aa r X i v : . [ m a t h . R A ] J u l Note on (De)homogenized Gr¨obner Bases ∗ Huishi Li † Department of Applied MathematicsCollege of Information Science and TechnologyHainan UniversityHaikou 570228, China
Abstract.
By employing the (de)homogenization technique in a relativelyextensive setting, this note studies in detail the relation between non-homogeneous Gr¨obner bases and homogeneous Gr¨obner bases. As a conse-quence, a general principle of computing Gr¨obner bases (for an ideal and itshomogenization ideal) by passing to homogenized generators is clarified sys-tematically. The obtained results improve and strengthen the work of [LWZ],[Li1], [Li2], [Li3], and very recent [SL] concerning the same topic.
Primary 16W70; Secondary 68W30 (16Z05).
Key words
Graded algebra, graded ideal, Gr¨obner basis, homogenization, dehomogenizationIn the computational Gr¨obner basis theory, though it is a well-known fact that by virtue of boththe structural advantage (mainly the degree-truncated structure) and the computational advan-tage (mainly the degree-preserving fast ordering), most of the popularly used commutative andnoncommutative Gr¨obner basis algorithms produce Gr¨obner bases by homogenizing generatorsfirst, it seems that in both the commutative and noncommutative case a general principle ofcomputing Gr¨obner bases (for an ideal and its homogenization ideal) by passing to homogenizedgenerators is still missing.Let K be a field, and let N be the additive monoid of nonnegative integers. Recall from [Li2]that if R = ⊕ p ∈ N R p is an N -graded K -algebra with an admissible system ( B , ≺ ), in which B is a skew multiplicative K -basis of R consisting of N - homogeneous elements (i.e., u, v ∈ B implies that u , v are homogeneous elements, uv = 0 or uv = λw for some nonzero λ ∈ K and w ∈ B ), and ≺ isa monomial ordering on B , then, theoretically every (two-sided) ideal I of R has a Gr¨obner basis G in the sense that if f ∈ I and f = 0 then there is some g ∈ G such that LM ( g ) | LM ( f ), where ∗ Project supported by the National Natural Science Foundation of China (10571038). † e-mail: [email protected] M ( ) denotes taking the leading monomial of elements in R , in particular, every graded ideal of R has a homogeneous Gr¨obner basis , i.e., a Gr¨obner basis consisting of N -homogeneous elements.Typical examples of such algebras include commutative polynomial K -algebra, noncommutativefree K -algebra, path algebra over K , the coordinate algebra of a quantum affine n -space over K , and exterior K -algebra (cf. [Bu], [BW], [Gr], [HT], [KRW], [Mor]). In this note, we employthe (de)homogenization technique to study in detail the relation between Gr¨obner bases in R and homogeneous Gr¨obner bases in the polynomial ring R [ t ], respectively the relation betweenGr¨obner bases in the free algebra K h X i = K h X , ..., X n i and homogeneous Gr¨obner bases inthe free algebra K h X, T i = K h X , ..., X n , T i . As a consequence, this makes a solid theoreticalfoundation for us to demonstrate, by passing to the graded ideal h S ∗ i , how to obtain a Gr¨obnerbasis for the ideal I = h S i generated by a subset S ⊂ R and hence a homogeneous Gr¨obner basisfor the central homogenization ideal h I ∗ i of I in R [ t ] with respect to t ; respectively, this enablesus to demonstrate, by passing to the graded ideal h e S i , how to obtain a Gr¨obner basis for theideal I = h S i generated by a subset S ⊂ K h X i and hence a homogeneous Gr¨obner basis for thenon-central homogenization ideal h e I i of I in K h X, T i with respect to T . The obtained resultsimprove and strengthen the work of [LWZ], [Li1], [Li3], [Li3], and very recent [SL] concerningthe same topic.Algebras considered in this paper are associative algebras with multiplicative identity 1.Unless otherwise stated, ideals considered are meant two-sided ideals. If S is a nonempty subsetof an algebra, then we use h S i to denote the two-sided ideal generated by S . Moreover, if K isa field, then we write K ∗ = K − { } .
