Scholarly article on topic 'Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules'

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules Academic research paper on "Mathematics"

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Mathematics in Computer Science
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Academic research paper on topic "Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules"


DOI 10.1007/s11786-017-0317-1

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Mathematics in Computer Science

Relative Reduction and Buchberger's Algorithm in Filtered Free Modules

Christoph Fürst • Alexander Levin

Received: 30 November 2016 / Revised: 2 March 2017 / Accepted: 21 March 2017 © The Author(s) 2017. This article is an open access publication

Abstract In this paper we develop a relative Grobner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and difference-differential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchberger-type algorithm for constructing relative Grobner bases of filtered free modules.

Keywords Filtered module • Admissible orders • Relative Grobner basis • Grobner reduction

Mathematics Subject Classification Primary 13P10; Secondary 12H05 • 12H10 • 68W30

1 Introduction

It is widely known that the classical Grobner basis method, first introduced in [1], is a powerful algorithmic technique for solving problems in commutative algebra and algebraic geometry (in particular, problems that can be formulated in terms of systems of multivariate polynomial equations). Moreover, this method has various applications in geometric theorem proving, graph theory (e.g., in problems of coloring of graphs), linear programming, theory of error-correcting codes, robotics and many other areas. One of the important algebraic application of Grobner bases is their use in the dimension theory, in particular, for the computation of Hilbert polynomials of graded and filtered modules. It turned out that the corresponding technique can be extended to the computation of differential, difference and difference-differential dimension polynomials via generalizations of the Grobner basis method to differential, difference and difference-differential modules, respectively. Such generalizations were obtained in [18] (for modules over ring of differential operators with power series coefficients) and in [12, Chapter 4] (for difference and difference-differential modules).

C. Fürst (B)

Research Institute for Symbolic Computation (RISC) Linz, Altenberger Straße 69, 4040 Linz, Austria e-mail:

A. Levin

Department of Mathematics, The Catholic University of America, Washington, DC 20064, USA e-mail:

Published online: 07 April 2017

^ Birkhäuser

In the last fifteen years the Grobner basis approach was applied to bifiltered and multifiltered modules over polynomial rings, rings of differential, difference and difference-differential polynomials, and Weyl algebras. The corresponding techniques use different types of reduction with respect to several term orderings; the resulting Grobner-type bases are called Grobner bases with respect to several term orderings ([13-15] and [5]) and relative Grobner bases ([20,22] and [4]). As applications of the generalized Grobner basis techniques, these works present proofs of the existence and methods of computation of multivariate difference-differential dimension polynomials, as well as bivariate Bernstein-type dimension polynomials of modules over Weyl algebras. Furthermore, a generalization of the relative Grobner basis technique to the case of difference-differential modules with weighted basic operators obtained by Donch in [3] allowed him to prove the existence and obtain methods of computation of Ehrhart-type dimension quasi-polynomials associated with filtrations of such modules.

In this paper we unify the theories of relative Grobner bases and Grobner bases with respect to several term orderings (including their "weighted" versions) by developing a generalized relative Grobner basis method for a wide class of filtered modules that includes modules over rings of differential, difference, inversive difference and difference-differential operators, Weyl algebras, and also multiparameter twisted Weyl algebras introduced and studied in [9]. (Note that the last class of algebras includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras that play an important role in the quantum group covariant differential calculus, see, for example, [19].)

In the next section we describe basic settings, give a characterization of ring filtrations considered in the rest of the paper and present a concept of Grobner reduction in a free module over a filtered ring with monomial filtration. The main results are presented in Sect. 3, where we introduce concepts of admissible orders and set-relative reduction in free modules over rings with monomial filtrations, define the notion of Grobner basis in this setting and obtain a Buchberger-type algorithm for its construction.

2 Algebraic Setup

Throughout the paper Z, N and Q denote the sets of integers, nonnegative integers and rational numbers, respectively. If p is a positive integer, then Np is treated as a commutative semigroup with componentwise addition and as a partially ordered set with the product order <n such that

r = (r i, ...,rp) <n s = (si, ...,sp) ^^ rt < st, Vi = 1,..., p.

In what follows, R denotes an arbitrary, left noetherian (non-commutative) ring containing a commutative ring K c R as a subring. Unless the contrary is indicated, by an R-module, we always mean a left R-module.

