Compatible rewriting of noncommutative polynomials for proving operator identities
Cyrille Chenavier, Clemens Hofstadler, Clemens G. Raab, Georg Regensburger
aa r X i v : . [ c s . S C ] F e b Compatible rewriting of noncommutativepolynomials for proving operator identities
Cyrille Chenavier, Clemens Hofstadler,Clemens G. Raab, and Georg Regensburger ∗ Abstract
The goal of this paper is to prove operator identities using equalitiesbetween noncommutative polynomials. In general, a polynomial expres-sion is not valid in terms of operators, since it may not be compatible withdomains and codomains of the corresponding operators. Recently, someof the authors introduced a framework based on labelled quivers to rigor-ously translate polynomial identities to operator identities. In the presentpaper, we extend and adapt the framework to the context of rewritingand polynomial reduction. We give a sufficient condition on the poly-nomials used for rewriting to ensure that standard polynomial reductionautomatically respects domains and codomains of operators. Finally, weadapt the noncommutative Buchberger procedure to compute additionalcompatible polynomials for rewriting. In the package
OperatorGB , we alsoprovide an implementation of the concepts developed.
Keywords
Rewriting, noncommutative polynomials, quiver representations,automated proofs, completion
Properties of linear operators can often be expressed in terms of identities theysatisfy. Algebraically, these identities can be represented in terms of noncom-mutative polynomials in some set X . The elements of X correspond to ba-sic operators and polynomial multiplication models composition of operators.Based on this, proving that a claimed operator identity follows from assumedidentities corresponds to the polynomial f , associated to the claim, lying inthe ideal generated by the set F of polynomials associated to the assumptions.However, ideal membership f ∈ ( F ) is not enough for proving an operator iden-tity in general, since computations with noncommutative polynomials ignorecompatibility conditions between domains and codomains of the operators.In order to represent domains and codomains of the operators, we use theframework introduced recently in [17]. So, we consider a quiver (i.e., a directed ∗ This work was supported by the Austrian Science Fund (FWF): P 27229, P 32301, andP 32952 { cyrille.chenavier,clemens.hofstadler,clemens.raab,georg.regrensburger } @jku.at Institute for Algebra, Johannes Kepler University, Altenberger Straße 69, Linz, Austria Q , where vertices correspond to functional spaces, edges corre-spond to basic operators between those spaces and are labelled with symbolsfrom X . Then, paths in Q correspond to composition of basic operators andinduce monomials over X that are compatible with Q . Note that we can allowthe same label for different edges if the corresponding operators satisfy the sameidentities in F . For instance, differential and integral operators can act on dif-ferent functional spaces, as illustrated in our running example below. For formaldetails and relevant notions, see Sections 2 and 4. Informally, a polynomial iscompatible with the quiver if it makes sense in terms of operators and f is calleda Q -consequence of F if it can be obtained from F by doing computations usingcompatible polynomials only. This means that these computations also makesense in terms of operators.Obviously, the claim f and the assumptions F have to be compatible with Q . In [17], it was shown that f is a Q -consequence of F if f ∈ ( F ) and eachelement of F is uniformly compatible, which means that all its monomials canbe assigned the same combinations of domains and codomains. This is in partic-ular the case when each edge has a unique label and polynomials do not have aconstant term. Note that ideal membership can be checked independently of Q and is undecidable in general. In practice, it can often be checked by computinga (partial) noncommutative Gr¨obner basis G of F and reducing f to zero by G , see [15]. The package OperatorGB [11] can check compatibitlity of polyno-mials with quivers and, based on partial Gr¨obner bases, can compute explicitrepresentations of polynomials in terms of generators of the ideal. Versions for
Mathematica and
SageMath can be obtained at: http://gregensburger.com/softw/OperatorGB
In this paper, we generalize the formal definition of Q -consequences to thecase when elements of F are compatible but not necessarily uniformly compat-ible, see Section 3. Then, we show in Section 4 that being a Q -consequenceimplies that the corresponding operator identity can indeed be proven by com-putations with operators. Since elements of F do not have to be uniformlycompatible, we impose in Section 5 restrictions on the polynomial rewriting, sothat it respects the quiver. For the same reason, we also impose restrictions onthe computation of partial Gr¨obner bases in Section 6. Based on such a partialGr¨obner basis, one often can prove algorithmically that f is a Q -consequenceof F just by standard polynomial reduction. To this end, we also extend thepackage OperatorGB .Gr¨obner bases for noncommutative polynomials have been applied to oper-ator identities in the pioneering work [10, 9], where Gr¨obner bases are used tosimplify matrix identities in linear systems theory. In [8, 13], the main strategyfor solving matrix equations, coming from factorization of engineering systemsand matrix completion problems, is to apply Gr¨obner bases with respect to anordering appropriate for elimination. The same approach was used in [18] tocompute Green’s operators for linear two-point boundary problems with con-stant coefficients.If edges of the quiver have unique labels, it has been observed in the liter-ature that the operations used in the noncommutative analog of Buchberger’salgorithm respect compatibility of polynomials with domains and codomains ofoperators, cf. [9, Thm. 25]. See also Remark 6 and Theorem 5 for a formal2tatement using the framework of the present paper. For an analogous obser-vation in the context of path algebras, see [16, Sec. 47.10]. We were informedin personal communication that questions related to proving operator identitiesvia computations of Gr¨obner bases are also addressed in [14].Alternatively, computations with operators can also be modelled by partialalgebras arising from diagrams, for which an analogous notion of Gr¨obner baseswas sketched in [1, Sec. 9] and developed in [3]. Moreover, generalizationsof Gr¨obner bases and syzygies are considered in [7], where higher-dimensionallinear rewriting systems are introduced for rewriting of operators with domainsand codomains.We conclude this section with a small running example that we use through-out the paper to illustrate the notions that we introduce from practical point ofview. A
Mathematica notebook that illustrates the use of the new function-ality of the package using this running example can be obtained at the webpagementioned above.
