∈ R ∗ M ∗ \{ 0 }∧ lp os( a ) = i } . Theorem 3.15. L e t M ∈ R g × g be of full r ank. F o r each 1 ≤ i ≤ g , let us define b i := min < ∗ { b | b ∈ G ( R ∗ M ∗ ) and lp os( b ) = i } . Since M is of full ra nk, the minimum ex ists for each 1 ≤ i ≤ g . Note that the set { b 1 , . . . , b g } corresp onds to the subset o f all rows with a b oxed entry in the blo c k triang ular form 2. Mo reov er R h b 1 , . . . , b g i = R M . Pr o of. Let f ∈ R M \{ 0 } . Due to Co rollary 3.14, there ex ists 1 ≤ k ≤ g such that lp os( b k ) = lp o s( f ) and deg( b k ) ≤ deg( f ). Thus there exists an element s k ∈ R such that deg( f − s k b k ) < deg( b k ). Since f − s k b k ∈ R M , Corolla ry 3.14 implies that we have lpo s( f − s k b k ) < lpos ( f ). Itera ting this reduction leads to the r e mainder zero a nd th us f = P k i =1 s i b i . 15 Using the notatio n of the previous theorem, let G ∗ ( R M ) := [ b 1 , . . . , b g ] T , which is by definition a low er tr iangular matrix . In the s equel, let M ∈ R g × g be of full r ank. Note that then o b viously G ∗ ( R M ) is a square matrix. Prop osition 3.16. Suppo s e M ∈ R g × g is a full rank matrix and there is U ∗ ∈ R l × g ∗ such that U ∗ M ∗ = G ( R ∗ M ∗ ). Let us select the indices { t 1 , . . . , t g } ⊆ { 1 , . . . , l } such that { ( U ∗ M ∗ ) t 1 , . . . , ( U ∗ M ∗ ) t g } = G ∗ ( R M ) (3) in the notation of Theor em 3.15. Then U := [( U ∗ ) t 1 , . . . , ( U ∗ ) t g ] T is R -unimo dular in R g × g and U M ∗ = G ∗ ( R M ). Pr o of. The eq ualit y U M ∗ = G ∗ ( R M ) follows by the definition of U . Now we show that U is R -unimodula r. Note that R ( U M ∗ ) = R G ∗ ( R M ) = R M = R M ∗ holds and U M ∗ ⊂ R g × g ⊃ M ∗ . Thus there exists V ∈ R g × g such that M ∗ = V ( U M ∗ ). Then V U = id g × g and a nalogously U V = id g × g since M ha s full r o w rank. Lemma 3.17. The equality of the fo llo wing left ideals ho lds: R h θ ( G ∗ ( R M ) g 1 ) , . . . , θ ( G ∗ ( R M ) gg ) i = R h G ∗ ( e θ ( G ∗ ( R M )) gg i . Pr o of. Using the argumentation given in the pro of of Lemma 3.4 we obtain R h θ ( G ∗ ( R M ) g 1 ) , . . . , θ ( G ∗ ( R M ) gg ) i = R hG ( e θ ( G ∗ ( R M )) gg i . Note the mo dule identities R G ∗ ( R M ) = R G ( R M ) ⇒ e θ ( G ∗ ( R M )) R = e θ ( G ( R M )) R ⇒ R G ∗ ( e θ ( G ∗ ( R M ))) = R G ( e θ ( G ( R M ))) . According to the latter identit y a nd to the fact that b oth G ( e θ ( G ( R M )) and G ∗ ( e θ ( G ∗ ( R M ))) are lower triangular matric e s, we obtain R hG ( e θ ( G ( R M )) gg i = R h G ∗ ( e θ ( G ∗ ( R M )) gg i . Now we a re ready to formulate the p olynomial version of Algo rithm 3.5. 16 Algorithm 3.18 ( Polyno mial di agonali zation with Gr¨ obner B ases ) . Input: M ∈ R g × g of full rank, θ an in volution on R ∗ and e θ as ab ov e . Output: R -unimo dular matric es U , V , D ∈ R g × g ∗ such that U · M · V = D = Diag( r 1 , . . . , r g ). Find T ∈ R g × g unimo dular such that T M ∈ R g × g ∗ M (0) ← T M , U ← T , V ← id g × g i ← 0 while M ( i ) is no t a diag o nal matrix or i ≡ 2 1 do i ← i + 1 Compute U ( i ) so that U ( i ) · M ( i − 1) = G ( R ∗ M ( i − 1) ) ∈ R l × g ∗ Select { t 1 , . . . , t g } ⊆ { 1 , . . . , l } as in (3) U ( i ) ← [( U ( i ) ) t 1 , . . . , ( U ( i ) ) t g ] T M ( i ) ← e θ ( G ∗ ( R M )) if i ≡ 2 0 then V ← V · e θ ( U ( i ) ) else U ← U ( i ) · U end if end whil e return ( U, V , M ( i ) ) Remark 3. 