Let M be a left R-module. A subset M of this module is said to be a set of monomials of M, if M is a free K -module with basis M, that is, every element f e M has a unique representation of the form

f = Y1 fmm, fm e K, only finitely many fm = 0.

the support of f, i.e. the set of monomials that appear in f with a non-zero coefficient. For example, if K is a field and A is a basis of R as a K-vector space (or K is a commutative ring and R is a free K-module with basis A), then A is a set of monomials of R.IfF is a free R-module with basis E, then it is easy to see that A E = {Xe : X e A, e e E} is the set of monomials of F. In this case we write R = K(A) and F = R(E) = K(AE

Definition 2.1 A family of K-submodules {Rr : r e Np} of R is called a p-fold filtration of a ring R if

1. Rr ç Rs, whenever r <n s e Np ;

2. Rr ■ Rs Ç Rr+s, r, s e Np ;

We denote by

3. R = Ure№ Rr;

4. 1 e R0 where R0 stands for R(0,..,0).

In this case we say that R is a p-filtered ring. (If p = 2 we also use the term bifiltration). A p-fold filtration of R is called monomial (and R is said to be a monomially p-filtered ring) if and only if the inclusion f e Rr implies the inclusion T( f) c Rr.

Definition 2.2 Let M be a left R-module. A p-fold filtration of the module M w.r.t. the p-fold filtered ring R is a family {Mr : r e Np} of K-submodules of M such that

• Mr c Ms, r <n s;

• Rr ■ Ms c Mr+s, for any r, s e Np;

• M = [J r sN pMr.

An R-module equipped with a p-fold filtration is said to be a p-filtered module.

Note that if R is a ring with p-fold filtration {Rr : r e Np} and M is a finitely generated R-module with generators {h1,...,hq}, then M can be naturally treated as a p-filtered module with the p-fold filtration

|Mr = ^Rrhi : r e Np J .

Clearly, this filtration is monomial if the filtration of R is monomial.

Given any mapping u : R ^ N, one can consider a family of additive subgroups {Rf,u) : k e N} of the ring R such that R^ := {r e R : u(r) < k} for every k e N. Clearly, Uk R^ = R. The following statement tells when a family of this kind is a (onefold) filtration of R.

Lemma 2.3 (Characterization of one-dimensional filtrations) With the above notation, the family {Rku) : k e N} is a (onefold) filtration of R if and only if the mapping u satisfies the following three conditions.

(i) Ifx e R, then u(x) = 0 if and only ifx e K;

(ii) u(x + y) < max{u(x), u(y)} for all x, y e R;

(iii) u(xy) < u(x) + u(y) for all x, y e R;

Furthermore, for any onefold filtration {Rr : r e N}, there exists a mapping u : R ^ N satisfying conditions (i)-(iii) such that Rr = R{ru) for all r e N.

Proof Clearly, if u : R ^ N is a mapping satisfying the above conditions and R^ = {x e R : u(x) < k} (k e N), then the family {Rku) : k e N} satisfies conditions 1-4 of Definition 2.1 (with p = 1). (Note that if x e R^ and c e K, then u(cx) < u(c) + u(x) = u(x); this observation and property (ii) imply that every R^ is a K-module.). For the converse, suppose that u is a mapping from R to N such that the family Rku) ={x e R : u (x) < k}, k e N, satisfies conditions 1-4 of Definition 2.1. Since r0u) = K and u(x) > 0 for any x e R, we obtain that x e K is equivalent to u (x) = 0. The other properties of the map u follow from the fact that every R^ is a K-module and from the first two conditions of Definition 2.1.

In order to prove the last part of the statement, consider a onefold filtration { Rr : r e N} of R and define the mapping u : R ^ N by setting u(x) = min{k : x e Rk}. It is easy to check that u satisfies conditions (i)-(iii). Indeed, since R0 = K, we have that u(a) = 0 for any a e K and, conversely, the equality u(x) = 0(x e R) implies that x e R0 = K. Furthermore, the fact that every Rk is a K-module and the first two properties of a filtration (see Definition 2.1) imply that the mapping u satisfies conditions (ii) and (iii).

