A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings

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1 Open Math. 25; 3: Open Mathematics Open Access Research Article Xiangui Zhao* and Yang Zhang A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings Abstract: Signature-based algorithms are efficient algorithms for computing Gröbner-Shirshov bases in commutative polynomial rings, and some noncommutative rings. In this paper, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings over fields. Keywords: Gröbner-Shirshov basis, Skew solvable polynomial ring, Signature-based algorithm MSC: 6S36, 3P DOI.55/math Received April 23, 24; accepted January 2, 25. Introduction Gröbner-Shirshov basis is a powerful tool in mathematics, science, engineering, and computer science. The theory of Gröbner-Shirshov bases was introduced independently by A.I. Shirshov [9] for Lie algebras in 962, and by B. Buchberger ([3]) for commutative algebras in 965. Buchberger([3]) gave the first algorithm to compute Gröbner- Shirshov bases in commutative polynomial rings. However, Buchberger s algorithm is not efficient since it has to reduce the S-polynomial for every pair of elements from the input set. There have been extensive efforts to improve the efficiency of Buchberger s algorithm in commutative polynomial rings, and several more efficient signaturebased algorithms have been proposed, such as F5 by Faugère ([6, 7]), G2V and GVW by Gao et al.([, ]). The essential idea in these algorithms is to detect useless S-polynomials, i.e., S-polynomials which can be reduced to zero (and thus the computations of these S-polynomials are redundant), in Buchberger s algorithm. Noncommutative Gröbner-Shirshov bases and their computations have also been widely investigated (see the survey [2]), especially for various skew polynomial rings, for example, Gröbner-Shirshov basis theory for Weyl algebras [9], solvable polynomial algebras [4], rings of differential operators [3, 6, 22], G-algebras [5], skew polynomial rings [5], differential difference algebras [7, 2], PBW algebras [4, 2] and skew PBW extensions [8]. Owing to the noncommutativity, it is difficult to detect and reject redundant computations effectively. In ISSAC 22, a signature-based algorithm was presented by Sun et al. [2] to compute Gröbner-Shirshov bases in solvable polynomial algebras. In this paper, we define skew solvable polynomial rings, which are generalizations of several well-known classes of rings such as solvable polynomial algebras and (skew) Poincaré-Birkhoff-Witt extensions (see Definition 2. and Examples 2.2 and 2.3). We extend the signature-based algorithm proposed in [2] to skew solvable polynomial rings *Corresponding Author: Xiangui Zhao: Department of Mathematics, Huizhou University, Huizhou, Guangdong, 567, China, xiangui.zhao@foxmail.com Yang Zhang: Department of Mathematics, University of Manitoba, Winnipeg, R3T 2N2, Canada, yang.zhang@umanitoba.ca 25 Xiangui Zhao and Yang Zhang licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3. License.

2 A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings 299 over fields. The signature-based algorithms for more general skew solvable polynomial rings will be investigated in the near future. This paper is organized as follows. In Section 2, we introduce basic definitions of skew solvable polynomial rings and Gröbner-Shirshov bases. Then we define and investigate strong Gröbner-Shirshov bases of skew solvable polynomial rings in Section 3. Finally a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings is given in Section 4. 2 Preliminaries 2. Skew solvable polynomial rings In order to define skew solvable polynomial rings, let us recall some basic definitions of orderings first. Let N be the set of nonnegative integers. Suppose < is a monomial ordering on N n, n 2 N, i.e., a total ordering on N n such that 2 N n is the smallest element in N n and < ˇ implies C < ˇ C for any ; ˇ; 2 N n. The set of (standard) monomials in n indeterminates fx ; : : : ; x n g is defined as fx x n n W i 2 N; i ng. We also denote x x n n by x and call the exponent of x (denoted by exp.x / D ), where D. ; : : : ; n/ 2 N n. We say x < xˇ if < ˇ. Thus, a monomial ordering on N n is also called a monomial ordering on the set of standard monomials. The multiple degree and the total degree of a monomial x are defined as mdeg.x / D and tdeg.x / D j j D C C n, respectively. For any nonzero f D P 2Nn c x, where only finitely many constants c are nonzero, the multiple degree and the total degree of f are defined as mdeg.f / D maxf W c g and tdeg.f / D maxfj j W c g, respectively. The monomial x D maxfx W c g is called the leading monomial of f and c is called the leading coefficient of f, denoted by lm.f / and lc.f /, respectively. From now on, we fix a monomial ordering < on N n. Throughout this paper, we suppose all rings considered are unitary and associative. If R is a ring and is a ring endomorphism of R, then a mapping ı W R! R is called a -derivation of R if for any a; b 2 R, ı.a C b/ D ı.a/ C ı.b/ and ı.ab/ D.a/ı.b/ C ı.a/b. Definition 2.. Let R and A be two rings with R A. Then A is called a skew solvable polynomial ring over R if the following conditions hold: (i) There exist finitely many elements x ; : : : ; x n 2 A such that A is a free left R-module with basis M D fx D x x n n W D. ; : : : ; n/ 2 N n g: (ii) For i n, there are an injective ring endomorphism i of R and a i -derivation ı i of R such that x i r D i.r/x i C ı i.r/ for any r 2 R. Furthermore, for any i; j n, i ı j D j ı i ; ı i ı ı j D ı j ı ı i ; i ı ı j D ı j ı i : (iii) For i < j n, there exist c ij 2 R and p ij 2 A with lm.p ij /<x i x j such that x j x i D c ij x i x j Cp ij : Then we write A D RhXI ; ı; c; pi. Clearly every nonzero element in A can be uniquely represented as f D P 2N n c x, where only finitely many c 2 R are nonzero. Skew solvable polynomial rings are generalizations of several well-known kinds of skew polynomial rings. Example 2.2. Let A D RhXI ; ı; c; pi be a skew solvable polynomial ring. (i) If R is a field, i D id R and ı i D for all i n, then A is a solvable polynomial algebra [4]. Any solvable polynomial algebra can be viewed as a skew solvable polynomial ring in this way. (ii) If tdeg.p ij / D for all i < j n, then A is a skew PBW extension of R [8]. Any skew PBW extension can be viewed as a skew solvable polynomial ring in this way.

3 3 X. Zhao, Y. Zhang Therefore, any ring belonging to the above two classes of rings is a skew solvable polynomial ring, for example, a Weyl algebra, the universal enveloping algebra of a finite dimensional Lie algebra, and an Ore extension of automorphism and/or derivation type. Example 2.3 ([8]). Let q 2 k. The coordinate ring of quantum Euclidean space, denoted by O q.ok 2nC /, is the k-algebra generated by 2n C variables w, x ; : : :, x n, y ; : : : ; y n with the following relations: where wy j D qy j w; j n; wx j D q x j w; j n; y j y i D q y i y j ; i < j; y j x i D q x i y j ; i j; x j x i D qx i x j ; i < j; y j x j D x j y j C f jj ; j n; f jj D q j.q =2 q =2 /w 2 C.q 2 / X l<j q l j y l x l : Let R D kœw. For i n, let x nci D y i, i be the k-algebra isomorphism over R determined by i.w/ D qw, nci be the k-algebra isomorphism over R determined by nci.w/ D q w, and ı j D be the zero mapping for any j 2n. Let 8 ˆ< q; i < j n; c ij D q ˆ: ; n < i < j; or i < n < j and i j n; ; i D j n; and p ij D ( ; i j; f ij ; i D j: It is easy to prove by induction that, for any j n, f jj (and thus p jj ) can be written as f jj D X c l x l y l C cw 2 ; c l ; c 2 k: l<j Let > be a monomial ordering on fx W 2 N 2n g such that x 2n > x 2n > > x. With the above notation, it is easy to check that O q.ok 2nC / D RhXI ; ı; c; pi is a skew solvable polynomial ring. In this paper, we consider Gröbner-Shirshov bases in a skew solvable polynomial ring over a field (i.e., R is a field) and the general case will be studied in the near future. From now on, let R D k be a field and A D RhXI ; ı; c; pi be a skew solvable polynomial ring. Let us fix more notation. Denote n n.c/ D.c/ and ı ı n n.c/ D ı.c/ for D. ; : : : ; n/ 2 N n and c 2 k. Suppose x ; xˇ 2 M. Then the least common multiple of x and xˇ is defined as lcm.x ; xˇ/ D x where D.maxf ; ˇg; : : : ; maxf n; ˇng/ 2 N n. We say that x is divisible by xˇ, or xˇ divides x, if x D lm.txˇ/ for some t 2 M. For convenience, denote x D x ˇ (but keep in mind that x ˇxˇ x in xˇ general in a skew solvable polynomial ring). We make the convention that lm./ D < t for any t 2 M. With the above notation, the proof of the following lemma is straightforward. Lemma 2.4. Suppose x ; xˇ; x 2 M. We have: (i) lm.x xˇ/ D x Cˇ D lm.xˇx /. (ii) xˇ divides x if and only if ˇ 2 N n. (iii) If x < xˇ then lm.x x / < lm.x xˇ/.