Example 1.
Consider the inhomogeneous linear differential equation y ′′ ( x ) + A ( x ) y ′ ( x ) + A ( x ) y ( x ) = r ( x ) and assume that it can be factored into the two first-order equations y ′ ( x ) − B ( x ) y ( x ) = z ( x ) and z ′ ( x ) − B ( x ) z ( x ) = r ( x ) . It is well-known that a particular solution is given by the nested integral y ( x ) = H ( x ) Z xx H ( t ) − H ( t ) Z tx H ( u ) − r ( u ) du dt, (1) where H i ( x ) is a solution of y ′ ( x ) − B i ( x ) y ( x ) = 0 such that H i ( x ) − exists.In order to translate this claim into an operator identity, let us consider thedifferentiation ∂ : y ( x ) y ′ ( x ) and the two integrations R : y ( x ) Z xx y ( t ) dt and R : y ( x ) Z xx y ( t ) dt. Moreover, any function F ( x ) induces a multiplication operator F : y ( x ) y ( x ) F ( x ) and · denotes the composition of operators. Thus, the factored differ-ential equation and the solution correspond to the following operators L := ( ∂ − B ) · ( ∂ − B ) , S := H · R · H − · H · R · H − and the claim corresponds to the identity L · S = id . In terms of functions, thismeans that y ( x ) = ( Sr )( x ) is a solution of ( Ly )( x ) = r ( x ) . (2) Using the Leibniz rule, H i being a solution of the factor differential equationcorresponds to ∂ · H i = H i · ∂ + B i · H i and the invertibility corresponds to H i · H − i = id . The last fact we use forproving the claim is the fundamental theorem of calculus, which corresponds to ∂ · R = id , ∂ · R = id . In Example 2, we will show how these operator identities can be translated intononcommutative polynomials that are compatible with a quiver. Preliminaries
In this section, we recall the main definitions and basic facts from [17] thatformalize compatibility of polynomials with a labelled quiver.We fix a commutative ring R with unit as well as a set X . We consider thefree noncommutative algebra R h X i generated by the alphabet X : it can be re-garded as the ring of noncommutative polynomials in the set of indeterminates X with coefficients in R , where indeterminates commute with coefficients butnot with each other. The monomials are words x . . . x n ∈ h X i , x i ∈ X , includ-ing the empty word 1. Every polynomial f ∈ R h X i has a unique representationas a sum f = X m ∈h X i c m m with coefficients c m ∈ R , such that only finitely many coefficients are nonzero,and its support is defined assupp( f ) := { m ∈ h X i | c m = 0 } , where c m are as above.Recall that a quiver is a tuple ( V, E, s, t ), where V is a set of vertices, E isa set of edges, and s, t : E → V are source and target maps, that are extendto all paths p = e n · · · e by letting s ( p ) = s ( e ) and t ( p ) = t ( e n ). For everyvertex v ∈ V , there is a distinct path ǫ v that starts and ends in v withoutpassing through any edge, and which acts as a local identity on paths p , that is ǫ t ( p ) p = p = pǫ s ( p ) . A labelled quiver, Q = ( V, E, X, s, t, l ) is a quiver equippedwith a label function l : E → X of edges into the alphabet X . We extend l intoa function from paths to monomials by letting l ( p ) = l ( e n ) · · · l ( e ) ∈ h X i , and l ( ǫ v ) = 1 is the empty word for every vertex v . From now on, we fix a labelledquiver Q = ( V, E, X, s, t, l ). Definition 1.
Given a labelled quiver and a monomial m , we define the set of signatures of m as σ ( m ) := { ( s ( p ) , t ( p )) | p a path in Q with l ( p ) = m } ⊆ V × V. A polynomial f ∈ R h X i is said to be compatible with Q if its set of signatures σ ( f ) is non empty, where: σ ( f ) := \ m ∈ supp( f ) σ ( m ) ⊆ V × V. Finally, we denote by s ( f ) and t ( f ) the images of σ ( f ) through the naturalprojections of V × V on V . Note that we have σ (0) = V × V and σ (1) = { ( v, v ) | v ∈ V } .Computing with compatible polynomials does not always result in compat-ible polynomials. However, under some conditions, the sum and product ofcompatible polynomials are compatible as well. The following properties ofsignatures are straightforward to prove; see also Lemmas 10 and 11 in [17]. Lemma 1.
Let f, g ∈ R h X i be compatible with Q . Then, . If σ ( f ) ∩ σ ( g ) = ∅ , then f + g is compatible with Q and σ ( f + g ) ⊇ σ ( f ) ∩ σ ( g ) .2. If s ( f ) ∩ t ( g ) = ∅ , then f g is compatible with Q and σ ( f g ) ⊇ { ( u, w ) ∈ s ( g ) × t ( f ) |∃ v ∈ s ( f ) ∩ t ( g ) : ( u, v ) ∈ σ ( g ) ∧ ( v, w ) ∈ σ ( f ) } . We use the following conventions when we draw labelled quivers: we do notgive names to vertices and edges, but denote them by a bullet and an arroworiented from its source to its target, respectively, and the label of an edge issimply written above the arrow representing this edge.
Example 2.