It remains to show that R(u) = Rr for any r e N. Let x e R(u) and u(x) = k0. Then k0 = min{k e N : x e Rk} < r, hence x e Rk0 c Rr. Conversely, if y e Rr, then u(y) = min{k e N : y e Rk} < r, so y e Rr"'1. Thus, r( ) = Rr. □

Remark 2.4 The first part of Lemma 2.3 can be generalized to p-fold filiations (p > 1) as follows. Let us consider a mapping u : R ^ Np and let ui = ni o u : R ^ N (1 < i < p) where ni is the projection of Np onto its i-th component: (ai,..., ap) ^ ai. For any r = (ri, ...,rp) e Np, let R(u) = {x e R : ui (x) < ri for 1 < i < p}. Then, one can mimic the corresponding part of the proof of Lemma 2.3 to obtain that {R(u) : r e Np} is a p-fold filtration of R if and only if the mapping u satisfies the following conditions:

(i) If x e R, then u(x) = 0 if and only if x e K;

(ii) u(x + y) <n (max{u1(x), u1(y)},..., max{up (x), up(y)}) for all x, y e R;

(iii) u(xy) <n (u1(x) + u1(y),..., up (x) + up (y)) for all x, y e R.

At the same time, if p > 1, then not every p-fold filtration is of the form {R(u) : r e Np} with a mapping u : R ^ Np satisfying the above conditions. It follows from the fact that the same element of R can belong to different components Rr and Rs with incomparable (with respect to <n) p-tuples r, s e Np. For example, let R = K [x1; x2] be a polynomial ring in two variables over a field K, equipped with a natural twofold filtration

Rr1,r2 ={f e K[x1, x2]: deg^ (f) < n A degx2(f) < r2}, (ru ri) e N2,

and let the factor ring R = K[x1; x2]/<x\ — xbe equipped with the canonical image Rr1,r2 of the filtration {Rr : r e N2}. Denoting the coset of a polynomial f e K [x1; x2] by f, we obtain that, say, the element t = x3 = x2 lies in R3,0 n R0,2. If there is a function u : R ^ N2 such that Rr1,r2 = Rifl]r2 for every r = (r1; r2) e N2, then one would have u1 (t) < 0 and u2 (t) < 0 (we use the notation of Remark 2.4). These inequalities imply that t e K, contrary to the obvious fact that x3 — a e <x3 — x|) and x| — be <x3 — x^) for any a, b e K.

Let R be a p-fold filtered ring with a p-fold filtration {Rr : r e Np}, and let F be a free R-module with set of free generators E. Assume that R is a free K-module with basis A (we still use the notation and conventions introduced at the beginning of this section), then the support T( f) of an element f e F is defined as its support with respect to the set of monomials AE = {Xe : X e A, e e E}.

In what follows, a binary relation p c F x F is said to be a reduction and the inclusion (f, g) e p is written as f —> g. Furthermore, for every f, h e F, we write f —>* h if there exists a finite chain

f = f0 f1 ----► fk = h.

Finally, Ip will denote the set of p-irreducible elements of F, that is Ip := {f e F : there is no h e F such that f —> h A h = f}.

With the above notation, one can consider the following concept of Grobner reduction first introduced in [7].

Definition 2.5 (Grobner Reduction) With the above notation, let the p-fold filtration of the ring R be monomial and let N be an R-submodule of the free R-module F. A reduction p c F x F is said to be a Grobner reduction for N if and only if it satisfies the following conditions

1. every reduction sequence f1 —> f2 —> ■ ■ ■ terminates in a finite number of steps;

2. K <Ip) C Ip and f e Ip ^ T( f) c Ip (where K <Ip) denotes the K-module generated by Ip);

3. f —> h implies that f = h (mod N)

4. Ip n N = 0, that is, every non-zero element in N is reducible,

5. f e Fr a f h ^ h e Fr, r e Np.

Remark2.6 Examples of Grobner reductions can be found in [6, Chapter 3] and [8]. If K is a field of zero characteristic, then the theory of Grobner reduction provides an algorithmic computation of the dimensions of components of p-fold filtrations of finitely generated R-modules. In the cases of p-dimensional filtrations of differential, difference and difference-differential modules (where R is the ring of the corresponding operators), the dimensions of the components are expressed by multivariate polynomials in p variables with rational coefficients (see [12, Theorem 4.3.39] and [17, Theorems 3.3.16]). Similar result for modules over Weyl algebras and rings of