1. Central (De)homogenized Gr¨obner Bases
Let R = ⊕ p ∈ N R p be an arbitrary N -graded K -algebra, and let R [ t ] be the polynomial ring in thecommuting variable t over R . Then R [ t ] has the mixed N -gradation, that is, R [ t ] = ⊕ p ∈ N R [ t ] p is an N -graded algebra with the degree- p homogeneous part R [ t ] p = X i + j = p F i t j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F i ∈ R i , j ≥ , p ∈ N . Considering the onto ring homomorphism φ : R [ t ] → R defined by φ ( t ) = 1, then for each f ∈ R ,there exists a homogeneous element F ∈ R [ t ] p , for some p , such that φ ( F ) = f . More precisely,if f = f p + f p − + · · · + f p − s with f p ∈ R p , f p − j ∈ R p − j and f p = 0, then f ∗ = f p + tf p − + · · · + t s f p − s is a homogeneous element of degree p in R [ t ] p satisfying φ ( f ∗ ) = f . We call the homogeneouselement f ∗ obtained this way the central homogenization of f with respect to t (for the reason2hat t is in the center of R [ t ]). On the other hand, for an element F ∈ R [ t ], we write F ∗ = φ ( F )and call it the central dehomogenization of F with respect to t (again for the reason that t isin the center of R [ t ]). Hence, if I is an ideal R , then we write I ∗ = { f ∗ | f ∈ I } and call the N -graded ideal h I ∗ i generated by I ∗ the central homogenization ideal of I in R [ t ] with respect to t ; and if J is an ideal of R [ t ], then since φ is a ring epimorphism, φ ( J ) is an ideal of R , so we write J ∗ for φ ( J ) = { H ∗ = φ ( H ) | H ∈ J } and call it the central dehomogenization ideal of J in R withrespect to t . Consequently, henceforth we will also use the notation ( J ∗ ) ∗ = { ( h ∗ ) ∗ | h ∈ J } .Since each f ∈ R has a unique decomposition by N -homogeneous elements, if if f = f p + f p − + · · · + f p − s with f p ∈ R p , f p − j ∈ R p − j and f p = 0, then we call f p the N -leadinghomogeneous element of f , denoted LH ( f ) = f p . Similarly, for an element F ∈ R [ t ], the N -leading homogeneous element LH ( F ) is defined with respect to the mixed N -gradation of R [ t ]. With every definition and notation made above, the following statements hold.(i) For
F, G ∈ R [ t ], ( F + G ) ∗ = F ∗ + G ∗ , ( F G ) ∗ = F ∗ G ∗ .(ii) For any f ∈ R , ( f ∗ ) ∗ = f .(iii) If F ∈ R [ t ] p and if ( F ∗ ) ∗ ∈ R [ t ] q , then p ≥ q and t r ( F ∗ ) ∗ = F with r = p − q .(iv) If f, g ∈ R are such that f g has nonzero LH ( f g ) ∈ R m and f ∗ g ∗ has nonzero LH ( f ∗ g ∗ ) ∈ R [ t ] q , then f ∗ g ∗ = t k ( f g ) ∗ with k = q − m .(v) If f, g ∈ R with nonzero LH ( f ) ∈ R p and nonzero LH ( g ) ∈ R q , then ( f + g ) ∗ = f ∗ + g ∗ incase p = q , and ( f + g ) ∗ = f ∗ + t ℓ g ∗ in case p > q , where ℓ = p − q .(vi) If I is a two-sided ideal of R , then each homogeneous element F ∈ h I ∗ i is of the form t r f ∗ for some r ∈ N and f ∈ I .(vii) If J is a graded ideal of R [ t ], then for each h ∈ J ∗ there is some homogeneous element F ∈ J such that F ∗ = h . Proof
Exercise. (cid:3)
Suppose that the N -graded K -algebra R = ⊕ p ∈ N R p has an admissible system ( B , ≺ gr ), where B is a skew multiplicative K -basis of R consisting of N - homogeneous elements , and ≺ gr is an N -graded monomial ordering on B , i.e., R has a Gr¨obner basis theory. Consider the mixed N -gradation of R [ t ] and the K -basis B ∗ = { t r w | w ∈ B , r ∈ N } of R [ t ]. Since B ∗ is obviouslya skew multiplicative K -basis for R [ t ], the N -graded monomial ordering ≺ gr on B extends to amonomial ordering on B ∗ , denoted ≺ t - gr , as follows: t r w ≺ t - gr