19. It is imp ortant to mention, that the matrices U, V , D (hence the elements r i as w ell) hav e entries from R ∗ , that is poly nomials. How ev er, U and V are o nly unimo dular over R and, in general, they need not be unimodular ov er R ∗ for obvious reasons . In each presented example we will inv estigate the case, whether U or V will b e unimo dular over R ∗ as well. After all, we co me up with the Conjecture 4.1 3. Theorem 3.20. Algorithm 3.18 terminates with the co rrect result. Pr o of. Using Pro positio n 3.16 and replacing Lemma 3 .4 by Lemma 3 .1 7 in the pro of of Theo rem 3.6 provides the claim. Example 3.21 . Supp ose R = K ( x )[ ∂ ; id , d dx ] and R ∗ = K [ x ][ ∂ ; id , d dx ]. Let us define a n in volution on R ∗ by θ ( ∂ ) = − ∂ and θ ( x ) = x . Let M =  ∂ 2 − 1 ∂ + 1 ∂ 2 + 1 ∂ − x  ∈ R 2 × 2 . Evidently T = id 2 × 2 and thus M (0) := M , U = V = id 2 × 2 and i = 0. 1: Since M (0) is no t diago nal, go into the while lo op • i ← 1. Since   − x∂ − ∂ + x 2 + x + 1 x∂ + ∂ + x − ∂ 2 + x∂ − ∂ + x + 2 ∂ 2 + 2 ∂ + 1 ∂ − x − ∂ − 1   M (0) = G ( R ∗ M (0) ) where G ( R ∗ M (0) ) =   x 2 ∂ 2 + 2 x∂ 2 + ∂ 2 + 2 x∂ + 2 ∂ − x 2 − 1 0 x∂ 3 + ∂ 3 + x∂ 2 + 5 ∂ 2 − x∂ + 3 ∂ − x − 1 0 − x∂ 2 − ∂ 2 − 2 ∂ + x − 1 1   and i ≡ 2 1 17 M (1) ←  x 2 ∂ 2 + 2 x∂ 2 + ∂ 2 + 2 x∂ + 2 ∂ − x 2 − 1 − x∂ 2 − ∂ 2 + x − 1 0 1  U ←  − x∂ − ∂ + x 2 + x + 1 x∂ + ∂ + x ∂ − x − ∂ − 1  2: Since M (1) is no t diago nal, go into the while lo op • i ← 2 . Since  1 x∂ 2 + ∂ 2 − x + 1 0 1  M (1) = G ( R ∗ M (1) ) and i ≡ 2 0 M (2) ← h x 2 ∂ 2 + 2 x∂ 2 + ∂ 2 + 2 x∂ + 2 ∂ − x 2 − 1 0 0 1 i , V ← h 1 0 ( x + 1) ∂ 2 + 2 ∂ − x + 1 1 i 3: Since i is even a nd M (2) is diago nal, the alg orithm returns U and V . And indeed, the algorithm outputs the c laimed res ult, since U M V =  x 2 ∂ 2 + 2 x∂ 2 + ∂ 2 + 2 x∂ + 2 ∂ − x 2 − 1 0 0 1  . In view of Remark 3.19, let us analy z e the tr ansformation matrices for R ∗ - unimo dularit y . Indeed, V is such since it admits an inv er se V ′ . O n the contrary , U is o nly unimodula r over R and not over R ∗ , since U · Z = W and W is first inv er tible in the lo calization: V ′ = h 1 0 − ( x + 1 ) ∂ 2 + x − 2 ∂ − 1 1 i , Z = h 2 ∂ + 2 ( x + 1) ∂ + x − 2 2( ∂ − x ) ( x + 1) ∂ − x 2 − x − 3 i , W = h 0 − 4 x 2 − 8 x − 4 2 5 x + 5 i . Algorithm 3.18 ca n b e extended to M ∈ R g × q along the lines alre ady pre- sented in Remark 3.7. Example 3.22. By ex ecuting the algor ithm in the 1 st rational shift alg ebra K h t, S | S t = tS + S i on the same matrix as in the previous example, where ∂ is repla c ed with the forward shift op era to r S , we obtain a diagonal for m Diag( ( t 2 + 3 t + 2) S 2 + 2 ( t + 1) S − t 2 − t + 2 , 1) = h − tS − S + t 2 + 2 t tS + S + t + 2 − S + t + 1 S + 1 i · h S 2 − 1 S + 1 S 2 + 1 S − t i · h 1 0 − tS 2 − 2 S 2 − 2 S + t 1 i . As in the previous example, it turns out that V (but not U ) is even R ∗ - unimo dular. 4 Implemen tation and Examples 4.1 Jacobson F or m Let R be a left and rig h t Euclidea n domain. Inspir ed by the Smith form, we will fo cus o n how to sha rpen the r e s ult of the alrea dy discussed diago nal form. 18 Theorem 4. 