4 A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings Gröbner-Shirshov bases of skew solvable polynomial rings In this subsection, we briefly introduce concepts related to Gröbner-Shirshov bases and Buchberger s algorithm for skew solvable polynomial rings. Definition 2.5. Let I be a left ideal of A. A (left) Gröbner-Shirshov basis (with respect to <) is a finite subset G I with the property that for every nonzero f 2 I, lm.f / is divisible by lm.g/ for some g 2 G. Let f; g 2 A be nonzero. Suppose that t f D lcm.lm.f /; lm.g// lm.f / and t g D The S-polynomial of f and g, denoted by SPoly.f; g/, is defined as lcm.lm.f /; lm.g// : lm.g/ SPoly.f; g/ D t f f ct g g 2 I; where c D lc.t f f / lc.t g g/ D exp.tf /.lc.f // exp.t g/.lc.g// : Suppose f; g 2 I. We say f is reducible by g if there exists t 2 M such that lm.tg/ D lm.f /. Moreover, if G A and g 2 G, then f! G f ctg is called one-step-reduction by G. If an element r 2 A is obtained from f by finitely many one-step-reductions by G and r is not reducible by G, then we say that r is a remainder of f modulo G. The following algorithm is an analogue of the Buchberger s Algorithm for (commutative) polynomial algebras. Algorithm 2.6 (Algorithm for (left) Gröbner-Shirshov bases). Input: F D ff ; : : : ; f m g A Output: A Gröbner-Shirshov basis G of the left ideal I of A generated by F. G WD F PairsWD f.f; g/ W f; g 2 Gg While Pairs ; Do Choose.f; g/ 2 Pairs PairsWDPairs nf.f; g/g h WD a remainder of SPoly.f; g/ modulo G If h Then Pairs:=Pairs [f.h; h / W h 2 Gg [ f.h ; h/ W h 2 Gg G WD G [ fhg End If End Do Return G The correctness and termination of the above algorithm can be proved in a way similar to the case of commutative polynomial algebras ([3]). 3 Strong Gröbner-Shirshov bases In this section, we introduce the definition of strong Gröbner-Shirshov bases and investigate their properties which will be used in the next section. Recall that A is a skew solvable polynomial ring over a field k. Let f D.f ; : : : ; f m / 2 A m ; m 2 N: We denote by I the left ideal of A generated by f ; : : : ; f m. Then an element f 2 I can be written as f D u f C C u m f m D u f ; where u D.u ; : : : ; u m / 2 A m :

5 32 X. Zhao, Y. Zhang Note that this expression for f is not unique, i.e., there may exist u 2 A m such that u u and f D u f. We write f Œu 2 I A m to indicate f 2 I, u 2 A m and f D uf. That is, f Œu is actually a pair.f; u/ 2 I A m such that f D u f. With this convention, for a 2 A, f Œu ; g Œv ; h Œw 2 I A m, by the equation h Œw D f Œu C ag Œv it means h D f C ag and w D u C av, particularly, h Œw D f Œu if and only if h D f and w D u. Note that A m is a left A-module with the standard basis fe i W i mg, where e i D.; : : : ; ; ; ; : : : ; / with the i-component and the other components. The set of (standard) monomials in A m is N D fx e i W 2 N n ; i mg: A (left) monomial ordering on N is a well-ordering on N such that if m > n then lm.tm/ > lm.tn/ for all m; n 2 N and t 2 M. A monomial ordering on M can be extended to a monomial ordering on N. Example 3.. Let < be a monomial ordering on M. Then < can be extended to monomial orderings on N as follows. (i) We say x e i < xˇe j if and only if x < xˇ, or x D xˇ and i < j. It is easy