t r w if and only if w ≺ gr w , or w = w and r < r . Thus R [ t ] holds a Gr¨obner basis theory with respect to the admissible system ( B ∗ , ≺ t - gr ).3s usual we call elements in B and B ∗ monomials and use LM ( ) to denote taking the leadingmonomial of elements with respect to the given monomial ordering.It follows from the definition of ≺ t - gr that t r ≺ t - gr w for all integers r > w ∈ B − { } (if B contains the identity element 1 of R ). Hence ≺ t - gr is not a graded monomial ordering on B ∗ . But noticing that ≺ gr is an N -graded monomial ordering on B , under taking the N -leading homogeneous element and central (de)homogenization the leading monomials behaveharmonically as described in the next lemma. With notation given above, the following statements hold.(i) If f ∈ R , then LM ( f ) = LM ( LH ( f )) w.r.t. ≺ gr on B . (ii) If f ∈ R , then LM ( f ∗ ) = LM ( f ) w.r.t. ≺ t - gr on B ∗ . (iii) If F is a nonzero homogeneous element of R [ t ], then LM ( F ∗ ) = LM ( F ) ∗ w.r.t. ≺ gr on B . Proof
The proof of (i) and (ii) is an easy exercise. To prove (iii), let F ∈ R [ t ] p be a nonzerohomogeneous element of degree p , say F = λt r w + λ t r w + · · · + λ s t r s w s , where λ, λ i , ∈ K ∗ , r, r i ∈ N , w, w i ∈ B , such that LM ( F ) = t r w . Since B consists of N -homogeneous elements and R [ t ] has the mixed N -gradation by the previously fixed assumption,we have d ( t r w ) = d ( t r i w i ) = p , 1 ≤ i ≤ s . Thus w = w i will imply r = r i and thereby t r w = t r i w i . So we may assume that w = w i , 1 ≤ i ≤ s . Then it follows from the definition of ≺ t - gr that w i ≺ gr w and r ≤ r i , 1 ≤ i ≤ s . Therefore LM ( F ∗ ) = w = LM ( F ) ∗ , as desired. (cid:3) The next result is a generalization of ([LWZ] Theorem 2.3.2).
With notions and notations as fixed before, let I = hGi be the ideal of R generated by a subset G , and h I ∗ i the central homogeneization ideal of I in R [ t ] with respect to t . The following two statements are equivalent.(i) G is a Gr¨obner basis for I in R with respect to the admissible system ( B , ≺ gr );(ii) G ∗ = { g ∗ | g ∈ G} is a Gr¨obner basis for h I ∗ i in R [ t ] with respect to the admissible system( B ∗ , ≺ t - gr ). Proof
In proving the equivalence below, without specific indication we shall use (i) and (ii) ofLemma 1.2 wherever it is needed.(i) ⇒ (ii) First note that LM ( G ∗ ) ⊂ LM ( I ∗ ). We have to prove that LM ( G ∗ ) generates h LM ( I ∗ ) i in order to see that G ∗ is a Gr¨obner basis for h I ∗ i . If F ∈ h I ∗ i , then since LM ( F ) =4 M ( LH ( F )), we may assume, without loss of generality, that F is a homogeneous element. So,by Lemma 1.1(vi) we have F = t r f ∗ for some f ∈ I . It follows from the equality LM ( f ∗ ) = LM ( f ) that LM ( F ) = t r LM ( f ∗ ) = t r LM ( f ) . Since G is a Gr¨obner basis for I , LM ( f ) = λv LM ( g i ) w for some λ ∈ K ∗ , g i ∈ G , and v, w ∈ B .Thus, LM ( F ) = t r LM ( f ) = λt r v LM ( g ∗ i ) w ∈ h LM ( G ∗ ) i . This shows that h LM ( h I ∗ i ) i = h LM ( G ∗ ) i , as desired.