Gröbner Bases over a Dual Valuation Domain

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1 International Journal of Algebra, Vol. 7, 2013, no. 11, HIKARI Ltd, Gröbner Bases over a Dual Valuation Domain André Saint Eudes Mialébama Bouesso Université Cheikh Anta Diop Département de Mathématiques et Informatique Laboratoire d Algèbre, de Cryptographie, de Géométrie algébrique et applications BP 5005 Dakar Fann, Sénégal sainteudes@gmail.com Copyright c 2013 André Saint Eudes Mialébama Bouesso. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Buchberger s algorithm was already studied over many kinds of rings such as principal ideal rings, noetherian valuation rings with zero divisors, Dedekind rings with zero divisors, Gaussian rings,.... In this paper, we propose the Buchberger s algorithm over V[ε] satisfying to ε 2 = 0 where V is any noetherian valuation domain and we give some applications in Z pz[ε] where p is a prime number and Z pz = { a b a Z, b/ pz}. Keywords: Gröbner bases, dual noetherian valuation domain, S-polynomials, Buchberger s algorithm 1 Introduction In 1965 Bruno Buchberger introduced the theory of Gröbner bases in polynomials ring over a field in order to solve the ideal membership problem (see [1, 2]). This theory was generalized by many authors in different ways such as [5, 8, 9],... for the noncommutative case and [4, 6, 7, 10, 11] for the commutative case. In [6, 10, 11] authors presented Buchberger s algorithm over noetherian valuation rings with zero divisors and over Dedekind rings with zero divisors.

2 540 André Saint Eudes Mialébama Bouesso Both methods don t cover the ring V[ε] since this ring is neither a valuation ring nor a Dedekind ring. It is also proposed in [4] an interesting method for computing a Gröbner basis over many kinds of rings with zero divisors, such rings cover Z n where n is not a prime number as well as Z n [i] satisfying to i = 0, this method cover many others rings with zero divisors but does not cover the ring of dual valuation domain V[ε] satisfying to ε 2 = 0 where V is a valuation domain. In this paper we propose a method for computing a Gröbner basis over the ring of dual noetherian valuation domain V[ε] and we present some applications in Z pz[ε] where p is a prime number. This paper is organized as follows: Section 1: We study some elementary properties of the ring V[ε] and we recall some necessary tools for computing a Gröbner basis. Section 2: We adapt the Buchberger s algorithm in V[ε]. 2 Basic notions Throughout this paper, we denote by V a noetherian valuation domain, V[ε] the ring of dual valuation domain satisfying to ε 2 = 0, whose elements are of the form a + εb with a, b Vand J ε = ε V[ε] ={εa/a V}the set of zero divisors of V[ε]. If R is a ring and E a subset of R, we denote by E the ideal generated by E. 1. Division in V[ε]: An element z 1 = a 1 + εb 1 divides z 2 = a 2 + εb 2 in V[ε] if and only if a 1 divides a 2 in V and a 1 divides b 2 b 1 a 2 a 1 in V. 2. Let R = V[ε][X 1,...,X m ] be the free associative algebra with commuting variables X 1,...,X m, defined over the ring V[ε]. A monomial in R is a multivariate polynomial of the form g = X α 1 1 Xm αm where α i N. We denote X α = X α 1 1 Xm αm where α =(α 1,...,α m ) and by M the set of all monomials in R. An element of the form zx α where z V[ε][X 1,...,X m ] is called a term. A term zx α divides z X β in R if and only if z divides z in V[ε] and X α divides X β in R. 3. Monomials order: A total order < in M is said to be a monomial order if it is a well ordering and If X α <X β then X α+γ <X β+γ for all α, β, γ N n.