1. [10, 2 0] Every matrix M ∈ R g × q is ass o cia ted to a certain diagonal matr ix, namely Diag ( m 1 , . . . , m ℓ , 0 , . . . , 0) such that additiona lly Rm i +1 R ⊆ m i R ∩ R m i (4) holds for all i = 1 , . . . , min { g , q } − 1. Due to [20, Theo rem 31 ] the elements m i are unique up to similar it y . Two elements m i and n i are called simil ar if a nd only if there exist a, b ∈ R such that am i = n i b, R = aR + n i R, R = Rb + Rm i . Using the nota tion of the previo us theorem, we call Diag ( m 1 , . . . , m ℓ , 0 , . . . , 0) a Jacobson n o rmal form o f M . Note that (4) is ha rd to ta ckle construc- tively in ge ne r al, s ince it requir es to work with the intersection of a le ft and a right ideal. This difficulty disapp ears if R has only trivial tw o-sided ideals, that is when R is simple. Then eac h matrix M p ossesses a Jac o bson form Diag(1 , . . . , 1 , m M , 0 , . . . , 0) with m M ∈ R . Lemma 4. 2. Let A ∗ be a G -alg ebra, A = Quo t( A ∗ ) a nd R = A [ ∂ ; σ , δ ]. Let U, V be unimo dular and a, b, c, d ∈ R \ { 0 } such that U Diag( a, b ) V = Diag ( c, d ) . (5) Then deg ( a ) + deg( b ) = deg( c ) + deg( d ). Pr o of. Due to (5 ) there exists a R - module isomorphism φ : R/aR ⊕ R /bR → R/cR ⊕ R /dR. Since A is a skew field, φ induces an A -v ector space iso mo rphism. Thus the A -dimensions of R /aR ⊕ R/ b R and R/cR ⊕ R/dR , which are no thing e ls e that the sums of degrees, coincide. Of cour se, inductive arg umen t implies that sums of degre e s of diag onal en- tries of t wo diagonal pres e n tation matrices of the same mo dule are the sa me. Jacobson normal form in the 1st W eyl algebra . Let R be the ra tional W eyl algebra K ( x )[ ∂ ; 1 , ∂ ∂ x ], which is a simple domain. Lemma 4.3. Consider a, b ∈ R with deg( a ) > 0, b 6 = 0 and deg ( b ) ≥ deg( a ). Then ther e exists i ∈ { 0 , . . . , deg( b ) − deg ( a ) + 1 } such that a is not a str ict right divisor of bx i . Pr o of. Suppose that for ev ery i ∈ { 0 , . . . , deg( b ) − deg ( a ) + 1 } ther e exists a q i ∈ R such that bx i = q i a . Let b = b n ( x ) ∂ n + · · · + b 1 ( x ) ∂ + b 0 ( x ). Note, that for any k ∈ N the e qualit y ∂ k x = x∂ k + k ∂ k − 1 . Thus we define r 1 := bx − xb = P n i =1 b i ( x ) i∂ i − 1 with deg ( r 1 ) = n − 1 < deg ( b ) and r 1 6 = 0 since deg ( b ) ≥ 1. Since b = q 0 a and bx = q 1 a , it follows that r 1 = b x − xb = ( q 1 − q 0 x ) a , that is 19 a is a strict right divisor of r 1 . By pro ceeding with b x 2 and so on, we o btain a sequence o f no n-zero po lynomials r i , such that deg ( b ) > deg( r 1 ) > . . . and a is a s trict right divisor of r i . Since the degr ee of r i decreases exactly by 1 a t each step, after a t mo st deg( b ) − deg ( a ) + 1 itera tions we obtain a p olynomial of degree deg( a ) − 1, which is non-zero . Such a po lynomial must contain a right factor o f degree deg ( a ), what is a contradiction. The Lemma (4.3) sugge s ts a n algorithm to compute the Jacobson form from a diagona l matrix over the ra tional W eyl alg ebra. Suppo se M ∈ R g × q , where g = q = 2. The extension to g , q ∈ N is evident. Algor ithm 3 .18 returns unimo dular matrices U, V such that U M V = Diag ( m 1 , m 2 ). Without loss of generality , assume that deg( m 2 ) ≤ deg( m 1 ). 1) If m 2 is a unit, we get the Jacobson form just by re pla cing U by Diag(1 , m − 1 2 ) U . Otherwise, choose a n exp onent i ∈ N (it exis ts by the Lemma 4 .3) s uch that m 1 x i = am 2 + b with deg( b ) < deg( m 2 ) and b 6 = 0. Then  1 − a 0 1  ·  m 1 0 0 m 2  ·  1 x i 0 1  =  m 1 b 0 m 2  . Replace U b y  1 − a 0 1  U and V by V  1 x i 0 1  . 