(ii) ⇒ (i) Suppose G ∗ is a Gr¨obner basis for the homogenization ideal h I ∗ i of I in R [ t ]. Let f ∈ I .Then LM ( f ∗ ) = λv LM ( g ∗ i ) w for some λ ∈ K ∗ , v, w ∈ B ∗ and g ∗ i ∈ G ∗ . Since LM ( f ) = LM ( f ∗ ),it follows that LM ( f ) = λv ∗ LM ( g i ) w ∗ ∈ h LM ( G ) i . This shows that h LM ( I ) i = h LM ( G ) i , i.e., G is a Gr¨obner basis for I in R . (cid:3) We call the Gr¨obner basis G ∗ obtained in Theorem 1.3 the central homogenization of G in R [ t ] with respect to t , or G ∗ is a central homogenized Gr¨obner basis with respect to t .By Lemma 1.2 and Theorem 1.3, we have immediately the following corollary. Let I be an arbitrary ideal of R . With notation as before, if G is a Gr¨obnerbasis of I with respect to the data ( B , ≺ gr ), then, with respect to the data ( B ∗ , ≺ t - gr ) we have B ∗ − h LM ( G ∗ ) i = { t r w | w ∈ B − h LM ( G ) i , r ∈ N } , that is, the set N ( h I ∗ i ) of normal monomials (mod h I ∗ i ) in B ∗ is determined by the set N ( I ) ofnormal monomials (mod I ) in B . Hence, the algebra R [ t ] / h I ∗ i = R [ t ] / hG ∗ i has the K -basis N ( h I ∗ i ) = { t r w | w ∈ N ( I ) , r ∈ N } . (cid:3) Theoretically we may also obtain a Gr¨obner basis for an ideal I of R by dehomogenizing ahomogeneous Gr¨obner basis of the ideal h I ∗ i ⊂ R [ t ]. Below we give a more general approach tothis assertion. Let J be a graded ideal of R [ t ]. If G is a homogeneous Gr¨obner basis of J withrespect to the data ( B ∗ , ≺ t - gr ), then G ∗ = { G ∗ | G ∈ G } is a Gr¨obner basis for the ideal J ∗ in R with respect to the data ( B , ≺ gr ). Proof If G is a Gr¨obner basis of J , then G generates J and hence G ∗ = φ ( G ) generates J ∗ = φ ( J ).For a nonzero f ∈ J ∗ , by Lemma 1.1(vii), there exists a homogeneous element H ∈ J such that5 ∗ = f . It follows from Lemma 1.2 that(1) LM ( f ) = LM ( f ∗ ) = LM (( H ∗ ) ∗ ) . On the other hand, there exists some G ∈ G such that LM ( G ) | LM ( H ), i.e.,(2) LM ( H ) = λt r w LM ( G ) t r v for some λ ∈ K ∗ , r , r ∈ N , w, v ∈ B . But by Lemma 1.1(iii) we also have t r ( H ∗ ) ∗ = H forsome r ∈ N , and hence(3) LM ( H ) = LM ( t r ( H ∗ ) ∗ ) = t r LM ( H ∗ ) ∗ ) . So, (1) + (2) + (3) yields λt r + r w LM ( G ) v = LM ( H )= t r LM (( H ∗ ) ∗ )= t r LM ( f ) . Taking the central dehomogenization for the above equality, by Lemma 1.2(iii) we obtain λw LM ( G ∗ ) v = λw LM ( G ) ∗ v = LM ( f ) . This shows that LM ( G ∗ ) | LM ( f ). Therefore, G ∗ is a Gr¨obner basis for J ∗ . (cid:3) We call the Gr¨obner basis G ∗ obtained in Theorem 1.5 the central dehomogenization of G in R with respect to t , or G ∗ is a central dehomogenized Gr¨obner basis with respect to t . Let I be an ideal of R . If G is a homogeneous Gr¨obner basis of h I ∗ i in R [ t ]with respect to the data ( B ∗ , ≺ t - gr ), then G ∗ = { g ∗ | g ∈ G } is a Gr¨obner basis for I in R withrespect to the data ( B, ≺ gr ). Moreover, if I is generated by the subset F and F ∗ ⊂ G , then F ⊂ G ∗ . Proof
Put J = h I ∗ i . Then since J ∗ = I , it follows from Theorem 1.5 that if G is a homogeneousGr¨obner basis of J then G ∗ is a Gr¨obner basis for I . The second assertion of the theorem is clearby Lemma 1.1(ii). (cid:3) Let S be a nonempty subset of R and I = h S i the ideal generated by S . Then, with S ∗ = { f ∗ | f ∈ S } , in general h S ∗ i ( h I ∗ i in R [ t ] (for instance, consider S = { y − x − y, y + 1 } in the commutative polynomial ring K [ x, y ] and the homogenization in K [ x, y, t ] with respectto t ). So, from both a practical and a computational viewpoint, it is the right place to set upthe procedure of getting a Gr¨obner basis for I and hence a Gr¨obner basis for h I ∗ i by producinga homogeneous Gr¨obner basis of the graded ideal h S ∗ i .6 .7. Proposition Let I = h S i be the ideal of R as fixed above. Suppose that Gr¨obner basesare algorithmically computable in R and hence in R [ t ]. Then a Gr¨obner basis for I and ahomogeneous Gr¨obner basis for h I ∗ i may be obtained by implementing the following procedure: Step 1.