Let us continue the running example. The Leibniz rule and invert-ibility for H and H and the fundamental theorem of calculus correspond to thefollowing noncommutative polynomials in Z h X i , where X = { h , h , b , b , ˜ h , ˜ h , i, d } . f = dh − h d − b h , f = dh − h d − b h ,f = h ˜ h − , f = h ˜ h − ,f = di − We collect these polynomials in the set F := { f , . . . , f } . Notice that we repre-sent the two integrals by a single indeterminate, so we only need one polynomialfor the fundamental theorem of calculus. The claim corresponds to f := ( d − b )( d − b ) h i ˜ h h i ˜ h − . Since integration and differentiation decrease and increase the regularity of func-tions, it is natural to consider the following labelled quiver with vertices (moredetails are given Section 4) with labels in the alphabet X . • • • d db b i ih h h h ˜ h ˜ h Either directly or by the package, we check that f and each element of F iscompatible with the quiver. Denoting the vertices from left to right by v , v , v ,we obtain the following signatures. σ ( f ) = { ( v , v ) } , σ ( f ) = { ( v , v ) } ,σ ( f ) = { ( v , v ) } , σ ( f ) = { ( v , v ) } ,σ ( f ) = { ( v , v ) , ( v , v ) } ,σ ( f ) = { ( v , v ) } To determine σ ( f ) , for example, notice that σ ( h i ˜ h h i ˜ h ) = { ( v , v ) } and that σ ( dd ) = σ ( b d ) = σ ( db ) = σ ( b b ) = { ( v , v ) } and recall that σ (1) containsall pairs of the form ( v i , v i ) . Q-consequences
The following definition characterizes the situations when a representation ofthe claim in terms of the assumptions is also valid in terms of operators. Thisgeneralizes the notion of Q -consequence given in [17]. Throughout the section,we fix a labelled quiver Q with labels in a set X . Definition 2. A Q - consequence of F ⊆ R h X i is a polynomial f ∈ R h X i ,compatible with Q , such that there exist g i ∈ F , a i , b i ∈ R h X i , ≤ i ≤ n , suchthat f = n X i =1 a i g i b i , (3) and for every ( u, v ) ∈ σ ( f ) and every i , there exist vertices u i , v i such that ( u, u i ) ∈ σ ( b i ) , ( u i , v i ) ∈ σ ( g i ) and ( v i , v ) ∈ σ ( a i ) . The conditions on the signatures mean that there exist three paths in thequiver as illustrated in the following diagram. u vu i v ifb i g i a i Proving that a given representation (3) satisfies the required conditions ofthe above definition is straightforward. In Proposition 1, we give an alternativecriterion for Q -consequences. This criterion will play an important role later inSection 5 on rewriting. Before, we need the following lemma. Lemma 2.
Let m ∈ h X i be a monomial and g ∈ R h X i be a polynomial such that σ ( m ) ⊆ σ ( g ) . Then, for all monomials a, b ∈ h X i , we have σ ( amb ) ⊆ σ ( agb ) .Moreover, for every ( u, v ) ∈ σ ( amb ) , there exist two vertices ˜ u, ˜ v such that ( u, ˜ u ) ∈ σ ( b ) , (˜ u, ˜ v ) ∈ σ ( g ) , and (˜ v, v ) ∈ σ ( a ) .Proof. For every ( u, v ) ∈ σ ( amb ), there exists a path from u to v with label amb . We split this path in 3 parts: the first part β has label b , the third part α has label a , and the second part has label m . Since σ ( m ) ⊆ σ ( g ), for every˜ m ∈ supp( g ), there also exists a path γ from ˜ u := t ( β ) to ˜ v := s ( α ) with label˜ m , as pictured on the following diagram u ˜ u ˜ v v b m ˜ m a Hence, a ˜ mb is the label of αγβ . Consequently, σ ( amb ) ⊆ σ ( a ˜ mb ) for every˜ m ∈ supp( g ), and (˜ u, ˜ v ) ∈ σ ( g ). Proposition 1.
Let F ⊆ R h X i be a set of polynomials such that for every g ∈ F , there exists m g ∈ supp( g ) such that σ ( m g ) ⊆ σ ( g ) . Let f ∈ R h X i bea compatible polynomial such that there exist λ i ∈ R , g i ∈ F , a i , b i ∈ h X i , ≤ i ≤ n , such that f = n X i =1 λ i a i g i b i , (4) and for each i , we have σ ( f ) ⊆ σ ( a i m g i b i ) . Then, f is a Q -consequence of F . roof. By hypotheses, f is compatible and for every ( u, v ) ∈ σ ( f ) and for every1 ≤ i ≤ n , we have ( u, v ) ∈ σ ( a i m g i b i ). Hence, using the hypothesis σ ( m g i ) ⊆ σ ( g i ), from Lemma 2, there exist vertices u i and v i such that ( u, u i ) ∈ σ ( b i ),( u i , v i ) ∈ σ ( g i ) and ( v i , v ) ∈ σ ( a i ). As a consequence, f is a Q -consequenceof F .Note that if for m g ∈ supp( g ), we have σ ( m g ) ⊆ σ ( g ), then σ ( m g ) = σ ( g )holds by definition. Example 3.