Ore polynomials are obtained in [14] and [16]. (Note that by a ring of Ore polynomials we mean a ring defined as follows. Let K be a field and A = {51,..., Sn}, a = {a1,..., an} sets of derivations and injective endomorphisms of K, respectively, such that any two mappings from the set A U a commute. Let © = ©(X) be a free commutative semigroup generated by a set X = {x1,..., xn} and let O denote the vector K-space with the basis © (elements of O are of the form qs© a$0 where a$ e K and only finitely many coefficients a$ are different from zero). Then O can be treated as a ring if one introduces the multiplication according to the rule xia = ai (a)xi + Si (a) (a e K, 1 < i < n) and the distributive laws. Then we say that O is a ring of Ore polynomials in the variables x1,...,xn over K.) In all these cases the results on the multivariate dimension polynomials were proved with the use of certain types of Grobner bases whose constructions are based on the corresponding reductions. All these reductions satisfy the conditions of Definition 2.5 and therefore are Grobner reductions. Note that the reduction with respect to several orderings defined in [16], which is a special instance of the Grobner reduction, can be naturally applied to algebras of a certain subclass of the class of algebras of solvable type (algebras of solvable type were

introduced and studied in [11]). This subclass consists of algebras of solvable type R — K{X1,..., Xn} (K is a

field) where the term ordering < is degree-respecting (that is, for any two power products t = X11 ... Xnn and t' = X1 ... Xln, the inequality deg t = Y!i=1 ki < deg t' = YH=i U implies that t < t') and the axiom 1.2(3) in [11, Section 1] is strengthened by the requirement that for all 1 < i < j < n, there exist 0 = cij e K and pij e R such that Xj * Xi = ctjXtXj + pij and deg pij < 1. (Algebras of this kind are considered in [6, Section 1.4] where, however, the above requirements on the term order and commutation of the generators are not explicitly formulated. Note also that one can mimic the proof of [16, Theorem 4.2] and obtain a theorem on a multivariate dimension polynomials for finitely generated modules over such algebras.) Finally, using the weight relative Grobner basis technique developed in [3, Section 2.3] (the corresponding reduction satisfies the conditions of Definition 2.5 as well) C. Donch showed that the dimensions of components of a multidimensional filtration of a finitely generated difference-skew-differential module over a difference-skew-differential field (that is a field K with the action of finitely many mutually commuting injective endomorphisms and skew derivations of K, see [3, Definition 2.1.1]) can be expressed by a multivariate quasi-polynomial (see [3, Theorem 3.1.22]).

3 Relative Reduction and Buchberger's Algorithm

Buchberger's algorithm, introduced in [1] and formulated for multivariate polynomial rings over a field of characteristic zero, is an algorithm for computing a generating set of an ideal, with the property that every non-zero element contained in the ideal can be reduced to zero in finitely many steps.

This was the starting point for generalizations of this algorithm towards many (non-commutative) ground domains, for example, to the ring of multivariate Ore-polynomials in [15], Weyl-Algebras in [14] and [5], or to the ring of difference-differential operators (see [12, Chapter 4], [13,20] and [22]).

With our considerations, we want to cover modules over rings, whose elements are K-linear combinations of monomials. The monomials should reflect the commutation properties of the considered operators, for example, generalized versions of derivations and automorphisms. Therefore, it is reasonable and appropriate to restrict our view to monomials in finite sets of symbols A := {a1,..., am} and B := {b1,..., bn} where all elements in A U B are pairwise commutative (which is not necessarily the case for elements in K). Then the monomials are defined as power products of the form A = Ak ■ Bl where k e Nm and l e Zn, using obvious multi-index notation.

Recall (see [22, Definition 2.3]) that a family of subsets {Z(p : j = 1,...,k} of Zn (n is a positive integer) is called an orthant decomposition of Zn if it satisfies the following conditions:

(i) For any j = 1,...,k, (0,..., 0) e z(n) and z(n does not contain any pair of nonzero mutually opposite elements of the form (c1,..., cn) and (-c1,..., -cn).

(ii) Every Z(jn) is a finitely generated subsemigroup of the additive group of Zn which is isomorphic to Nn as a semigroup.

(iii) For any j = 1,...,k, the group generated by zj'1 is Zn.

Given such an orthant decomposition of Z" and m e N, the family {Nm x Z^ : j = l,...,k} is said to be an orthant decomposition of Nm x Z" .A standard example of an orthant decomposition of Z" is a family {Z^,..., Z2")} of all distinct Cartesian products of n sets each of which is either N or Z- = {a e Z : a < 0}.