Starting with the initial subset S ∗ = { f ∗ | f ∈ S } , compute a homogeneous Gr¨obnerbasis G for the graded ideal h S ∗ i of R [ t ]. Step 2.
Noticing h S ∗ i ∗ = I , use Theorem 1.5 and dehomogenize G with respect to t in orderto obtain the Gr¨obner basis G ∗ for I . Step 3.
Use Theorem 1.3 and homogenize G ∗ with respect to t in order to obtain the homoge-neous Gr¨obner basis ( G ∗ ) ∗ for the graded ideal h I ∗ i . (cid:3)
2. Non-central (De)homogenized Gr¨obner Bases
In a similar way, we proceed now to consider the free algebra K h X i = K h X , ..., X n i of n genera-tors as well as the free algebra K h X, T i = K h X , ..., X n , T i of n + 1 generators, and demonstratehow Gr¨obner bases in K h X i are related with homogeneous Gr¨obner bases in K h X, T i if the non-central (de)homogenization with respect to T is employed.Let K h X i be equipped with a fixed weight N -gradation , say each X i has degree n i > ≤ i ≤ n . Assigning to T the degree 1 in K h X, T i and using the same weight n i for each X i as in K h X i , we get the weight N -gradation of K h X, T i which extends the weight N -gradationof K h X i . Let B and e B denote the standard K -bases of K h X i and K h X, T i respectively. Tobe convenient we use lowercase letters w, u, v, ... to denote monomials in B as before, but usecapitals W, U, V, ... to denote monomials in e B .In what follows, we fix an admissible system ( B , ≺ gr ) for K h X i , where ≺ gr is an N - gradedlexicographic ordering on B with respect to the fixed weight N -gradation of K h X i , such that X i ≺ gr X i ≺ gr · · · ≺ gr X i n . Then it is not difficult to see that ≺ gr can be extended to an N - graded lexicographic ordering ≺ T - gr on e B with respect to the fixed weight N -gradation of K h X, T i , such that T ≺ T - gr X i ≺ T - gr X i ≺ T - gr · · · ≺ T - gr X i n , and thus we get the admissible system ( e B , ≺ T - gr ) for K h X, T i . With respect to ≺ gr and ≺ T - gr weuse LM ( ) to denote taking the leading monomial of elements in K h X i and K h X, T i respectively.Consider the fixed N -graded structures K h X i = ⊕ p ∈ N K h X i p , K h X, T i = ⊕ p ∈ N K h X, T i p ,and the ring epimorphism ψ : K h X, T i −→ K h X i ψ ( X i ) = X i and ψ ( T ) = 1. Then each f ∈ K h X i is the image of some homogeneouselement in K h X, T i . More precisely, if f = f p + f p − + · · · + f p − s with f p ∈ K h X i p , f p − j ∈ K h X i p − j and f p = 0, then e f = f p + T f p − + · · · + T s f p − s is a homogeneous element of degree p in K h X, T i p such that ψ ( e f ) = f . We call the homogeneouselement e f obtained this way the non-central homogenization of f with respect to T (for the reasonthat T is not a commuting variable). On the other hand, for F ∈ K h X, T i , we write F ∼ = ψ ( F )and call F ∼ the non-central dehomogenization of F with respect to T (again for the reason that T is not a commuting variable). Furthermore, if I = h S i is the ideal of K h X i generated by asubset S , then we write e S = { e f | f ∈ S } ∪ { X i T − T X i | ≤ i ≤ n } , e I = { e f | f ∈ I } ∪ { X i T − T X i | ≤ i ≤ n } , and call the graded ideal h e I i generated by e I the non-central homogenization ideal of I in K h X, T i with respect to T ; while if J is an ideal of K h X, T i , then since ψ is a surjective ringhomomorphism, ψ ( J ) is an ideal of K h X i , so we write J ∼ for ψ ( J ) = { H ∼ | H ∈ J } and call it the non-central dehomogenization ideal of J in K h X i with respect to T . Consequently, henceforthwe will also use the notation( J ∼ ) ∼ = { ( h ∼ ) ∼ | h ∈ J } ∪ { X i T − T X i | ≤ i ≤ n } . It is straightforward to check that with resspect to the data ( e B , ≺ T - gr ), the subset { X i T − T X i | ≤ i ≤ n } of K h X, T i forms a homogeneous Gr¨obner basis with LM ( X i T − T X i ) = X i T ,1 ≤ i ≤ n . In the latter discussion we will freely use this fact without extra indication. With notation as fixed before, the following properties hold.(i) If