to see that < is a monomial ordering on N. We call < the TOP extension of <, where TOP stands for term over position, following terminology in []. (ii) Similarly, we can introduce the POT ( position over term ) extension []: Define x e i < 2 xˇe j if and only if i < j or i D j and x < xˇ. It is easy to see that < 2 is also a monomial ordering on N. From now on, we fix a monomial ordering, also denoted by <, on N such that it is compatible with the monomial ordering on M, i.e., for any x ; xˇ 2 M and i m, if x < xˇ in M then x e i < xˇe i in N. Every element u 2 A m can be written uniquely as a k-linear combination of monomials: u D P iq c im i where q 2 N; c i 2 k; m i 2 N. Then, as what we did for elements in A, we can define the leading monomial lm.u/ and leading coefficient lc.u/ of u. Definition 3.2. Given f Œu 2 I A m, the leading monomial lm.u/ is called the signature of f Œu. Definition 3.3. A finite set G D fg Œv ; : : : ; g Œv s s g I A m is called a strong Gröbner-Shirshov basis (cf. [, 2]) of I if, for any f Œu 2 I A m, there exist g Œv i 2 G and t 2 M such that lm.tv i i / D lm.u/ and lm.tg i / lm.f /. We call G a t-strong Gröbner-Shirshov basis for a monomial t in A m if for any f Œu 2 I A m with lm.u/ < t, there exist g Œv i 2 G and t 2 M such that lm.tv i i / D lm.u/ and lm.tg i / lm.f /. Lemma 3.4. Suppose that G D fg Œv ; : : : ; g Œv s s g is a strong Gröbner-Shirshov basis of I. Then G D fg ; : : : ; g s g is a Gröbner-Shirshov basis of I. Proof. By way of contradiction, we suppose that G is not a Gröbner-Shirshov basis, i.e., the following set E is not empty: E D ff Œu 2 I A m W f ; lm.f / is not divisible by lm.g/ for any g 2 G g: We choose f Œu 2 E such that f Œu has minimal signature among E. By the definition of a strong Gröbner- Shirshov basis, there exist g Œv 2 G and t 2 M such that lm.tv/ D lm.u/ and lm.tg/ lm.f /. If lm.tg/ D lm.f /, then lm.f / is divisible by lm.g/, contradicting our assumption that f 2 E. Hence lm.tg/ < lm.f /. Let h Œw D f Œu ctg Œv lc.u/ where c D. Then lm.h/ D lm.f / and thus lm.h/ is not divisible by exp.t/.lc.v// lm.g / for any g 2 G. Hence h Œw 2 E. By the choice of c, we have that lm.u/ D lm.ctv/ and thus lm.w/ < lm.u/, which contradicts the minimality of f Œu in E. In a similar way, we can prove the following Lemma 3.5. Suppose that G is a t-strong Gröbner-Shirshov basis of I. Then, for any f Œu lm.u/ < t, there exist g Œv 2 G and t 2 M such that lm.tg/ D lm.f / and lm.tv/ lm.u/. 2 I A m with

6 A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings 33 Suppose that f Œu ; g Œv 2 I A m are nonzero and lm.t f u/ lm.t g v/ where t f D lcm.lm.f /; lm.g// ; t g D lm.f / lcm.lm.f /; lm.g// : lm.g/ Then the ordered 4-tuple.t f ; f Œu ; t g ; g Œv / is called the critical pair of f Œu and g Œv. Furthermore, the critical pair.t f ; f Œu ; t g ; g Œv / is said to be regular if lm.t f u/ > lm.t g v/. With the above notation, we have the following Lemma 3.6. Suppose that.t f ; f Œu ; t g ; g Œv / is the critical pair of f Œu and g Œv. Then lm.spoly.f; g// < lm.t f f / D lm.t g g/. Recall that a non-strict partial order on a set P is a binary relation P over P which is reflexive, antisymmetric, and transitive. We call < P a (strict) partial