To extend our concept to free modules F = R( E) = K(A E) with finite generating set E := {e1,...,et}, let {Z^n) : j = 1,...,k} be an orthant decomposition of Z" .A total order -< on the set Nm x Z" x E (where m e N) is said to be a ge"eralized term order on Nm x Z" x E if the following conditions hold:

(a) For every i = 1,...,t, (0,..., 0, ei) is the smallest element of Nm x Z" x {ei}.

(b) If a, b, c e Nm x Z", (a, ei) — (b, ej) (1 < i, j < t) and c and b lie in the same orthant Z("), then (a + c, et) — (b + c, ej).

As we have seen, AE = {lei : A e A, 1 < i < t} is a set of monomials of F, which is in natural one-to-one correspondence with the set Nm x Z" x E (obviously, af ... a^b! . ..b^ei — (k1,...,km, l1,...,l", ei)). A total order — of the set of monomials A E is called a ge"eralized term order on A E if the corresponding order of the set Nm x Z" x E is a generalized term order in the above sense.

Let j = Ak ■ Bl and v = Ak' ■ Bl', where k, k' e Nm and l, l' e Z", and let j be such that l e Zj]. Then, we

say that j divides v if and only if (k', l') e (k, l) + Nm x Z^.If ti = jei and t2 = vej are elements of AE, we say that t1 divides t2 and write t1|t2 if and only if j\v and i = j.

Since AE is a free basis of F as a K-module, every element f e F has a unique representation of the form

f = a1l1ej1 +-----+ adAdejd, ai e K, 1 < i < d,

where A1ej1,..., Adejd are distinct elements of AE. Given a generalized term order — on AE, the greatest monomial with respect — among A1ej1,..., Adejd is called the leadmg mo"omial of f; it is denoted by LT—( f). The coefficient of the leading monomial is called the leadmg coefficie"t of f and denoted by LC— (f). It is easy to see, that if LT f) = jei and the monomials j and v lie in the same orthant, then LT—(v ■ f) = v ■ LT f), while the converse does not to hold in general.

Example A difference-differential ring is a commutative ring K equipped with two sets of operators A :={S1,...,Sm}, £ := {<71,...,a"},

consisting of derivations and automorphisms of K respectively, such that every two operators from the set A U £ commute. The corresponding ring of difference-differential operators D is defined as a free K-module whose free basis consists of all monomials of the form 8k ... ikn &11 . , where (k1,...,km) e Nm and (l1,...,l") e Z". The set of all such monomials is denoted by Am,". The multiplication in D is defined by the relationships

8ia = a8i + 8i(a), oja = oj(a)oj, a e K, 1 < i < m, 1 < j <

extended by distributivity. Furthermore, for any v := 8^ ...8mm o^ ...o"" e Am,", we define the orders of v relative to A and £ as

|v |1 = k1 +-----+km, |v |2 = \11+-----+\l" |, respectively.

Let F be the free D-module generated by a finite set E := {e1,..., et}. Then, the orders of a monomial vei e F (v e Am,", 1 < i < t) with respect to A and £ are defined as ^|1 and |v|2 respectively. In this case F can be considered as a bifiltered D-module with twofold filtration {Fr,s : (r, s) e N2} defined as follows:

Fr,s :={ f e F : | f |1 < r A^ ^ < s}, r, s e N,

where for any f e F, | f h := max{|v|;; 3 j e{1,...,t}: vej e T( f)}, (i = 1, 2). Clearly, if F = D, then the functions Ui = | ■ | : D ^ N satisfy the conditions of Lemma 2.3 and Dr,s = D^ where u = (u1, u 2) : D ^ N2.

The proof of the following theorem can be obtained by mimicking the proof of [22, Theorem 3.1] that states a similar result for rings of difference-differential operators.

Theorem 3.1 (Relative reduction) Let R = K(Л) bearing, E := {e1, ...,et} be a set of free generators of the free R-module F = R(e ), and let <1 and <2 be two generalized term orders on ЛE. Let G := [g\,..., gq} С F\{0} and f e F. Then f can be represented as

f = h 1 gl + --- + hqgq + r, (3.1)

for some elements h1, ...,hq e R and r e F such that

1. hi = 0 or LT <1 (higi) ^iLT <1 (f ),i = 1,...,q;

2. r = 0 orr = 0 л LT<1 (r) <1 LT<1 (f) such that

LT <1 (r) e {LT <1 (Xgi) : LT<2 (kgi) <2 LT< (r), X e Л, i = 1,...,q}. (3.2)

We say that f <1-reduces modulo G relative to <2 to r, and the transition from f to r is said to be the relative reduction.