Let us continue Example 2. We show that f is a Q -consequenceof F by considering the following representation: f = f i ˜ h + ( d − b ) f i ˜ h h i ˜ h + f + ( d − b ) f h i ˜ h + ( d − b ) h f ˜ h h i ˜ h + h f ˜ h . (5) Such a representation can be obtained with the package by tracking cofactorsin polynomial reduction w.r.t. a monomial order. Here, we consider a degree-lexicographic order such that d is greater than h i ’s and b i ’s. Then, f can bereduced to zero using F , which gives (5) . Now, we have to check assumptions onsignatures, either by checking Definition 2 or the assumptions of Proposition 1,both options are implemented in the package. For applying Proposition 1 byhand, we can choose m f = dh , m f = h d, m f = h ˜ h , m f = h ˜ h , and m f = di , which satisfy m f i ∈ supp( f i ) and σ ( m f i ) = σ ( f i ) . Expanding (5) in the form (4) , we may check that σ ( a i m g i b i ) = { ( v , v ) } = σ ( f ) for everysummand in the representation (4) , which proves that f is a Q -consequence of F . To conclude this section, we prove that the property of being a Q -consequenceis transitive, which we will exploit in Section 6. Theorem 1.
Let
F, G ⊆ R h X i be sets of polynomials such that each elementof G is a Q -consequence of F . Then, any Q -consequence of G is also a Q -consequence of F .Proof. Let h be a Q -consequence of G , so that it is compatible with Q . More-over, h = P i a i g i b i , with g i ∈ G and a i , b i ∈ R h X i such that for every ( u, v ) ∈ σ ( h ) and every i , there exist vertices u i , v i such that ( u, u i ) ∈ σ ( b i ), ( u i , v i ) ∈ σ ( g i ) and ( v i , v ) ∈ σ ( a i ). Since every element of G is a Q -consequence of F , foreach g i , there exist a i,j , b i,j ∈ R h X i and f i,j ∈ F such that g i = P j a i,j f i,j b i,j and for every ( u i , v i ) ∈ σ ( g i ) and every j , there exist ( u i,j , v i,j ) ∈ σ ( f i,j ) suchthat ( u i , u i,j ) ∈ σ ( b i,j ) and ( v i,j , v i ) ∈ σ ( a i,j ). All together, we have h = X i X j a i a i,j f i,j b i,j b i . For every j , u i and v i belong to s ( b i,j ) ∩ t ( b i ) and s ( a i ) ∩ t ( a i,j ), respectively,so that from Point 2 of Lemma 1, ( u, u i,j ) ∈ σ ( b i,j b i ) and ( v i,j , v ) ∈ σ ( a i a i,j ),respectively. Hence, h is a Q -consequence of F . In this section, we formalize the translation of polynomials to operators bysubstituting variables by basic operators. In particular, we show in Theorem 27hat being a Q -consequence is enough to ensure that the corresponding operatoridentity can be inferred from the assumed operator identities. To this end, wesummarize the relevant notions and basic facts from [17, Section 5].For a quiver ( V, E, s, t ) and a ring R , ( M , ϕ ) is called a representation ofthe quiver ( V, E, s, t ), if M = ( M v ) v ∈ V is a family of R -modules and ϕ is amap that assigns to each e ∈ E a R -linear map ϕ ( e ) : M s ( e ) → M t ( e ) , see e.g.[5, 6]. Not that any nonempty path e n . . .e in the quiver induces a R -linear map ϕ ( e n ) · . . . · ϕ ( e ), since the maps ϕ ( e i +1 ) and ϕ ( e i ) can be composed for every i ∈ { , . . . , n − } by definition of ϕ . Similarly, for every v ∈ V , the empty path ǫ v induces the identity map on M v . Remark 1.
All notions and results of this section naturally generalize to R -linear categories by considering objects and morphisms in such a category insteadof R -modules and R -linear maps, respectively. For more details, see Section 5.2in [17]. Definition 3.
Let R be a ring and let Q be a labelled quiver with labelling l .We call a representation ( M , ϕ ) of Q consistent with the labelling l if for anytwo nonempty paths p = e n . . .e and q = d n . . .d in Q with the same source andtarget, equality of labels l ( p ) = l ( q ) implies ϕ ( e n ) · . . . · ϕ ( e ) = ϕ ( d n ) · . . . · ϕ ( d ) as R -linear maps. Remark 2.
If all paths with the same source and target have distinct labels,then every representation of that labelled quiver is consistent with its labelling.In particular, this holds if for every vertex all outgoing edges have distinct labelsor analogously for incoming edges. These sufficient conditions can be verifiedwithout the need for considering all possible paths.
For Definition 4 and Lemma 3, we fix a ring R , a labelled quiver Q =( V, E, X, s, t, l ) and a consistent representation R = ( M , ϕ ) of Q . In order todefine realizations of a polynomial, we first need to introduce some notations.Given two vertices v, w , we write R h X i v,w for the set of polynomials f ∈ R h X i such that ( v, w ) ∈ σ ( f ). From Point 1 of Lemma 1, R h X i v,w is a module, andit is clear that this module is free with basis the set of monomials m such that( v, w ) ∈ σ ( m ). We also denote by Hom R ( M v , M w ) the set of R -linear mapsfrom M v to M w . Definition 4.
For v, w ∈ V , we define the R -linear map ϕ v,w : R h X i v,w → Hom R ( M v , M w ) by ϕ v,w ( l ( e n . . .e )) := ϕ ( e n ) · . . . · ϕ ( e ) for all nonempty paths e n . . .e in Q from v to w and, if v = w , also by ϕ v,v (1) :=id M v . For all f ∈ R h X i v,w , we call the R -linear map ϕ v,w ( f ) a realization of f w.r.t. the representation R of Q . Notice that the map ϕ v,w is well-defined since, by consistency of R , for everymonomial m ∈ R h X i v,w , its realization ϕ v,w ( m ) does not depend on the pathfrom v to w with label m .In the proof of Theorem 2, we use an intermediate result given in [17, Lemma31], whose statement is the following. 8 emma 3. Let u, v, w ∈ V . Then, for all f ∈ R h X i v,w and g ∈ R h X i u,v , wehave that f g ∈ R h X i u,w and ϕ u,w ( f g ) = ϕ v,w ( f ) · ϕ u,v ( g ) . Theorem 2.