order over P. Now we are in a position to introduce the rewriting criterion. Definition 3.7 (Rewriting Criterion). Let S D ff Œu j W j 2 J g I A m where J N is a nonempty index set, j and < S be a partial order on S. Suppose that f Œu 2 S and t 2 M. Then t.f Œu / is called rewritable by S (with respect to < S ) if there exist g Œv 2 S and t 2 M such that lm.t v/ D lm.tu/ and g Œv < S f Œu. In particular, a critical pair.t f ; f Œu ; t g ; g Œv / of S is called rewritable by S if either t f.f Œu / or t g.g Œv / is rewritable by S. Lemma 3.8. Suppose that t.f Œu / is rewritable by a finite subset S I A m with respect to a partial order < S on S. Then there exist g Œv 2 S and t 2 M such that lm.t v/ D lm.tu/, g Œv < S f Œu and t.g Œv / is not rewritable by S. Proof. Since t.f Œu / is rewritable by S, by definition, there exist g Œv 2 S and t 2 M such that lm.t v/ D lm.tu/ and g Œv < S f Œu. If t.g Œv / is not rewritable by S, then g Œv WD g Œv is as required. We assume that t.g Œv / is rewritable by S. Then there exist g Œv 2 S and t 2 M such that lm.t v / D lm.t u / and g Œv < S g Œv. If t.g Œv / is not rewritable by S, then g Œv WD g Œv is as required. Otherwise, if t.g Œv / is rewritable by S then we repeat the above process and obtain a chain where each g i is rewritable by g Œv ic ic ends with g Œv n n for some n 2 N. Then g Œv WD g Œv n n f Œu > S g Œv > S g Œv > S g Œv 2 2 > S 2 S. Since S is finite, the above chain contains only finitely many g i, say it is as required. The following is a key lemma for deriving the criterion for strong Gröbner-Shirshov bases (Theorem 3.). Roughly speaking, this lemma tells us that, among elements with the same signature, a nonrewritable one has a minimal leading monomial. Lemma 3.9. Let t be a monomial in A m and let G be a t-strong Gröbner-Shirshov basis of I with a partial order < G. Suppose that every regular critical pair of G is rewritable by G. For any t 2 M and f Œu 2 G with lm.t u / t, if t.f Œu / is not rewritable by G, then lm.t f / lm.f / for any f Œu 2 I A m with lm.u/ D lm.t u /: Proof. By way of contradiction, we assume that N ;, where N D f.t ; f Œu / 2 M G W lm.t u / t; t.f Œu / not rewritable by G; lm.t f / > lm.f / for some f Œu 2 I A m with lm.u/ D lm.t u /g: Suppose.t ; f Œu / 2 N with minimal lm.t u / among N and suppose t D x. Then there exists f Œu 2 I A m such that lm.u/ D lm.t u / and lm.t f / > lm.f /. Let f Œu D t.f Œu / cf Œu 2 I A m where c D.lc.u //. lc.u/ Then lm.f / D lm.t f / and lm.u/ < lm.t u / t:

7 34 X. Zhao, Y. Zhang Since G is a t-strong Gröbner-Shirshov basis, by Lemma 3.5, there exist g Œv 2 G and t g 2 M such that lm.t g g/ D lm.f / and lm.t g v/ lm.u/. Let D D f.t g ; g Œv / 2 M G W lm.t g g/ D lm.f /; lm.t g v/ lm.u/g: Suppose.t g ; g Œv / is a minimal pair in D, i.e., there is no pair.t g ; g Œv / 2 D such that either lm.t g v / < lm.t g v/, or lm.t g v / D lm.t g v/ and g Œv < G g Œv. We claim that.t ; f Œu ; t g ; g Œv / is a regular critical pair of f Œu and g Œv, where t D lcm.lm.f /; lm.g// lm.f / and t g D lcm.lm.f /; lm.g// : lm.g/ To obtain a contradiction, we assume that lm.t u / lm.t g v/. Note that lm.t f / D lm.t g g/ is a multiple of lcm.lm.f /; lm.g// D lm.t f / D lm.t g g/. Thus t divides