2) Now we apply Algorithm 3.18 to the ma tr ix  m 1 r 0 m 2  . The res ult is then Diag( m ′ 1 , m ′ 2 ), wher e deg( m ′ 2 ) < deg( m 2 ). Thu s, by iterating 1) a nd 2) w e co mpute U and V , such that U M V = Diag (1 , m M ). Remark 4.4. It seems to us, that the pro cess of obtaining J acobson norma l form from an appropria te diago nal matr ix can b e genera lized to a n y construc- tive simple Euclidean PID. Moreov er, it ca n be applied even ov er non-s imple domains. Ther e, it is not guara n teed, that the r esult is so nice as Ja cobson form, but the pr ocedur e ab ov e ca n simplify dia gonal matrices. Example 4.5. Over the first r ational shift algebra A = K ( t ) h s | st = ts + s i (whic h is a not a s imple domain), we provide a counterexample for a statement, similar to 4.3. Co nsider the 2 × 2 diagonal matrix D 1 = Diag ( s, s ). Then the left mo dule M 1 = A 2 / A 2 D 1 (it is of dimensio n 2 over K ( t )) is annihilated by the tw o - sided ideal h s i and hence D 1 is not equiv a le n t to a matrix of the form D 2 = Diag(1 , p ). If it were so, due to the K ( t )-dimension of M 1 and hence M 2 = A 2 / A 2 D 2 , we see that lm( p ) = s 2 . Since M 2 = A 2 / A 2 Diag(1 , p ) ∼ = A/ Ap , we have Ann A M 2 = h p i . Since it is not e q ual to Ann A M 1 = h s i , M 1 6 ∼ = M 2 . Hence, unlike ov er the W eyl algebra (or a simple domain [1 0]), ther e are man y differ en t types of nor mal forms even for diagonal matric e s. 20 Example 4.6. Consider the ra tio nal q -W ey l algebra , cf. 2.3. It is not simple since e. g. the ideal h ( q − 1) ∂ + x − 1 i is a prop er tw o-sided ideal. Denote the generator by f , then, by the same arg umen tation as in the previous exa mple we can show, that Diag( f , f ) is not equiv alent to an y matrix o f the t yp e Diag(1 , g ). Since little is known ab out normal forms of non-simple do mains, this appr oach is very interesting to investigate in the future. Cyclic v ector m e tho d . Indeed, the existence of Jaco bson form in simple Euclidean PID is very strong result. In par ticular, it tells us tha t any finitely generated mo dule is cyclic and its presentation is a principal ideal. The metho d of finding a cyclic vector in a mo dule and o bta in a left ide a l, annihilating it, is used in D -module theory . J . Middeke in [28] did so me inv estigations of this question. Conjecture 4.7. W e conjecture, that the Ja cobson form for, say , square matr ix M ov er a simple Euclidean domain R can b e computed from the given dia go- nal for m in the following wa y . Le t M = Dia g( m 1 , . . . , m r ). Since P deg( m i ) is inv ariant of the mo dule R r × r / M , this n umber ca n b e used as a cer tificate for pr obabilistic a ppr oach. Namely , consider p olynomials p i of degree at most deg( m i ) − 1 with r andom co efficien ts in A . Compute a g e nerator c ∈ R of the left annihilator ideal of a vector [ p 1 , . . . , p r ] T in R r × r / M . If deg c = P deg( m i ), then Diag(1 , . . . , 1 , c ) is a Jacobson form of M . Otherwise o ne takes another set of rando m po lynomials p i and rep eats the pro cedure. One needs the probabilistic estimations on the length of r andom coefficients like in [2 1]. 