F, G ∈ K h X, T i , then ( F + G ) ∼ = F ∼ + G ∼ , ( F G ) ∼ = F ∼ G ∼ .(ii) For each nonzero f ∈ K h X i , ( e f ) ∼ = f .(iii) Let C be the graded ideal of K h X, T i generated by { X i T − T X i | ≤ i ≤ n } . If F ∈ K h X, T i p , then there exists an L ∈ C and a unique homogeneous element of the form H = P i λ i T r i w i , where λ i ∈ K ∗ , w i ∈ B , such that F = L + H ; moreover there is some r ∈ N suchthat T r ( H ∼ ) ∼ = H , and hence F = L + T r ( F ∼ ) ∼ .(iv) Let C be as in (iii) above. If I is an ideal of K h X i , F ∈ h e I i is a homogeneous element,then there exist some L ∈ C , f ∈ I and r ∈ N such that F = L + T r e f .(v) If J is a graded ideal of K h X, T i and { X i T − T X i | ≤ i ≤ n } ⊂ J , then for each nonzero h ∈ J ∼ , there exists a homogeneous element H = P i λ i T r i w i ∈ J , where λ i ∈ K ∗ , r i ∈ N , and w i ∈ B , such that for some r ∈ N , T r ( H ∼ ) ∼ = H and H ∼ = h .8 roof (i) and (ii) follow from the definitions of non-central homogenization and non-centraldehomogenization directly.(iii) Since the subset { X i T − T X i | ≤ i ≤ n } is a Gr¨obner basis in K h X, T i with respect to( e B , ≺ T - gr ), such that LM ( X i T − T X i ) = X i T , 1 ≤ i ≤ n , if F ∈ K h X, T i p , then the division of F by this subset yields F = L + H , where L ∈ C , and H = P i λ i T r i w i is the unique remainderwith λ i ∈ K ∗ , w i ∈ B , in which each monomial T r w i is of degree p . By the definition of ≺ T - gr , thedefinitions of non-central homogenization and the definition of non-central dehomogenization,it is not difficult to see that H has the desired property.(iv) By (iii), F = L + T r ( F ∼ ) ∼ with L ∈ C and r ∈ N . Since by (ii) we have F ∼ ∈ h e I i ∼ = I ,thus f = F ∼ is the desired element.(v) Using basic properties of homogeneous element and graded ideal in a graded ring, this followsfrom the foregoing (iii). (cid:3) As in the case using central (de)homogenization, before turning to deal with Gr¨obner bases,we are also concerned about the behavior of leading monomials under taking the N -leadinghomogeneous element and non-central (de)homogenization. Below we use LH ( ) to denotetaking the N -leading homogeneous element (i.e., the highest-degree homogeneous component)of elements in both K h X i and K h X, T i with respect to the fixed N -gradation. With the assumptions and notations as fixed above, the following statementshold.(i) If f ∈ K h X i , then LM ( f ) = LM ( LH ( f )) w.r.t. ≺ gr on B ;If F ∈ K h X, T i , then LM ( F ) = LM ( LH ( F )) w.r.t. ≺ T - gr on e B . (ii) For each nonzero f ∈ K h X i , we have LM ( f ) = LM ( e f ) w.r.t. ≺ T - gr on e B . (iii) If F is a homogeneous element in K h X, T i such that X i T LM ( F ) with respect to ≺ T - gr for all 1 ≤ i ≤ n , then LM ( F ) = T r w for some r ∈ N and w ∈ B , such that LM ( F ∼ ) = w = LM ( F ) ∼ w.r.t. ≺ gr on B . Proof