With this modified reduction, we can now consider the notion of relative Grobner basis.

Definition 3.2 (Relative Grobner basis [22, Definition 3.3]) As above, let F be a finitely generated free R-module, and N С F a submodule. Further, let <1 and <2 be a pair of generalized term orders, G := [g1,..., gq} с N\{0}. The set G is called a <1-Grobner basis relative to <2 if and only if every f e N\{0} can be <1-reduced modulo G relative to <2 to zero. Then, if no confusion is possible, G is called a relative Grobner basis for N.

We consider two generalized term orders <1 and <2 and relative reduction for monomially filtered rings R. It is obvious that G is a (relative) Grobner basis for N if and only if N = R (G) and every non-zero element in N can be <1-reduced modulo G relative to <2 to zero, which is equivalent to Ip П N = 0, i.e. axiom 4 of Grobner reduction.

It is shown in [7, Section 3], that for the bivariate filtration Frs, using an appropriate choice of term orders <1 and <2 it is possible to ensure condition f e Fr,s and f —> h ^ h e Fr,s. In particular, this property holds with respect to the orders given in [22, Section 3].

To ensure that every non-zero element can be reduced to zero, we need to exploit Buchberger's algorithm, as presented in [22, Theorem 3.4]. At the same time, Donch [4] gave an example of a certain pair of term-orders such that the Buchberger's algorithm, which takes into account relative reduction, does not terminate. This termination property was further investigated in [10] where one can find a condition on the considered generalized orders that guarantees the termination of the Buchberger's algorithm (it is called the difference-differential degree compatibility condition, see [10, Definition 3.1]).

In what follows, a ring R is assumed to be a free K-module with a basis Л. Furthermore, we suppose that R is equipped with a bivariate filtration induced by a mapping u : R ^ N2 satisfying conditions (i)—(iii) of Remark 2.4. In other words,

Rr,s = {f e R : u 1 (f) < r л u2(f) < s}, (r, s) e N2,

where the mappings ui := ж, о u (i = 1, 2) satisfy the conditions of Lemma 2.3. Let F be a free R-module with basis E and let {Fr,s = Rr,sE : (r, s) e N2} be the induced filtration of F.

Definition 3.3 (Admissible orders) With the above notation, a pair of generalized term orders <1 and <2 on the set Л E is said to be admissible, if for any two terms t1 , t2 e Л E

• t1 <1 t2 when u1(t1) < u 1(t2), or u 1(t1) = u1(t2) and u2(t1) < u2(t2);

• t1 <2 t2 when u2(t1) < u2(t2), or u2(t1) = u2(t2) and u1(t1) < u 1(t2);

• t1 <1 t2 ^ t1 <2 t2 if u1(t1) = u 1 (t2) and u2(t1) = u2(t2).

Example Let y : AE ^ Ns (s is some appropriate chosen positive integer) uniquely identify a monomial. An example is given in [8, Paragraph after Proposition 6.]. The pair

t1 —1 t2 (U1(t1), U2(t1), (p(t\)) <lex (u 1(t2), U2(t2), pfo)): t1 —2 t2 (U2(t1), U1(t1), y(t2)) <lex (U2(t2), U^), pfo)),

is an example for a pair of admissible orders on the monomials A E in F. The following Lemma is proven in [13, Lemma 4.1].

Lemma 3.4 Let R = K(A), where A = Ak ■ B1 with (k, l) e Nm x Z". FUrther, let S be a" mftmte seqUe"ce of mommials AE (where E := {e1,..., et}). The", there exists a" i"dex 1 < j < t, a"d a" mftmte sUbseqUe"ce

{l1ej,A2ej,...,lkej,...} c S,

sUch that lk divides Ak+1 for k > 1.

As next step, we generalize [10, Lemma 3.2] to arbitrary rings with that particular kind of monomials.

Lemma 3.5 Let F be a free R-modUle, — 1 a"d — 2 be a pair of admissible term orders o" AE, a"d Gi := {g1,..., gq, r1,...,ri} c F\{0}. Ifr,+1 is —1-redUced modUlo Gi relative to — 2 (see Theorem 3.1), a"d if

for a"y l e A, h e Gi : LT—1 (rt+1) = LT—1 (A ■ h), (3.3)

the" the asceMrng chai" G1 c G2 c ■ ■■ stabilizes.