4.2 Examples, A pplications and Comparison Implementations of Jacobson normal form. T o the best of our kno wledge, Jacobso n normal for m algorithm has b een implemented in Maple by Culia nez and Quadr at [11], b y Rob ertz et a l. [4, 8 ], by Middeke [28] a nd b y Cheng et al [3, 6, 12]. W e could not lo cate the download version of the implementation of [11]. The pack a ges FFreduce [3] and Mo dreduce [6] are av ailable via p ersonal request to their a uthors. The implementation of J. Middeke [28] was, a c cording to its author, merely a check of ideas a nd was not supp osed to bec ome a freely dis- tributed pack age for Maple . This pack age is a ble to compute in the 1st W eyl algebra with co efficients in a differential field. D. Ro bertz informed us, that his implementation [4] directly fo llows the clas- sical alg o rithm and it has not b een sp ecially optimized. Nevertheless, in what follows, we compa re o ur implementation with the one in the Maple pack a ge Janet [4 ] on so me nontrivial ex a mples. This pack age is av aila ble to general 21 public. In pa c k ages b y H. Cheng et al. mo dular ( Modreduce ) a nd fraction-free ( FFreduce ) versions of an order basis o f a po lynomial matrix M from a n O re algebra A ar e implemented. In particular , such a basis is used to compute the left nullspace of M , and indir ectly the Popov form of M . Examples. As w e a lready p ointed out in the introduction, b ehind diago nalized matrices and normal forms there are v ario us applicatio n- driven motiv atio ns , see e. g. [11]. Example 4 . 8. F or insta nce, co nsider a double p endulum with lengths ℓ 1 and ℓ 2 . Thus ℓ 1 , ℓ 2 and g ar e consta n ts, that is non-zero elements of K (for details see [11], Example 3.2.2). The linearizatio n of this problem leads to the sys tem of linear partial differential e quations in ∂ = ∂ ∂ t , which can b e wr itten in the matrix for m with the matrix M =  ℓ 1 ∂ 2 + g 0 − g 0 ℓ 2 ∂ 2 + g − g  . Since the v a r iable t do es not app ear in M , the ground ring for the diagonalizatio n pro cess can be thoug ht as of A = Q ( ℓ 1 , ℓ 2 , g )[ ∂ ]. Thus, indeed one ca n compute the Smith normal form. Our implementation o f the diag onal form of M o n this e x ample returns U =  − 1 /g 0 − 1 /g 1 /g  and V =   0 g ℓ 2 − g ℓ 2 ∂ 2 − g 2 0 g ℓ 1 − g ℓ 1 ∂ 2 − g 2 1 ℓ 1 ℓ 2 ∂ 2 + gℓ 2 − ℓ 1 ℓ 2 ∂ 4 − gℓ 1 − gℓ 2 ∂ 2 − g 2   such that U M V =  1 0 0 0 g ℓ 1 − gℓ 2 0  . This result ag rees with results, obtained in [11]. Note, that a purely fractio nal metho d (as well a s co efficient no rmalization pro cedure) will return 1 ins tea d of g ( ℓ 1 − ℓ 2 ). With our p olynomia l approa ch we o btain a p olynomia l matrix, which is us e ful for further inv estigations . In particular, in the current example we see, that setting ℓ 1 = ℓ 2 implies the drop of the r ank of the Smith form from 2 to one, thus the prop erties of the cor r espo nding system will change. In control theory one establishes quite different pro perties of the mo dule in the non-generic ca se ℓ 1 = ℓ 2 . Remark 4.9. In [25] the algorithm for finding so-called “obstructions to gener- icity” was derived and discuss e d. A lesson learned from that pap er can b e applied for an implement ation of Jaco bson form as follows. It is r ecommended to split the algorithm (resp. the implementation) int o tw o parts. In the fir st part one co mputes a diagona l matrix, where the inv e r tibles of the gr ound ring are not canceled a rtificially . The sec o nd pa rt applies the no rmalization on the inv er tibles; this par t is triv ia l to a c hieve. Note, that o ur p olynomial alg orithm allows o ne to keep a close track on suspicious in vertibles due to this