The proof of (i) and (ii) is an easy exercise. To prove (iii), let F ∈ K h X, T i p be a nonzerohomogeneous element of degree p . Then by the assumption F may be written as F = λT r w + λ T r X j W + λ T r X j W · · · + λ s T r s X j s W s , where λ, λ i , ∈ K ∗ , r, r i ∈ N , w ∈ B and W i ∈ e B , such that LM ( F ) = T r w . Since B consistsof N -homogeneous elements and the N -gradation of K h X i extends to give the N -gradation of9 h X, T i , we have d ( T r w ) = d ( T r i X j i W i ) = p , 1 ≤ i ≤ s . Also note that T has degree 1.Thus w = X j i W i will imply r = r i and thereby T r w = T r i X j i W i . So we may assume that w = X j i W i , 1 ≤ i ≤ s . Then it follows from the definition of ≺ T - gr that r ≤ r i , 1 ≤ i ≤ n .Hence X j i W i ≺ T - gr w , 1 ≤ i ≤ s . Therefore ( X j i W i ) ∼ ≺ gr w , 1 ≤ i ≤ n , and consequently LM ( F ∼ ) = w = LM ( F ) ∼ , as desired. (cid:3) The next result strengthens ([Li2], Theorem 8.2), in particular, the proof of (i) ⇒ (ii) givenbelow improves the argument given in loc. cit. With the notions and notations as fixed above, let I = hGi be the ideal of K h X i generated by a subset G , and h e I i the non-central homogenization ideal of I in K h X, T i withrespect to T . The following two statements are equivalent.(i) G is a Gr¨obner basis of I with respect to the admissible system ( B , ≺ gr ) of K h X i ;(ii) e G = { e g | g ∈ G} ∪ { X i T − T X i | ≤ i ≤ n } is a homogeneous Gr¨obner basis for h e I i withrespect to the admissible system ( e B , ≺ T - gr ) of K h X, T i . Proof
In proving the equivalence below, without specific indication we shall use (i) and (ii) ofLemma 2.2 wherever it is needed.(i) ⇒ (ii) Suppose that G is a Gr¨obner basis for I with respect to the data ( B , ≺ gr ). Let F ∈ h e I i .Then since ≺ T - gr is a graded monomial ordering and hence LM ( F ) = LM ( LH ( F )), we mayassume that F is a nonzero homogeneous element. We want to show that there is some D ∈ e G such that LM ( D ) | LM ( F ), and hence e G is a Gr¨obner basis.Note that { X i T − T X i | ≤ i ≤ n } ⊂ e G with LM ( X i T − T X i ) = X i T . If X i T | LM ( F ) forsome X i T , then we are done. Otherwise, X i T LM ( F ) for all 1 ≤ i ≤ n . Thus, by Lemma2.2(iii), LM ( F ) = T r w for some r ∈ N and w ∈ B and(1) LM ( F ∼ ) = w = LM ( F ) ∼ . On the other hand, by Lemma 2.1(iv) we have F = L + T q e f , where L is an element in the ideal C generated by { X i T − T X i | ≤ i ≤ n } in K h X, T i , q ∈ N , and f ∈ I . It turns out that(2) F ∼ = ( e f ) ∼ = f and hence LM ( F ∼ ) = LM ( f ) . Since G is a Gr¨obner basis for I , there is some g ∈ G such that LM ( g ) | LM ( f ), i.e., there are u, v ∈ B such that(3) LM ( f ) = u LM ( g ) v = u LM ( e g ) v. Combining (1), (2), and (3) above, we have w = LM ( F ∼ ) = LM ( f ) = u LM ( e g ) v. Therefore, LM ( e g ) | T r w , i.e., LM ( e g ) | LM ( F ), as desired.10ii) ⇒ (i) Suppose that e G is a Gr¨obner basis of the graded ideal h e I i in K h X, T i . If f ∈ I , thensince e f ∈ e I , there is some H ∈ e G such that LM ( H ) | LM ( e f ). Note that LM ( e f ) = LM ( f ) andthus T LM ( e f ). Hence H = e g for some g ∈ G , and there are w, v ∈ B such that LM ( f ) = LM ( e f ) = w LM ( e g ) v = w LM ( g ) v. This shows that G is a Gr¨obner basis for I in R . (cid:3) We call the Gr¨obner basis e G obtained in Theorem 2.3 the non-central homogenization of G in K h X, T i with respect to T , or e G is a non-central homogenized Gr¨obner basis with respect to T . By Lemma 2.1 and Theorem 2.3, the following Corollary is straightforward. Let I be an arbitrary ideal of K h X i . With notation as before, if G is a Gr¨obnerbasis of I with respect to the data ( B , ≺ gr ), then, with respect to the data ( e B , ≺ T - gr ) we have e B − h LM ( e G ) i = { T r w | w ∈ B − h LM ( G ) i , r ∈ N } , that is, the set N ( h e I i ) of normal monomials (mod h e I i ) in e B is determined by the set N ( I )of normal monomials (mod I ) in B . Hence, the algebra K h X, T i / h e I i = K h X, T i / h e Gi has the K -basis