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.

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.

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 0 2 N n is the smallest element in N n and˛<ˇimplies˛C <ˇC for any˛;ˇ; 2 N n . The set of (standard) monomials in n indeterminates fx 1 ; : : : ; x n g is defined as fx˛1 1 x˛n n W˛i 2 N; 1 Ä i Ä ng. We also denote x˛1 1 x˛n n by x˛and call˛the exponent of x˛(denoted by exp.x˛/ D˛), where˛D .˛1; : : : ;˛n/ 2 N n . We say x˛< xˇif˛<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˛1 C C˛n, respectively. For any nonzero f D P˛2 N n c˛x˛, where only finitely many constants cą re nonzero, the multiple degree and the total degree of f are defined as mdeg.f / D maxf˛W c˛¤ 0g and tdeg.f / D maxfj˛j W c˛¤ 0g, respectively. The monomial x D maxfx˛W c˛¤ 0g 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 2.1. Let R and A be two rings with R Â A. Then A is called a skew solvable polynomial ring over R if the following conditions hold: (i) There exist finitely many elements x 1 ; : : : ; x n 2 A such that A is a free left R-module with basis M D fx˛D x˛1 1 x˛n n W˛D .˛1; : : : ;˛n/ 2 N n g: (ii) For 1 Ä i Ä 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 1 Ä i; j Ä n, Then we write A D RhX I ; ı; c; pi. Clearly every nonzero element in A can be uniquely represented as f D P˛2 N 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 RhX I ; ı; c; pi be a skew solvable polynomial ring.
(i) If R is a field, i D id R and ı i D 0 for all 1 Ä i Ä n, then A is a solvable polynomial algebra [14]. Any solvable polynomial algebra can be viewed as a skew solvable polynomial ring in this way. (ii) If tdeg.p ij / D 1 for all 1 Ä i < j Ä 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.
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 ([18]
). Let 0 ¤ q 2 k. The coordinate ring of quantum Euclidean space, denoted by O q .ok 2nC1 /, is the k-algebra generated by 2n C 1 variables w, x 1 ; : : :, x n , y 1 ; : : : ; y n with the following relations: nCi be the k-algebra isomorphism over R determined by nCi .w/ D q 1 w, and ı j D 0 be the zero mapping for any 1 Ä j Ä 2n. Let i < j Ä n; q 1 ; n < i < j; or i < n < j and i ¤ j n; 1; i D j n; and It is easy to prove by induction that, for any 1 Ä j Ä n, f jj (and thus p jj ) can be written as c l x l y l C cw 2 ; c l ; c 2 k: Let > be a monomial ordering on fx˛W˛2 N 2n g such that x 2n > x 2n 1 > > x 1 . With the above notation, it is easy to check that O q .ok 2nC1 / D RhX I ; 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 RhX I ; ı; c; pi be a skew solvable polynomial ring. Let us fix more notation. Denote ˛1 1 ˛n n .c/ D ˛. c/ and ı˛1 1 ı˛n n .c/ D ı˛.c/ for˛D .˛1; : : : ;˛n/ 2 N n and c 2 k. Suppose x˛; xˇ2 M. Then the least common multiple of x˛and xˇis defined as lcm.x˛; xˇ/ D x where D .maxf˛1;ˇ1g; : : : ; maxf˛n;ˇng/ 2 N n . We say that x˛is divisible by xˇ, or xˇdivides x˛, if x˛D lm.txˇ/ for some t 2 M. For convenience, denote xxˇD x˛ ˇ( but keep in mind that x˛ ˇxˇ¤ x˛in general in a skew solvable polynomial ring). We make the convention that lm.0/ D 0 < t for any 0 ¤ t 2 M.
With the above notation, the proof of the following lemma is straightforward.