t, t g divides t g, and t =t D t g =t g. Hence lm..t =t / lm.t u // D lm..t g =t g / lm.t u // lm..t g =t g / lm.t g v//; i.e., lm.t u // lm.t g v/, contradicting the fact lm.t g v/ lm.u/ < lm.t u /. Hence our claim is true. By the hypothesis of the lemma, the regular critical pair.t ; f Œu ; t g ; g Œv / is rewritable by G, i.e., either t.f Œu / or t g.g Œv / is rewritable by G. Since t divides t, if t.f Œu / is rewritable by G then so is t.f Œu /, which is a contradiction. Thus t g.g Œv / is rewritable by G and hence so is t g.g Œv /. By Lemma 3.8, there exist g Œv 2 G and t 2 M such that lm.t v / D lm.t g v/, g Œv < G g Œv, and t.gœv / is not rewritable by G. Now we claim that lm.t g / lm.t g g/. Otherwise, if lm.t g / > lm.t g g/, then it is easy to see that.t ; gœv / 2 N (note that t g.g Œv / 2 I and lm.t v / D lm.t g v/ lm.u/ < t). But, lm.t v / D lm.t g v/ < lm.t u /, contradicting the fact that.t ; f Œu / has minimal lm.t u / among N. Thus our claim holds true. Now we have two cases to consider: lm.t g / D lm.t g g/ and lm.t g / < lm.t g g/. Our goal is to deduce a contradiction for each case and thus end the proof of the lemma. If lm.t g / D lm.t g g/ D lm.f /, since lm.t v / D lm.t g v/ lm.u/, we have.t ; gœv / 2 D. By the minimality of.t g ; g Œv / among D, we have g Œv G g Œv, which contradicts the fact that g Œv < G g Œv. Now consider the second case: lm.t g / < lm.t g g/. Let g Œv D t g.g Œv / ct.gœv / 2 G; c D exp.t g/ lc.v/ exp.t / lc.v / : Then lm.g/ D lm.t g g/ and lm.v/ < lm.t g v/ < lm.t u / t. Since G is a t-strong Gröbner-Shirshov basis, by Lemma 3.5, there exist h Œw 2 G and t h 2 M such that lm.t h h/ D lm.g/ D lm.t g g/ D lm.t f / and lm.t h w/ lm.v/ < lm.t g v/ lm.u/. Thus, h Œw 2 D. But then the inequality lm.t h w/ < lm.t g v/ implies that.t g ; g Œv / is not minimal in D, which is a contradiction. The following theorem gives a criterion for strong Gröbner-Shirshov bases. Theorem 3.. Let A be a skew solvable polynomial ring and let I be a left ideal of A generated by m 2 N elements. Suppose that G D fg Œu i W i sg I A m with a partial order < i G, where s 2 N. Then G is a strong Gröbner-Shirshov basis of I if the following conditions hold: (i) For any i m, there exists g Œu 2 G such that lm.u/ D e i. (ii) Every regular critical pair of elements from G is rewritable by G with respect to < G. Proof. (By contradiction.) Suppose G is not a strong Gröbner-Shirshov basis, i.e., N ; where N D ff Œu 2 I A m W there exists no.t; g Œv / 2 M G such that lm.tv/ D lm.u/; lm.tg/ lm.f /g: Let f Œu 2 N with a minimal signature lm.u/ among N. Suppose lm.u/ D te j for some t 2 M and j m. Then G is a te j -strong Gröbner-Shirshov basis. By condition (i) of the theorem, there exists g Œv 2 G such that lm.v/ D e j. Then lm.tv/ D lm.u/. If t.g Œv / is rewritable by G, then there exist f Œv 2 G and t 2 M such that lm.t v / D lm.tv/ D lm.u/ and f Œv < G g Œv. Repeating this process and by a similar argument as in the proof of Lemma 3.8, we can prove that there exist h Œw 2 G and s 2 M such that lm.sw/ D lm.u/ and s.h Œw / is not rewritable by G. Applying Lemma 3.9 gives that lm.sh/ lm.f /. But the facts lm.sw/ D lm.u/ and f u 2 N imply that lm.sh/ > lm.f /, which is a contradiction.