Proof Since for all l e A and h e Gj we have LT—1 (ri+1) = LT—1 (l ■ h), the element ri+1 is irreducible with respect to Gi. The second condition (3.2) involving the order —2 would apply only if ri+1 would be reducible, hence, we are considering the usual notion of reduction with respect to Gi for the order —1. If the chain G1 c G2 C ■■■ does not stabilize, then there would be an infinite sequence of monomials {LT (ri) : i = 1, 2,...} in A E such that LT (ri) does not divide LT(r;+1) for all i, contrary to the statement of Lemma 3.4. □

Assume given a fixed orthant decomposition of Nm x Z" consisting of k orthants and A j (1 < j < k) is a subset of A (we use the notation of Lemma 3.4) consisting of all power products whose exponent vectors lie in the j-th orthant. Furthermore, K [A j ] will denote the subring of R generated by the set A j. If f and g are non-zero elements of a free R-module F with a finite set of free generators E and — a generalized term order of AE, then V (j, f, g) will denote a finite system of generators of the K [A j ]-module

k[Aj](LT —(lf) e AjE : l e A) n k[aj]<LT — (ng) e AjE : n e A).

(The idea of considering such modules and their systems of generators is due to F. Winkler and M. Zhou, see [22].) As it is shown in [21, Lemma 3.5], for any h e F and j = 1,...,k, there exists some l e A and a monomial U j in h such that LT— (lh) = lUj e A jE. Moreover, this term Uj in h is unique; it is denoted by LTj, — (h). With the above notation, for every generator v e V (j, f, g), the element

v f v f

S<(j, f, g,v) = J J

LT j, — (f) LC j, — ( f) LT j, — (g) LCj—g

is said to be an S-polynomial of f and g with respect to j, v and —. The following algorithm is applied for each orthant Z(f.

Algorithm 1 Buchberger Algorithm for bifiltered Rings, [22, Algorithm 1] Require: F is a free R-module, G := {f1,..., fr}c F\{0}; The ring R is 2-fold filtered; <1 and <2 are generalized term orders on AE. Ensure: G" :={g1,..., gs} c F\{0} where R (G") = R (G} such that Ip n (RG") = 0. G' G;

while there exist f, g e G' and v e V(j, f, g) such that

S<2 (j, f, g, v) is <2-reduces relative to <2 to r = 0 by G'; do G' G'U {r}; G" ^ G';

while there exist f, g e G" and v e V (j, f, g) such that

S<1 (j, f, g, v) <1-reduces relative to <2 to r = 0 by G"; do G" G" U {r}; return G".

Theorem 3.6 IfR denotes a bifiltered ring, where R is builtfrommonomials of the form Ak ■ Bl with (k, l) e Nm xZn, and the orders <1 and <2 are chosen to be admissible, then Buchberger's algorithm for filtered rings terminates in a finite number of steps.

Proof We start with G := {g1;..., gq} and already assume that it is a Grobner basis with respect to <2. Suppose that the relative reduction proceeds by generating the sequence of sets Gi := {g1;..., gq, r1,...,ri} for i > 1 and let ri+1 be reduced with respect to Gi. Then, either

• LT<1 (ri+1) = LT<1 (kh) for any X e A and h e Gi, or

• LT<1 (ri+1) = LT<1 (kh) for some k e A, h e Gi such that LT<2 (ri+1) <2 LT<2 (kh).

By Lemma 3.5, the first case cannot occur infinitely many times (that is, if all Gi+1 are obtained from Gi via a transition of the first type, then the ascending chain G1 c G 2 c ■■■ stabilizes). For the second case, we have that

LT <1 (n+1) = LT <1 (kh),

LT<2 (ri+1) <2 LT<2 (kh) u2(LT<2 (ri+1)) < u2(LT<2 (kh)).

If the algorithm does not terminate, then (Gi )i>1 is a strictly increasing sequence. Therefore, we can assume that there are infinitely many pairs (i, j) e N2 with i > j such that

LT<1 (ri) = LT<1 (krj) A LT<2 (ri) <2 LT<2 (krj).

We obtain a strictly descending (with respect to the order <2) infinite sequence of monomials in A E that contradicts the fact that AE is well-ordered with respect to <2. □

Consider now the following situation: Let ui e NR satisfy the conditions of Lemma 2.3, let R be p-fold filtered

Rr := f|{f e R : ^(f) < n}, r = (ru ...,rp) e Np, i=1

We are going to give a formulation of Buchberger's algorithm that corresponds to the Grobner reduction in this case.