3 Gröbner bases over a dual Noetherian valuation domain 541 Lexicographic order: we say that X α > lex X β if the first left non zero component of α β is > 0. Graded lexicographic order: We say that X α > grlex X β if α > β or if α = α i = β = β i and X α > lex X β for α = (α 1,...,α n ),β=(β 1,...,β n ). 4. Let f = α z α X α be a nonzero polynomial in R. Let I = f 1,...,f s be a finitely generated ideal of R and let us fix a monomial order < in M, then: The multidegree of f is mdeg(f) := max{α/z α 0}. The leading coefficient of f is Lc(f) :=z mdeg(f). The leading monomial of f is Lm(f) :=X mdeg(f). The leading term of f is Lt(f) :=Lc(f) Lm(f). Lt(I) := Lt(g)/g I \{0}. 5. Division algorithm (see [3, 10]) We recall the division algorithm in R. Let < be a monomial order and f 1,...,f s R. Then there exists q 1,...,q s,r R such that f = q i f i + r with mdegf mdeg(q i f i ) for q i f i 0 and r =0or each term occurring in r is not divisible by any of Lt(f i ) 1 i s. Input: f 1,...,f s,f and <. Output: q 1,...,q s,r. Initialization: q 1 := 0,...,q s := 0; r := 0 and p := f. While p 0 do: i := 1 DIVOCCUR:=False while i s and DIVOCCUR:=False do If Lt(f i ) divides Lt(p) then q i := q i + Lt(p) Lt(f i )

4 542 André Saint Eudes Mialébama Bouesso p := q i Lt(p) Lt(f i ) f i DIVOCCUR:=True Else i := i +1 If DIVOCCUR:=False then r := r + Lt(p); p := p Lt(p) Example 2.1. In Z 2Z[ε][x, y] with x< lex y, let us divide f =(3+ 5ε)xy 2 +3εy by f 1 =(5 2ε)x εy 2 and f 2 =(7 3ε)xy. We find: f =( ε) f 1 +( 3 5 εy4 +3εy) if we start the division by f 1. f =( ε)y f 2 +(3εy) if we start the division by f Gröbner basis: A subset G = {g 1,...,g t } of an ideal I R is called Gröbner basis for I with respect to a monomial order < if I = G and Lt(I) = Lt(G). 3 Buchberger s algorithm Definition 3.1. S-polynomials: Let f,g R, and let us consider a monomial order >. Denoting by Lt(f) =(a 1 +εb 1 ) X α, Lt(g) =(a 2 +εb 2 ) X β where a 1,a 2,b 1,b 2 V[ε]; α, β N n. Let γ N n with γ i = max(α i,β i ) for each i, the S-polynomial of f and g is given by: 1. Suppose that f g: Lc(f) J ε and Lc(g) J ε then: S(f,g) = If Lc(f) J ε and Lc(g) / J ε then: S(f,g) = b 2 b 1 X f Xγ α X g if b 1/b β 2 X f b 1 α b 2 X g if b 2/b β 1. a 2 b 1 X α f Xγ X β (εg) if b 1/a 2 f b 1 X α a 2 X (εg) if a 2/b β 1. If Lc(f) / J ε and Lc(g) J ε then replace f by g and vice versa.

5 Gröbner bases over a dual Noetherian valuation domain 543 If Lc(f) / J ε and Lc(g) / J ε then: S(f,g) = 2. Suppose that f = g then: a 2 Xγ (εf) a 1 Xα X (εg) if a 1/a β 2 X (εf) a 1 α a 2 X (εg) if a 1/a β 2. { εf if Lc(f) Jε S(f,f) = 0 if not. Example 3.2. Let us compute in R = Z 2Z[ε][x, y] the S-polynomials of f = (3+5ε)xy 2 +3εy, g =(7 3ε)xy, h 1 =5εx (1+ε)y 2 and h 2 =2εx 2 +(3 2ε)y with respect to X< lex Y : S(f,g) = 7 (εf) y(εg) =0 3 S(f,h 1 )= 5 3 (εf) y2 h 1 =(1+ε)y 4 S(h 1,h 2 )= 2 5 xh 1 h 2 = 2 5 xy2 (3 2ε)y S(h 1,h 1 )= εy 2. Lemma 3.3. Let < be a monomial order, and f 1,...,f s R = V[ε][X 1,...,X m ] such that mdeg(f i )=γ N n for each 1 i s. Suppose that mdeg( z i f i ) < γ for some z 1,...,z s V[ε]: 1. If Lc(f i ) J ε 1 i s then z i f i is a linear combination with coefficients in V[ε] of S-polynomials S(f i,f j ) for 1 i j s. 2. If there exists i 0 such that Lc(f i0 ) / J ε then ε z i f i is a linear combination with coefficients in V[ε] of S-polynomials S(f i,f j ) for 1 i j s. Furthermore, each S-polynomial has multidegree <γ. Proof: 1. Suppose that Lc(f i )=εb i J ε 1 i s and we can assume that / 1 /.../b 1 since V is a valuation domain.