Let F ⊆ R h X i be a set of polynomials and let Q be a labelledquiver with labels in X . If a polynomial f ∈ R h X i is a Q -consequence of F ,then for all consistent representations of the quiver Q such that all realizationsof all elements of F are zero, all realizations of f are zero.Proof. Assume that f is a Q -consequence, so that it is compatible with Q andit can be written in the form P a i g i b i , such that for each ( u, v ) ∈ σ ( f ) andeach i , there exist vertices u i , v i such that ( u, u i ) ∈ σ ( b i ), ( u i , v i ) ∈ σ ( g i ) and( v i , v ) ∈ σ ( a i ). Let us fix a consistent representation R = ( M , ϕ ) of Q . Bylinearity of ϕ u,v and from Lemma 3, we have ϕ u,v ( f ) = X ϕ u,v ( a i g i b i ) = X ϕ v i ,v ( a i ) · ϕ u i ,v i ( g i ) · ϕ u,u i ( b i ) . Hence, if all realizations of all elements of F are zero, then ϕ u,v ( f ) = 0, whichmeans that all realizations of f w.r.t R are zero. Example 4.
We finish our proof of (2) by considering certain representationsof the quiver of Example 2. For a nonnegative integer k and an open interval I ⊆ R , we assign the spaces C k ( I ) , C k +1 ( I ) , and C k +2 ( I ) to the the verticesfrom right to left. Hence, differentiation and integration induce operators ∂ : C k +1 ( I ) → C k ( I ) , ∂ : C k +2 ( I ) → C k +1 ( I ) , R : C k ( I ) → C k +1 ( I ) , and R : C k +1 ( I ) → C k +2 ( I ) . We also assume the following regularity of functions: B is C k , H and B are C k +1 and H is C k +2 on I . Then, the naturalrepresentation associated with these operators is consistent. Moreover, we haveseen in Example 3 that f is a Q -consequence of F . Since all realizations of f i ’sare zero, by Theorem 2, all realizations of f are zero. In particular, for everynonnegative integer k and every r ( x ) ∈ C k ( I ) , the function y ( x ) defined by (1) is a solution of the inhomogeneous differential equation (2) .Instead of considering scalar differential equations we could consider dif-ferential systems of the form (2) for vector-valued functions y ( x ) of arbitrarydimension n . More explicitly, we can also consider coefficients B ( x ) , B ( x ) as n × n matrices, r ( x ) as a vector of dimension n , and H ( x ) and H ( x ) as fun-damental matrix solutions of the homogeneous systems y ′ ( x ) − B i ( x ) y ( x ) = 0 .We still obtain consistent representations of the quiver where the vertices aremapped to C k ( I ) n , C k +1 ( I ) n and C k +2 ( I ) n , respectively. Then, Theorem 2 im-mediately proves that the function y ( x ) defined by (1) is a solution of the inho-mogeneous differential equation (2) . Similarly, analogous statements for othersuitable functional spaces can be proven just by choosing different representa-tions of the quiver. In this section, we give conditions on polynomials such that rewriting to zeroof a compatible polynomial by them proves that it is a Q -consequence. First,we recall from [17, Definition 2] a general notion of rewriting one polynomial by9nother in terms of an arbitrary monomial division. Notice that the standardpolynomial reduction is a particular case, where m is the leading monomial of g w.r.t. a monomial order and λ is such that amb is cancelled in (6). Definition 5.
Let g ∈ R h X i be a polynomial and let m ∈ supp( g ) . Let f ∈ R h X i be a polynomial such that m divides some monomial m f ∈ supp( f ) , i.e., m f = amb for monomials a, b ∈ h X i . For every λ ∈ R , we say that f can berewritten to h := f + λagb, (6) using ( g, m ) . We fix a labelled quiver Q with labels in X . It turns out that to obtain Q -consequences using rewriting (Theorem 3), we need to choose suitable di-visor monomials such that signatures only increase. In particular, this is thecase when divisor monomials have minimal signature, as stated in the followinglemma. Lemma 4.
Let g ∈ R h X i be a polynomial and let m ∈ supp( g ) be such that σ ( m ) = σ ( g ) . If f can be rewritten to h = f + λagb using ( g, m ) , then σ ( f ) ⊆ σ ( h ) and σ ( f ) ⊆ σ ( amb ) . Proof.
By definition of signatures, σ ( f ) ⊆ σ ( amb ). By Lemma 2 and from σ ( m ) = σ ( g ), we have σ ( amb ) ⊆ σ ( agb ). Altogether, σ ( f ) is included in σ ( agb ),which itself is contained in σ ( λagb ). From 1. of Lemma 1, we deduce σ ( f ) ⊆ σ ( h ).Now, we define the rewriting relation induced by a fixed choice of divisormonomials and its compatibility with a quiver. For any rewriting relation wedenote single rewriting steps by → and the reflexive transitive closure by ∗ → . Definition 6.