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.
The S-polynomial of f and g, denoted by SPoly.f; g/, is defined as 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 1 ; : : : The correctness and termination of the above algorithm can be proved in a way similar to the case of commutative polynomial algebras ( [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 Note that this expression for f is not unique, i.e., there may exist u 0 2 A m such that u 0 ¤ u and f D u 0 f . We where e i D .0; : : : ; 0; 1; 0; : : : ; 0/ with the i -component 1 and the other components 0. The set of (standard) monomials in A m is 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 3.1. Let < 0 be a monomial ordering on M. Then < 0 can be extended to monomial orderings on N as follows.
(i) We say x˛e i < 1 xˇe j if and only if x˛< 0 xˇ, or x˛D xˇand i < j . It is easy to see that < 1 is a monomial ordering on N . We call < 1 the TOP extension of < 0 , where TOP stands for "term over position", following terminology in [1].
(ii) Similarly, we can introduce the POT ("position over term") extension [1]: Define x˛e i < 2 xˇe j if and only if i < j or i D j and x˛< 0 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˛; xˇ2 M and 1 Ä i Ä m, if x˛< xˇin 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: Then, as what we did for elements in A, we can define the leading monomial lm.u/ and leading coefficient lc.u/ of u.  Proof. By way of contradiction, we suppose that G 0 is not a Gröbner-Shirshov basis, i.e., the following set E is not empty: In a similar way, we can prove the following  Then the ordered 4-tuple .t f ; f OEu ; t g ; g OEv / is called the critical pair of f OEu and g OEv . Furthermore, the critical pair .t f ; f OEu ; t g ; g OEv / 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 OEu ; t g ; g OEv / is the critical pair of f OEu and g OEv . 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 Ä 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
OEu j j W j 2 J g Â I A m where J Â N is a nonempty index set, and < S be a partial order on S . Suppose that f OEu 2 S and t 2 M. Then t .f OEu / is called rewritable by S (with respect to < S ) if there exist g OEv 2 S and t 0 2 M such that lm.t 0 v/ D lm.t u/ and g OEv < S f OEu . In particular, a critical pair .t f ; f OEu ; t g ; g OEv / of S is called rewritable by S if either t f .f OEu / or t g .g OEv / is rewritable by S. / is rewritable by S then we repeat the above process and obtain a chain where each g i is rewritable by g OEv i C1 iC1 2 S. Since S is finite, the above chain contains only finitely many g i , say it ends with g OEv n n for some n 2 N. Then g OEv WD g OEv n n is as required.
The following is a key lemma for deriving the criterion for strong Gröbner-Shirshov bases (Theorem 3.10). Roughly speaking, this lemma tells us that, among elements with the same signature, a nonrewritable one has a minimal leading monomial. Since G is a t-strong Gröbner-Shirshov basis, by Lemma 3.5, there exist g OEv 2 G and t g 2 M such that lm.t g g/ D lm.f / and lm.t g v/ Ä lm.u/. Let D D f.t g ; g OEv / 2 M G W lm.t g g/ D lm.f /; lm.t g v/ Ä lm.u/g: Suppose .t g ; g OEv / is a minimal pair in D, i.e., there is no pair .t g 0 ; g 0OEv 0 / 2 D such that either lm.
We claim that .t 0 ; f OEu 0 0 ; t g ; g OEv / is a regular critical pair of f OEu 0 0 and g OEv , where t 0 D lcm.lm.f 0 /; lm.g// lm.f 0 / and t g D lcm.lm.f 0 /; lm.g// lm.g/ : To obtain a contradiction, we assume that lm Thus t 0 divides t 0 , t g divides t g , and t 0 =t 0 D t g =t g . Hence i.e., lm.t 0 u 0 // Ä lm.t g v/, contradicting the fact lm.t g v/ Ä lm.u/ < lm.t 0 u 0 /. Hence our claim is true. By the hypothesis of the lemma, the regular critical pair .t 0 ; f OEu 0 0 ; t g ; g OEv / is rewritable by G, i.e., either which is a contradiction. Thus t g .g OEv / is rewritable by G and hence so is t g .g OEv /. By Lemma 3.8, there exist g OEv 0 0 2 G and t 0 Otherwise, if lm.t 0 0 g 0 / > lm.t g g/, then it is easy to see that .t 0 0 ; g OEv 0 0 / 2 N (note that t g .g OEv / 2 I and lm. But then the inequality lm.t h w/ < lm.t g v/ implies that .t g ; g OEv / is not minimal in D, which is a contradiction.
The following theorem gives a criterion for strong Gröbner-Shirshov bases. < G g OEv . Repeating this process and by a similar argument as in the proof of Lemma 3.8, we can prove that there exist h OEw 2 G and s 2 M such that lm.sw/ D lm.u/ and s.h OEw / is not rewritable by G. Applying Lemma 3.9 gives that lm.sh/ Ä 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.

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 1 ; : : : ; f m 2 A, m 2 N, and let I be the left ideal of A generated by f 1 ; : : : ; f m .
Suppose f OEu ; g OEv 2 I A m . We say f OEu is reducible by g OEv if there exists t 2 M such that lm.tg/ D lm.f / and lm.t v/ < lm.u/. Moreover, if G Â A A m and g OEv 2 G, then f OEu ! G f OEu ct .g OEv / is said to be a onestep-reduction by G where c D lc.f /= lc.tg/. If f 0OEu 0 is obtained from f OEu by finitely many one-step-reductions by G and f 0OEu 0 is not reducible by G, then we say that f 0OEu 0 is a remainder of f OEu modulo G. With the above definitions, we have the following Proof. Note that in a one-step-reduction f OEu ! G f OEu ct .g OEv / D .f ctg/ u ctv , since lm.tg/ D lm.f / and lm.t v/ < 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 0OEu 0 from f OEu . Now we are in a position to state our main algorithm.  Note that in Line 6 of the above algorithm, a partial order on G is needed. The GVW-orders implied by the criterion in [11] 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 OEw g in the algorithm. Definition 4.3. A partial order < G on G Â I A m is called a GVW-order if, for any f OEu ; g OEv 2 G, we have f OEu < G g OEv whenever one of the following conditions hold:  Proof. (Correctness.) If the algorithm terminates after finitely many steps, then its correctness follows clearly from Theorem 3.10.