s c heme. 22 Example 4.10. Over the fir st rational W eyl algebra Q ( t )[ ∂ ; id , d dt ], consider the matr ix R =   ∂ 2 ∂ + 1 0 ∂ + 1 0 ∂ 3 − t 2 ∂ 2 ∂ + 1 ∂ 3 + ∂ 2 ∂ 2   . A typical implemen tation of the Ja cobson normal form r e turns the matrix D = Diag ( g , 1 , 1) to gether with transfor mation matrices U, V ∈ Q [ t ][ ∂ ; id , d dt ] 3 × 3 such that U RV = D . Below, we write do wn just the leading term of ea c h matr ix ent ry and moreover, we wr ite “l.o .t.” for “low er order terms” with res pect to degree lexicographica l o rdering on Q [ t ][ ∂ ; id , d dt ]. The implement ation of the Algorithm 3.1 8 in Singular returns D = Dia g (2 t 2 d 8 + 33 l.o .t. , 1 , 1). The transformatio n ma tr ices are U =   1 2 t∂ 13 + 24 l.o.t. 1 2 t∂ 10 + 19 l.o.t. 1 2 t∂ 11 + 44 l.o.t. 1 2 0 0 − 1 4 ∂ 5 + 2 l.o.t. − 1 4 ∂ 2 1 4 + 2 l.o.t.   , and V =   2 t∂ 2 + 3 l.o.t. 2 ∂ 2 2 ∂ 2 + 1 l.o.t. − 2 t∂ 3 + 2 l.o.t. − 2 ∂ 3 + 3 l.o.t. − 2 ∂ 3 t∂ 8 + 28 l.o.t. ∂ 8 + 11 l.o.t. ∂ 8 + 16 l.o.t.   . In vie w of 3.1 9, V (but not U ) is unimo dular over R ∗ = Q [ t ][ ∂ ; id , d dt ]. Janet returns a matrix Diag(1 , 1 , (27 9936 t 14 + 14 l.o.t. ) − 1 (27993 6 t 14 ∂ 8 + 145 l.o.t. )), U = " 1 0 0 (6 t 2 + 2 l.o.t.) − 1 ( ∂ 2 + 1 l.o.t.) (6 t 2 + 2 l.o.t.) − 1 ( ∂ 3 + 3 l.o.t.) (6 t 2 + 2 l.o.t.) − 1 u 31 u 32 u 33 # , where g = (5598 72 t 14 +14 l.o.t.) , u 31 = g − 1 ( − 279936 t 14 ∂ 9 +158 l.o.t.) , u 32 = g − 1 (279936 t 14 ∂ 10 + 182 l.o.t.) , u 33 = g − 1 (279936 t 14 ∂ 7 + 127 l.o.t.) . The right transfo rmation matrix V =   1 1 2 ∂ 6 + 15 l.o.t. (279936 t 14 + 14 l.o.t. ) − 1 (46656 t 12 ∂ 7 + 110 l.o.t. ) ∂ + 1l.o.t. − 1 2 ∂ 7 + 15 l.o.t. ( − 1679614 t 16 + 16 l.o.t. ) − 1 (279936 t 14 ∂ 8 + 138 l.o.t. ) 0 1 (6 t 2 + 2 l.o.t. ) − 1 (2 ∂ 2 + 1 l.o.t )   . Example 4.11. Conside r the matrix from the E x ample 4 .10, replacing ∂ b y S , the forward shift o pera tor in the firs t rational shift algebra in t, s . Then the diagonal form, computed with our algorithm is Diag ( t 12 S 8 + 101 l.o.t. , 1 , 1). No- tably , the leading co efficient in t factorizes completely . T r ansformation matrices are, as exp ected, mor e co mplicated a s in the Example 4 .10. U has only thr ee ent ries of leng th bigger than 1; their lengths a re 113, 116, 15 0. In the matrix V , the lengths o f entries are 22 , 1 1, 58, 20 , 1 4, 60, 26 , 17, 64 with deg ree in S up to 7. Co efficient s, ha ving more than 7 digits appea r only in one en try , and gr ow up to 12 digits. The situation in the first r ational difference algebra is similar , as a reader can see b y computing with our implementation. W e hav e computed all the examples from this pap er in the shift and difference settings as well. 23 Example 4.12. Let R = Q ( y , x )[ ∂ ; id , d dx ] and th us R ∗ = Q [ y , x ][ ∂ ; id , d dx ]. The matrix M b elow comes from the s ystem of pa r tial differential eq uations. With our a lgorithm we o btain trans fo rmation ma tr ices and a diag onal one: M =  y 2 ∂ 2 + ∂ + 1 1 x∂ x 2 ∂ 2 + ∂ + y  ,  − x 2 ∂ 2 − ∂ − y 1 1 0  M  1 0 − y 2 ∂ 2 − ∂ − 1 1  =  g 0 0 1  , where g = − y 2 x 2 ∂ 4 − x 2 ∂ 3 − x 2 ∂ 2 − y 2 ∂ 3 + x∂ + ( − y 3 − 1) ∂ 2 + ( − y − 1 ) ∂ − y . If w e consider M ∈ Z 2 ( y , x )[ ∂ ; id , d dx ] 2 × 2 , w e obtain the single ex ample from [28]. Then the r ational form of o ur result is ex a ctly the r esult obtained in [2 8], namely Dia g (1 , − g x 2 y 2 mo d 