Let G ⊆ R h X i be a set of polynomials and let DM : G → P ( h X i ) be a function from G to the power set of h X i , such that DM( g ) ⊆ supp( g ) , forevery g ∈ G .1. For g ∈ G , we say that m ∈ DM( g ) is a divisor monomial of g w.r.t. DM .2. We say that f rewrites to h by ( G, DM) , denoted as f → G, DM h , if thereexists g ∈ G and a divisor monomial m ∈ DM( g ) such that f can berewritten to h using ( g, m ) .3. We say that DM is compatible with a labelled quiver Q if for every g ∈ G and every m ∈ DM( g ) , we have σ ( m ) = σ ( g ) . From now on, we fix a set of polynomials G ⊆ R h X i as well as a map DMselecting divisor monomials. Remark 3.
Notice that there exist two extreme cases for the definition of DM :1. DM selects exactly one monomial for each g ∈ G , for instance, the leadingmonomial LM( g ) w.r.t. a monomial order, see the example in Section 6. . All monomials in supp( g ) are divisor monomials. Then, → G, DM coincideswith the rewriting relation introduced in [17, Definition 2], for which idealmembership is equivalent to reduction to zero [17, Lemma 4]. Moreover, ifsuch a DM is compatible with Q , then all polynomials in G are uniformlycompatible, i.e., every monomial of a polynomial has the same signature.The following theorem gives a generalization of Corollary 17 in [17]. Theorem 3.
Let G ⊆ R h X i be a set of polynomials and let DM be a functionselecting divisor monomials as in Definition 6. Let f ∈ R h X i be a polynomialsuch that f ∗ → G, DM . Then, for every labelled quiver Q with labels X such that DM is compatible with Q , we have that f is compatible with Q ⇐⇒ f is a Q -consequence of G. Proof.
Since f rewrites to zero, there exists a sequence f = h → h → · · · → h n = 0. Hence, there exist λ i ∈ R , a i , b i ∈ h X i , g i ∈ G , and m i ∈ DM( g i )such that h i = h i − + λ i a i g i b i and a i m i b i ∈ supp( h i − ). Hence, f can bewritten as f = P ni =1 − λ i a i g i b i . From Lemma 4, we conclude inductively that σ ( f ) ⊆ σ ( h i − ) ⊆ σ ( a i m i b i ). Hence, if f is compatible with Q , then f is a Q -consequence of G by Proposition 1. Conversely, if f is a Q -consequence of G ,then it is compatible by definition. Example 5.
Let us translate Example 3 in the language introduced in thissection. The leading monomials w.r.t. the degree-lexicographic order used in thatexample can be understood as the divisor monomials selected by the function DM defined on F such that DM( f i ) = { LM( f i ) } holds for all i . In particular, DM( f ) = { dh } , DM( f ) = { dh } , DM( f ) = { h ˜ h } , DM( f ) = { h ˜ h } , DM( f ) = { di } . Then, DM is not compatible with Q , since σ ( f ) = { ( v , v ) } is not equal to σ ( dh ) = { ( v , v ) , ( v , v ) } . Hence, we cannot apply Theorem 3 to show that f is a Q -consequence of F even though f ∗ → F, DM . So, we need to look at theexplicit representation of f induced by this reduction, which was already donein Example 3. In order to apply Theorem 3, we need to redefine DM so that itis compatible with Q . In particular, we need to impose DM( f ) ⊆ { h d, b h } .If b h ∈ DM( f ) , then f ∗ → F, DM , which gives another proof that f is a Q -consequence of F based on Theorem 3. Otherwise, if DM( f ) = { h d } , then f isirreducible w.r.t. ∗ → F, DM . Therefore, we need to complete F with Q -consequencesof it such that DM remains compatible with Q and f reduces to zero, which isthe topic of the next section. In this section, we discuss standard noncommutative polynomial reduction asa special case of the rewriting approach from the previous section. Since inthe noncommutative case, Gr¨obner bases are not necessarily finite, see [15], wealso have to work with partial Gr¨obner bases which are obtained by finitely11any iterations of the Buchberger procedure. We adapt the noncommutativeBuchberger procedure for computing (partial) Gr¨obner bases that can be usedfor compatible rewriting.In what follows, R is assumed to be a field K and we fix a monomial order ≤ on h X i , that is, a well-founded total order compatible with multiplication on h X i . We also fix a labelled quiver Q with labels in X and a set of polynomials F ⊆ K h X i . Given a set of polynomials G ⊆ K h X i , one step of the standardpolynomial reduction w.r.t. G is denoted by f → G h .As explained in Remark 3, the monomial order induces the DM function thatselects leading monomials of a set G ⊆ K h X i . This DM function is compatiblewith Q if and only if all elements of G are Q -order compatible in the followingsense. Definition 7.
A compatible polynomial f is said to be Q - order compatible if σ (LM( f )) = σ ( f ) . By transitivity of Q -consequences, see Theorem 1, and Theorem 3, we obtainthe following statement. Corollary 1.
Let F ⊆ K h X i , G ⊆ ( F ) , and f ∈ K h X i such that f ∗ → G . Then,for all labelled quivers Q such that all elements of G are both Q -consequences of F and Q -order compatible, we have f is compatible with Q ⇐⇒ f is a Q -consequence of F. Remark 4.