N ( h e I i ) = (cid:8) T r w (cid:12)(cid:12) w ∈ N ( I ) , r ∈ N (cid:9) . (cid:3) As with the central (de)homogenization with respect to the commuting variable t in section1, theoretically we may also obtain a Gr¨obner basis for an ideal I of K h X i by dehomogenizinga homogeneous Gr¨obner basis of the ideal h e I i ⊂ K h X, T i . Below we give a more generalapproach to this assertion. Let J be a graded ideal of K h X, T i , and suppose that { X i T − T X i | ≤ i ≤ n } ⊂ J . If G is a homogeneous Gr¨obner basis of J with respect to the data ( e B , ≺ T - gr ), then G ∼ = { G ∼ | G ∈ G } is a Gr¨obner basis for the ideal J ∼ in K h X i with respect to the data( B , ≺ gr ). Proof If G is a Gr¨obner basis of J , then G generates J and hence G ∼ = φ ( G ) generates J ∼ = φ ( J ). We show next that for each nonzero h ∈ J ∼ , there is some G ∼ ∈ G ∼ such that LM ( G ∼ ) | LM ( h ), and hence G ∼ is a Gr¨obner basis for J ∼ .Since { X i T − T X i | ≤ i ≤ n } ⊂ J , by Lemma 2.1(v) there exists a homogeneous element H ∈ J and some r ∈ N such that T r ( H ∼ ) ∼ = H and H ∼ = h . It follows that(1) LM ( H ) = T r LM (( H ∼ ) ∼ ) = T r LM ( e h ) = T r LM ( h ) .
11n the other hand, there is some G ∈ G such that LM ( G ) | LM ( H ), i.e., there are W, V ∈ e B such that(2) LM ( H ) = W LM ( G ) V. But by the above (1) we must have LM ( G ) = T q w for some q ∈ N and w ∈ B . Thus, by Lemma2.2(iii),(3) LM ( G ∼ ) = w = LM ( G ) ∼ w.r.t. ≺ gr on B . Combining (1), (2), and (3) above, we then obtain LM ( h ) = LM ( H ) ∼ = ( W LM ( G ) V ) ∼ = W ∼ LM ( G ) ∼ V ∼ = W ∼ LM ( G ∼ ) V ∼ . This shows that LM ( G ∼ ) | LM ( h ), as expected. (cid:3) We call the Gr¨obner basis G ∼ obtained in Theorem 2.5 the non-central dehomogenization of G in K h X i with respect to T , or G ∼ is a non-central dehomogenized Gr¨obner basis with respectto T . Let I be an ideal of K h X i . If G is a homogeneous Gr¨obner basis of h e I i in K h X, T i with respect to the data ( e B , ≺ T - gr ), then G ∼ = { g ∼ | g ∈ G } is a Gr¨obner basis for I in K h X i with respect to the data ( B, ≺ gr ). Moreover, if I is generated by the subset F and e F ⊂ G , then F ⊂ G ∼ . Proof
Put J = h e I i . Then since J ∼ = I , it follows from Theorem 2.5 that if G is a homogeneousGr¨obner basis of J then G ∼ is a Gr¨obner basis for I . The second assertion of the theorem isclear by Lemma 2.1(ii). (cid:3) Let S be a nonempty subset of K h X i and I = h S i the ideal generated by S . Then, with e S = { e f | f ∈ S } ∪ { X i T − T X i | ≤ i ≤ n } , in general h e S i ( h e I i in K h X, T i (forinstance, consider S = { Y − XY − X − Y, Y − X + 3 } in the free algebra K h X, Y i and thehomogenization in K h X, Y, T i with respect to T ). Again, as we did in the case dealing with(de)homogenized Gr¨obner bases with respect to the commuting variable t , we take this place toset up the procedure of getting a Gr¨obner basis for I and hence a Gr¨obner basis for h e I i byproducing a homogeneous Gr¨obner basis of the graded ideal h e S i . Let I = h S i be the ideal of K h X i as fixed above. Suppose the ground field K is computable. Then a Gr¨obner basis for I and a homogeneous Gr¨obner basis for h e I i maybe obtained by implementing the following procedure:12 tep 1. Starting with the initial subset e S = { e f | f ∈ S } ∪ { X i T − T X i | ≤ i ≤ n } , compute a homogeneous Gr¨obner basis G for the graded ideal h e S i of K h X, T i . Step 2.
Noticing h e S i ∼ = I , use Theorem 2.5 and dehomogenize G with respect to T in orderto obtain the Gr¨obner basis G ∼ for I . Step 3.
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