2). Note, that in our method no co mputations w ith 4 × 4 matrices a s in [28] ar e needed. As demonstrated, our implemen ta tion works ov er finite fields as well. And, as be fo re, the r ight tra nsformation matrix V is unimo dular even ov er R ∗ = Q [ y , x ][ ∂ ; id , d dx ]. As we hav e seen, in all the ex a mples ab ov e the right transfor mation matrix V was indeed unimo dular ov er R ∗ . W e obs erve this phenomeno n for even more examples over W eyl and shift algebras. Conjecture 4.13 . Let A ∗ be a G -algebr a a nd A = Q uot( A ∗ ). Moreover, let R = A [ ∂ ; σ , δ ], s uc h that R ∗ = A ∗ [ ∂ ; σ , δ ] is a G -alg e bra. F o r a matrix M ∈ R p × p there exist square matrices U, V , D with ent ries from R ∗ , such that U M V = D , where D is diagona l a nd U, V unimo dular ov er R . If D has o nly one po lynomial non-constant en try , then V can b e chosen to be unimo dular ov er R ∗ . Application . Over R , the decomp osition a s ab ov e can b e applied as follows. W e s tart with a sys tem o f equations M ω = 0 in unknown functions ω = ( ω 1 , . . . , ω p ). Since U and V a r e unimo dular over R a nd U M V = Diag( d 11 , . . . , d pp ), we obta in a decoupled system { d ii z i = 0 } , where z = V − 1 ω , which is equiv alent to M ω = 0 over R . Note, that d ii = 0 is p ossible, then one calls z i a free v aria ble o f the system in the literature (e. g. in [31]). Let us analyze what ca n be done ov er R ∗ . Suppose, that V is unimo dular ov er R ∗ . Then U M ω = 0 ⇔ D V − 1 ω = 0. Ho wev er, since U is no t unimo dular ov er R ∗ , we have implication M ω = 0 ⇒ U M ω = 0 only . Let T b e a matrix, such that T U = id R , then, by a re a soning, similar to Lemma 3.8 ther e exists a diagonal matrix Q = Diag( . . . , q ii , . . . ) such that Q resp. QT have with entries from A ∗ resp. R ∗ . F or simplicity , assume that A ∗ is co mmutative. Denote by S the multiplicativ ely clos ed set, genera ted by q , the least common m ultiple o f { q ii } . If S happ ens to b e an Ore set in R ∗ , then the lo caliza tion S − 1 R ∗ exists and U will b e unimo dular over S − 1 R ∗ . F urther computations happ en in different branches: first in the generic S − 1 R ∗ , wher e by U M ω = 0 ⇒ M ω = 0 and then in the ca se, determined by the relation q = 0. In the latter, one can a pply the algorithm Generi city from [25], which delivers a disjoint decomp osition of the set of zero s of q into lo cally closed s ets L i . One can pr o ceed w ith analys is of sys tems U M w = 0 alo ng L i and obtain s pecial solutions on each L i . This 24 shows, that the left transformation matrix U in this setting carries essential information ab out the so -called singula rities of a system. Note, that working ov er R we compute only gener ic informa tion, while following the p olynomial strategy over R ∗ allows us to make a complete des cription of the sy stem. Clearly the decoupling, provided by a diago nal form, is of big impor ta nce for solving systems of op erator equations with ratio na l co efficien ts and fo r the structural ana lysis, p erformed in the algebr a ic system and control theor y (see e. g. Theorem 8 of [31]). 