For polynomials, being Q -order compatible can also be interpretedin terms of a partial monomial order. Given m, m ′ ∈ h X i , we define m ≤ Q m ′ if m ≤ m ′ and σ ( m ′ ) ⊆ σ ( m ) . The partial order ≤ Q respects multiplication ofmonomials since, by Lemma 2, σ ( m ′ ) ⊆ σ ( m ) implies σ ( am ′ b ) ⊆ σ ( amb ) forall a, b ∈ h X i . Then, f is Q -order compatible if and only if supp( f ) admits agreatest element for ≤ Q . Candidates for G as in Corollary 1 are partial Gr¨obner bases that arecomputed by the noncommutative Buchberger procedure [4, 15]. However, inview of the assumptions, we only add reduced S -polynomials that are both Q -consequences of F and Q -order compatible in each iteration. Checking Q -ordercompatibility is easy. Selecting Q -consequences is harder since we do not wantto use explicit representations as in Definition 2. Instead, we propose a simplercriterion based on the following lemma and discussion.First, we recall some terminology and fix notations for S -polynomials. Let G ⊆ K h X i . Ambiguities of G defined in [1], also called compositions in [2], aregiven by minimal overlaps or inclusions of the two leading monomials LM( g ) andLM( g ′ ), where g and g ′ belong to G . Formally, each ambiguity can be describedby a 6-tuple a = ( g, g ′ , a, b, a ′ , b ′ ), where a, b, a ′ , b ′ are monomials such that,among other conditions, we have a LM( g ) b = a ′ LM( g ′ ) b ′ . This monomial is called the source of a and the S -polynomial of a is SP( a ) := agb − a ′ g ′ b ′ , cf. [15]. Lemma 5.
Let G ⊆ K h X i be a set of Q -order compatible polynomials and let s be a S -polynomial of G with source a compatible monomial m ∈ h X i . Then ( m ) ⊆ σ ( s ) . If moreover, s ∗ → G ˆ s with σ (ˆ s ) ⊆ σ ( m ) , then σ ( s ) = σ (ˆ s ) = σ ( m ) and ˆ s is a Q -consequence of G .Proof. Since s is a S -polynomial of G of source m , there exist g, g ′ ∈ G andmonomials a, a ′ , b, b ′ ∈ h X i such that s = ( m − agb ) − ( m − a ′ g ′ b ′ ) with a LM( g ) b = a ′ LM( g ′ ) b ′ = m .Let us prove the first assertion. The polynomials g and g ′ being Q -ordercompatible, we have σ (LM( g )) = σ ( g ) and σ (LM( g ′ )) = σ ( g ′ ). Hence, fromLemma 2, we have σ ( m ) = σ ( a LM( g ) b ) ⊆ σ ( agb ) ,σ ( m ) = σ ( a ′ LM( g ′ ) b ′ ) ⊆ σ ( a ′ g ′ b ′ ) . (7)From this and s = a ′ g ′ b ′ − agb , we get that σ ( m ) ⊆ σ ( s ).Now, we assume that s ∗ → G ˆ s , that is there is a rewriting sequence s = s → G s → G · · · → G s n = ˆ s, so that there exist g i ∈ G , a i , b i ∈ h X i and λ i ∈ K , 1 ≤ i ≤ n , such that a i LM( g i ) b i ∈ supp( s i ) and s i +1 = s i + λ i a i g i b i , so that we have ˆ s − s = P i λ i a i g i b i and ˆ s = n X i =1 λ i a i g i b i + a ′ g ′ b ′ − agb. (8)Using inductively s i → G s i +1 and Lemma 4, we get σ ( s ) ⊆ σ (ˆ s ) and σ ( s ) ⊆ σ ( a i LM( g i ) b i ) , (9)so that we have σ ( m ) ⊆ σ ( s ) ⊆ σ (ˆ s ) . If, moreover σ (ˆ s ) ⊆ σ ( m ), then we get the following sequence of inclusions: σ (ˆ s ) ⊆ σ ( m ) ⊆ σ ( s ) ⊆ σ (ˆ s ) . Hence, the equality σ (ˆ s ) = σ ( s ) = σ ( m ) holds. Now, we show that ˆ s isa Q -consequence using Proposition 1. Since the elements of G are Q -ordercompatible, we have σ (LM(˜ g )) = σ (˜ g ), for all ˜ g ∈ G . Moreover, since m iscompatible and σ (ˆ s ) = σ ( m ), ˆ s is compatible. Finally, from (7) and (9), wehave the following inclusions: σ (ˆ s ) ⊆ σ ( a LM( g ) b ), σ (ˆ s ) ⊆ σ ( a ′ LM( g ′ ) b ′ ) and σ (ˆ s ) ⊆ σ ( a i LM( g i ) b i ). As a conclusion, ˆ s is a Q -consequence of G .Starting with a set of Q -order compatible polynomials F , we apply thislemma in the case where G is the partial Gr¨obner basis computed in the currentiteration of the completion procedure. In particular, if a reduced S -polynomialˆ s satisfies σ (ˆ s ) ⊆ σ ( m ) as in the lemma, then it is a Q -consequence of G .By transitivity, it is then also a Q -consequence of F , which follows from thefollowing observation. Remark 5.
Consider a set F ⊆ K h X i of compatible polynomials and a familyof sets G i inductively defined by G = ∅ and G i +1 = G i ∪ { g i +1 } , where g i +1 is a Q -consequence of F ∪ G i . Using inductively transitivity of Q -consequencesproven in Theorem 1, we obtain that, for each i , all elements of F ∪ G i are Q -consequences of F .