8 A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings 35 4 A Signature-based algorithm In this section, we present a signature-based algorithm for computing a strong Gröbner-Shirshov basis in skew solvable polynomial rings. As before, let A D khxi ; ı; c; pi be a skew solvable polynomial ring over k, f ; : : : ; f m 2 A, m 2 N, and let I be the left ideal of A generated by f ; : : : ; f m. Suppose f Œu ; g Œv 2 I A m. We say f Œu is reducible by g Œv if there exists t 2 M such that lm.tg/ D lm.f / and lm.tv/ < lm.u/. Moreover, if G A A m and g Œv 2 G, then f Œu! G f Œu ct.g Œv / is said to be a onestep-reduction by G where c D lc.f /= lc.tg/. If f Œu is obtained from f Œu by finitely many one-step-reductions by G and f Œu is not reducible by G, then we say that f Œu is a remainder of f Œu modulo G. With the above definitions, we have the following Lemma 4.. Suppose that f Œu is a remainder of f Œu modulo G. Then lm.f / < lm.f / and lm.u / D lm.u/. Proof. Note that in a one-step-reduction f Œu! G f Œu ct.g Œv / D.f ctg/ u ctv, since lm.tg/ D lm.f / and lm.tv/ < lm.u/, we have lm.g/ < lm.f / and lm.v/ D lm.u/. Thus the lemma follows by induction on the number of one-step-reductions to obtained f Œu from f Œu. Now we are in a position to state our main algorithm. Algorithm 4.2 (Algorithm for strong Gröbner-Shirshov bases). Input: F D ff Œe ; : : : ; f Œe m m g A A m Output: A strong Gröbner-Shirshov basis G of the left ideal I of A generated by F.. G WD F 2. CPairsWD fregular critical pair.t f ; f Œu ; t g ; g Œv / W f Œu ; g Œv 2 Gg 3. While CPairs ; Do 4. Choose.t f ; f Œu ; t g ; g Œv / 2 CPairs 5. CPairsWDCPairs nf.t f ; f Œu ; t g ; g Œv /g 6. If.t f ; f Œu ; t g ; g Œv / is not rewritable by G then 7. h Œw WD a remainder of SPoly.f Œu ; g Œv / modulo G 8. CPairs WD CPairs [ fregular critical pair.t h ; h Œw ; t h ; h Œw / W h Œw 2 Gg [ fregular critical pair.t h ; h Œw ; t h ; h Œw / W h Œw 2 Gg 9. G WD G [ fh Œw g. End If. End Do 2. Return G Note that in Line 6 of the above algorithm, a partial order on G is needed. The GVW-orders implied by the criterion in [] will be used in our algorithm, which can be updated automatically when a new element is added to G, i.e., when G WD G [ fh Œw g in the algorithm. Definition 4.3. A partial order < G on G I A m is called a GVW-order if, for any f Œu ; g Œv 2 G, we have f Œu < G g Œv whenever one of the following conditions hold: (a) lm.t g/ < lm.tf / where t D lcm.lm.u/;lm.v// and t D lcm.lm.u/;lm.v//. lm.v/ lm.u/ (b) lm.t g/ D lm.tf / and g Œv is added to G later than f Œu. Theorem 4.4. If a GVW-order is used in the Rewriting Criterion of Algorithm 4.2, then the algorithm terminates after finitely many steps and returns a strong Gröbner-Shirshov basis of I. Proof. (Correctness.) If the algorithm terminates after finitely many steps, then its correctness follows clearly from Theorem 3..