5 Conclusion and F uture W ork Indeed, this pap er is a part o f a gener al progr am o n providing effectiv e co mputa- tions within O r e lo calized G -a lgebras. Notably , p olynomial strateg y , whic h we describ ed in details for the case of o ne polyno mia l v ariable, is one of the key ele- men ts of the progra m. Ther e is ongoing work on the implemen tation of Gr ¨ obner bases for Ore lo calize d G -algebra s under the co dename Singular:: Locap al . Polynomial strategy br ings us several adv antages in pr actical computations. One of them is the generality of the ov erall a ppr oach. Namely , as so on as there is an implement ation of Gr ¨ obner ba ses for mo dules (and hence syzygies) ov er a G -algebra A , under so me mild a ssumptions we a r e able to work effectively with Ore lo calization A B ∗ of A with re s pect to a multiplicatively clo sed Ore set B ∗ , where B is a suitable G -subalgebra of A (cf. Theor em 2 .6). The question, whether dire ct computations with fractions of A B ∗ will b e alwa ys outp erformed by the p olynomial strateg y , is still ope n. Consider , for instance, the s ituation, wher e the input matr ix M is given a lready with r atio- nal non-c omm utative co efficients. Then br ing ing M to the frac tion-free form is already a nontrivial op eration (as so on as we w ork with non-commutativ e algebra), a s indicated e. g. in the pro of of Le mma 3.8. In our o pinion the answer to the ab ov e questio n depends b oth on the algebra A B ∗ and on the pre s en tation matr ix M . How ever, in general nontrivial com- putation directly using fractions in the alg orithm might cause the app earance of enormous co efficien ts, a s several examples demonstra te. W e wan t to stres s, that these examples have not b een sp ecially sele c ted for this purp ose; instead, we picked a couple o f them from a bigger family of ex a mples. In our opinion, this pheno menon is quite ubiquitous. Our implementation of the Jaco bson normal form w ill b e developed further to pr ovide a user with the p ossibility to compute in mo r e ge ner al algebr as. A t the momen t, the stable version of the library [30] supp orts first W eyl, shift a nd difference algebras . Investigation of normal for ms ov er non-simple domains (as in 4 .5, 4.6) is an imp ortant future task. Middeke [2 8] has rep orted, that the classica l algor ithm, computing Jacobson form of a matr ix over the W eyl a lgebra ov er a differential field is p olynomial- time. Ho wev er, it seems to us (due to p olynomial strategy a pproach), that the 25 subalgebra of inv ertible elements must b e involv ed in the complexity analys is. Perhaps one should consider different mo dels for studying complexity , s inc e ex - per ience with pra ctical applications sugg ests, that the imp ortant ro le, pla yed b y the co efficien t ar ithmetics (which is not the a rithmetics ov er a numerical field anymore!) must be a ppr opriately reflected in the ov erall complexit y . O ther wise the complexity o f o pera tions over the skew field of inv ertible elements rema ins hidden. Recently , Mark Gies brech t a nd George La bahn suggested the use of another techn ique from [21], namely the r andomization. Starting with a matrix M , one m ultiplies M with random square (hence unimodular) matrices from bo th sides, in or der to reduce the num b er of iterations in the Algo rithms 1 and 2 . So me exp erimen ts co nfirm that this might b e g eneralized to the setting of lo calized G -algebra s . How ever, the computations be come muc h har der in practice due to increa sed size of p olynomials to deal with. 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