13n summary, we obtain the following adaptation of the noncommutative ver-sion of Buchberger’s procedure for computing a partial Gr¨obner basis composedof elements that are both Q -consequences of F and Q -order compatible. At eachstep, we select an S -polynomial s whose source m is a compatible monomial,and we keep a reduced form ˆ s only if it is Q -order compatible and σ (ˆ s ) ⊆ σ ( m ).This procedure is implemented in the Mathematica package
OperatorGB . Notethat since the Buchberger procedure does not terminate in general for noncom-mutative polynomials, also our adaptation of it is not guaranteed to terminate.Notice that the completion procedure described above can be slightly gen-eralized by not necessarily computing reduced forms of S -polynomials. Instead,we only reduce an S -polynomial as long as it remains a Q -consequence, see thediscussion above, and it remains Q -order compatible. This is stated formally inProcedure 1. Procedure 1 Q -order compatible completion Input: F ⊆ K h X i , a labelled quiver Q with labels in X , and a monomial order ≤ such that every f ∈ F is Q -order compatible Output: G ⊇ F a set of Q -consequences of F that are Q -order compatible P := ambiguities of F ; G := F while P = ∅ do choose a ∈ P P := P \ { a } ; s := SP( a ); m := the source of a if m is compatible and σ ( s ) ⊆ σ ( m ) and s is Q -order compatible then while ∃ s ′ : s → G s ′ do if s ′ = 0 then go to if statement) else if σ ( s ′ ) ⊆ σ ( m ) and s ′ is Q -order compatible then s := s ′ else go to
15 (i.e., break the inner while loop) end if end while G := G ∪ { s } P := P ∪ { ambiguities created by s } end if end while return G Due to the checks in line 9, each element g of the output G of the procedureis both a Q -consequence of F and Q -order compatible. In summary, we haveshown that our procedure is correct. Theorem 4.
Let F ⊆ K h X i be a set of polynomials, let Q be a labelled quiverwith labels in X and let ≤ be a monomial order such that each element of F is Q -order compatible. Then, each element of the output G of Procedure 1 is botha Q -consequence of F and Q -order compatible. Example 6.
Let us continue Example 5 in the case
DM( f ) = { h d } . Forthat, we consider the field K = Q and a degree-lexicographic order such that d < b < h and d is greater than b and h . Then, choosing the ambiguity f , f , , i, h , , the first iteration of the outer loop in Procedure 1 yields G := F ∪ { b h i − dh i + h } . With this G , we have f ∗ → G . From this reduction to , and since f is compatible with Q , f is a Q -consequence of F by Corollary 1.These computations can also be done by the package. Remark 6.
We consider the special case when all edges of Q have unique labels.Then, all non-constant monomials have at most one element in their signature.Therefore, every compatible polynomial is Q -order compatible for any monomialorder, since the monomial is the smallest. Moreover, one can show easilythat the source of an ambiguity of two polynomials is compatible whenever thesetwo polynomials are compatible with Q . In addition, from Lemma 4, it followsthat polynomial reduction of compatible polynomials by compatible ones doesnot change the signature unless the result of the reduction lies in K . Altogether,Procedure 1 reduces to the standard Buchberger procedure (i.e., without checkingsignatures and compatibility during computation) as long as no S -polynomial is(or is reduced to) a nonzero constant. In other words, we have the followingtheorem, which, together with Theorem 2, gives a generalization of Theorem 1in [17]. Theorem 5.
Assume that edges of Q have unique labels. Let F ⊆ K h X i be aset of compatible polynomials and let G be a (partial) Gr¨obner basis computed bythe standard Buchberger procedure (i.e., disregarding Q during computation). If G does not contain a constant polynomial, then for every polynomial f ∈ K h X i such that f ∗ → G , we have f is compatible with Q ⇐⇒ f is a Q -consequence of F. Moreover, if ( F ) , then this equivalence holds for every f ∈ ( F ) . By Theorem 2, for proving new operator identities from known ones, it suffices toshow that the corresponding polynomials are Q -consequences. In practice, thereare several options to prove that a compatible polynomial f is a Q -consequenceof some set F of compatible polynomials. Each of these options can be turnedinto a certificate that f is a Q -consequence of F . Given an explicit representationof f in terms of F of the form (3), one can either check Definition 2 directly, orexpand cofactors into monomials and apply Proposition 1. Alternatively, usingcompatible rewriting, if f ∗ → F, DM Q , then f is a Q -consequence by Theorem 3. Altogether,from Theorems 2 and 3, we immediately obtain the following. Corollary 2.
Let F be a set of polynomials, DM a function selecting divisormonomials, and f a polynomial such that f ∗ → F, DM . Then, for all labelledquivers Q such that f , F , and DM are compatible with Q and for all consistentrepresentations of Q such that all realizations of all elements of F are zero, allrealizations of f are zero. Note that rewriting to zero w.r.t. F and DM is independent of the quiver Q .In particular, if the above corollary is interpreted in terms of R -linear categories,the main result of [17], Theorem 32, is obtained as a special case by Remark 3.15ore generally, if one cannot verify that f can be rewritten to zero by F ,there still might exist a set G of Q -consequences of F with divisor monomialsselected by some DM such that f ∗ → G, DM G such that Corollary 1 can be used to prove that f is a Q -consequenceof F by standard polynomial reduction.Notice that Procedure 1 can be extended in various directions. For ex-ample, in order to systematically generate more Q -consequences, reduced S -polynomials that are not Q -order compatible could be collected in a separateset, which should not be used for constructing and reducing new S -polynomials.Instead of fixing a monomial ordering from the beginning, one might start with apartial ordering that is then extended during the completion procedure in orderto make obtained S -polynomials Q -order compatible. More generally, withoutany partial ordering on monomials, one might even consider compatible func-tions DM which not necessarily select only one divisor monomial per polynomialand aim at completing the induced rewriting relation. However, termination ofsuch rewriting relations is an issue. Finally, another topic for future researchis to generalize the results of this paper to tensor reduction systems used formodelling linear operators as described in [12]. References [1] George M. Bergman. The diamond lemma for ring theory.
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