9 36 X. Zhao, Y. Zhang (Termination.) Similarly to [2], we introduce a map ' W A A m! kœy; Z; W to investigate the termination of the algorithm, where Y D fy ; : : : ; y n g, Z D fz ; : : : ; z m g, W D fw ; : : : ; w n g, and kœy; Z; W denotes the polynomial algebra over k with indeterminates Y [ Z [ W. Define ' W A A m! kœy; W; Z as follows: for.f; u/ 2 A A m with lm.u/ D x e i and lm.f / D xˇ, where ; ˇ 2 N n and i m, let '.f; u/ D ( y z i D y y n n z i ; if f D ; y z i w wˇ D y y n n z i w wˇ wˇn n ; if f : Suppose that.t f ; f Œu ; t g ; g Œv / is a regular critical pair and not rewritable by G with respect to a GVW-order < G, and that h Œw is a remainder of h Œw modulo G where h Œw D SPoly.f Œu ; g Œv / (and thus w D t f u t g v). We claim that '.h Œw / is not divisible by '.h Œw / for any h Œw 2 G. Otherwise, we assume that '.h Œw / is divisible by '.h Œw / for some h Œw 2 G. Now we have the following three cases and we deduce a contradiction for each case, which ends the proof of the claim. Case (): h D. In this case, it is easy to see from the definition of ' that lm.w / divides lm.w/. Then lm.sw / D lm.w/ for some monomial s 2 M. Note that lm.w / D lm.t f u/ since.t f ; f Œu ; t g ; g Œv / is regular. Thus, from Lemma 4., lm.sw / D lm.w/ D lm.w / D lm.t f u/. Recall that the GVW-order < G is used in the algorithm and lm.sh / D < lm.t f f /. Thus, t f.f Œu / is rewritable by h Œw, which is a contradiction. Case (2): h and h D. From the definition of ', it is impossible that '.h Œw / divides '.h Œw /. Case (3): h and h. Let s D lm.w/= lm.w / and t D lm.h/= lm.h /. If s t, then, by Lemmas 3.6 and 4., we have that lm.sh / lm.th / D lm.h/ < lm.t f f / and lm.sw / D lm.w/ D lm.w / D lm.t f u/. Suppose that s D lcm.lm.w /; lm.u// lm.w / and t f D lcm.lm.w /; lm.u// : lm.u/ Then s divides s, t f divides t f, and s=s D t f =t f. Hence lm.s h / < lm.t f f / (otherwise, lm.s h / < lm.t f f / implies lm..s=s /s h / < lm..t f =t f /t f f /, i.e., lm.sh / < lm.t f f /, a contradiction). Since < G is a GVW-order, it follows from Definition 4.3 that h Œw < G f Œu. Thus t f.f Œu / is rewritable by h Œw, which is a contradiction. Now suppose s > t. Then lm.th / D lm.h/ and lm.tw / < lm.sw / D lm.w/. Hence, h Œw is reducible by the set fh Œw g G and thus by G, which contradicts the assumption that h Œw is a remainder modulo G. Now return to the proof of the termination of the algorithm. In the While Loop of the algorithm (Lines 3 ), if.t f ; f Œu ; t g ; g Œv / is regular and not rewritable by G with respect to < G, then h Œw is added to G, and the new G is denoted as G D G [ fh Œw g. From the above claim, the ideal of kœy; Z; W generated by '.G/ is strictly contained in the ideal generated by '.G /. Repeating the While Loop gives a chain of ideals of kœy; Z; W : h'.g/i h'.g /i h'.g 2 /i By the Hilbert basis theorem of polynomial algebras, the above chain has a finite length (we may suppose the chain ends with h'.g L /i, L 2 N). Hence after finitely many repeats of the While Loop, every regular critical pair in CPairs is rewritable by G L and thus the algorithm terminates. In summary, a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings over a field has been presented in this paper. This algorithm can detect redundant S-polynomials (i.e., the S-polynomials corresponding to nonregular critical pairs) and therefore it is more efficient than Buchberger s algorithm. The implementation of the proposed algorithms in a computer algebra system (e.g., Maple, and Singular) and computing examples will be discussed in the near future. Acknowledgement: We would like to thank the anonymous referees for their careful reading and valuable comments that have improved the readability of this article. The majority of this work was done during the stay of the first author at the University of Manitoba. This work was supported in part by NSERC and MGS.

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