Let us consider two monomial orders <f and <n on the set AE defined as follows. If t1; t2 e AE, then

t1 <m t2 (un (t\), um (t1), (p(h)) <lex (un (t2), um (t2), V(t2)), t1 <n t2 (u n (t1), u(t1),V(t1)) <lex (un (t2), u(t2), ip(t2)),

where U(ti) = U1(ti) +----+ Up(ti)(1 < m, " < p) and y is a fixed map y : A ^ Ns that uniquely identifies the

monomials (as demonstrated in [8, Paragraph after Proposition 6.]). Obviously, the pair of generalized term orders —"m and —"m are admissible.

Theorem 3.7 (Set-relative reduction) Let F be the free R-modUle, where R is a ("ot "ecessarily commUtative) "oetheria" ri"g a"d the fixed commUtative sUbri"g K of R is a field. Let f e F a"d G = {g1,..., gq} c F\{0}. Let A be a sUbsetofthe orders {—1,..., — p}, a"da" order —m (1 < m, " < p) deft"ed above be fixed. The" there exist eleme"tsh1, ...,hq e R a"d r e F sUch that

f = h 1 g1 +-----+hqgq + r a"d

• hi = 0 or for all — i" A 1 < i < q: LT—(higi) < LT — (f)

• r = 0 or for all — i" A: LT — (r) < LT — (f) sUch that

LT—» (r) e {LT—» (lgi) : 1 < i < q ,l e A, LT—m (lgt) LT—m (f)}. I" this case we say that the eleme"t f A-redUces tor by G relative to —m

Proof We give a constructive proof, along the lines of [22, Theorem 3.1]. First, we initialize r = f and hi = 0 for 1 < i < q. The next steps are repeated until r = 0 or there exists no l and gi that satisfy the conditions of the theorem. If there exists l e A such that for all — in A we have LT- (r) < LT - (f) and

LT(r) = LT(lgi) A LT—m (lgi) LT—m (r),

we are allowed to perform the reduction step, and update the quantities r to r', respectively hi to h', as follows:

LC—m (r) , ,, , LC—m (r)

r' = r--m-lgi hi = hi +--m--l.

LC—"m (lgi) 51 1 1 LC—« (lgi)

Obviously, we have

LT—m (r') —m LT—m (r), while for all — i"Awe have LT— (lgi) ^ LT—(f).

Since the set of monomials AE c F is well-ordered, this can only be repeated finitely often. Summing up the ls to hi we obtain that for all —e A, one has LT —(higi) < LT -(f). □

Based on set-relative reduction, we can now consider a p-step procedure for computing Grobner bases in this setting.

Algorithm 2 Buchberger Algorithm for Filtered Rings Require: F is a free R-module, V := {f1,..., fr }c F\{0};

The ring R is p-fold filtered. Ensure: G := {g1,..., gs}c F\{0} where R (G) = R (V) such that Ip n (RG) = 0. (Ip is the set of all irreducible elements with respect to the set-relative reduction)

A ^ {—p}; G <°> ^{/1,..., fr};

while there exist j e{1,..., k}, f, g e G(0) and v e V(j, f, g) such that S<p (j, f, g,v) A-reduces to r = 0 relative to — p by G(0) do G<® ^ G(0) U {r}; G« ^ G«»; for t = p — 1,..., 1 do

(A, —) ^({—p1+1}, —1+1);

while there exist j e{1,...,k}, f, g e G(p—tt and v e V(j, f, g) such that S—t (j, f, g,v) is A-reduces to r = 0 relative to — by G(p—t) do G(p—t G(p—t) U {r}; G(p—t+1) G(p—t); return G (p>

The termination of the algorithm is justified by the following facts. First, a generalized term order is a well-order and the first loop terminates by Buchberger's Theorem (it is proven in [1]). For the second loop, in each step we have a pair of admissible orders <lg-1and <ti~l, and hence one can apply Theorem 3.6. The set-relative reduction ensures condition 5 of Definition 2.5. Finally, property 4 in Definition 2.5 follows from a straight-forward generalization of [22, Theorem 3.3]. An implementation of a the algorithm for the ring of difference-differential operators and a bi-filtered ring has been obtained in [2].

Acknowledgements Open access funding provided by Johannes Kepler University Linz.

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