6 544 André Saint Eudes Mialébama Bouesso z i f i = z 1 (f 1 b 1 f s )+z 2 (f 2 b 2 f s )+...+ z s 1 (f s 1 1 f s )+ (z 1 b 1 + z 2 b z s z s )f s. b 1 b 2 1 By hypothesis, we have (z 1 + z z s 1 + z s ) ε = 0 then, b 1 b 2 1 (z 1 + z z s 1 + z s )f s V[ε]S(f s,f s ). We deduce that z i f i = z i,j S(f i,f j ). i,j 2. Let Lc(f i )=a i + εb i and suppose that there exists at least i 0 for which a i0 0 then: ε z i f i = z i (εf i )+ (εf i ). Lc(f i )/ J ε Lc(f i ) J ε Without loss of generalities we set: Lc(f i ) J ε z i (εf i )= i=k+1 z i (εf i )= i=k+1 i=k+1 z i (εf i ), then: z i S(f i,f i ) (*). Lc(f i )/ J ε z i (εf i ) = k z i (εf i ) and Assume that Re(Lc(f k ))/Re(Lc(f k 1 ))/.../Re(Lc(f 1 )) then: k z i (εf i )=z 1 [(εf 1 ) Re(Lc(f 1)) Re(Lc(f k )) (εf k)] + z 2 [(εf 2 ) Re(Lc(f 2)) Re(Lc(f k )) (εf k)] +...+z k 1 [(εf k 1 ) Re(Lc(f k 1)) Re(Lc(f k )) (εf Re(Lc(f 1 )) k)]+(z 1 Re(Lc(f k )) +z Re(Lc(f 2 )) 2 Re(Lc(f k )) + Re(Lc(f k 1 ))...+ z k 1 + z k )(εf k ). Re(Lc(f k )) Since by hypothesis 0 = z 1 Lc(f 1 )+...+ z s Lc(f s )=ε(z 1 Lc(f 1 )+...+ z k Lc(f k )) = ε(z 1 Re(Lc(f 1 )) z k Re(Lc(f k ))) then: k z i (εf i )= z i,j S(f i,f j ) (**). Thus from (*) and (**) we get the i,j desired result.

7 Gröbner bases over a dual Noetherian valuation domain 545 Theorem 3.4. Let < be a monomial order and G = {g 1,...,g s } be a finite set of polynomials of R = V[ε][X 1,...,X m ]. Let I = G be an ideal of R, then G is a Gröbner basis for I if and only if all remainder of S-polynomials S(g i,g j ) by G is zero for 1 i j s. Proof (a) S(g i,g j ) g i,g j G S(g i,g j ) G = 0 where S(g i,g j ) G is the remainder of S(g i,g j ) under the division by G. (b) Conversely, we need to prove that Lt(I) = Lt(G). Let f I then f = h i g i (1) where h i R and mdeg(f) max{mdeg(h i g i )} = γ. Let γ be the i smallest multidegree satisfying to (1), then mdeg(f) γ. If mdeg(f) <γ, from (1) we have: f = h i g i = h i g i + h i g i (2) Notice that: h i g i = mdeg(h i g i )<γ Lt(h i )g i + (3). Let Lt(h i )=m i X α i with m i V[ε], then Lt(h i )g i = m i (X α i g i ) (*). (h i Lt(h i ))g i Suppose that Lc(g i ) J ε i then from the previous lemma (*) become: z ij S(X α i g j ) where z ij Lt(h i )g i = i,j V[ε]. Since S(X α i g j )= Im(Lc(g i)) Im(Lc(g j )) X α i Lm(gi ) (Xα i g i ) X α j Lm(gj ) (Xα j g j )= γ ij S(g i,g j ) with ij = lcm(lm(g i ),Lm(g j )). Therefore z ij γ ij S(g i,g j ) (4). Lt(h i )g i = i,j By hypothesis for 1 i j s, S(g i,g j )= q ijk g k k where mdeg(q ijk g k ) mdeg(s(g i,g j )), then Lt(h i )g i =

8 546 André Saint Eudes Mialébama Bouesso z ij γ ij q ijk g k. i,j,k Since mdeg(s(g i,g j )) γ ij then mdeg( γ ij q ijk g k ) mdeg( γ ij S(g i,g j )) <γ. By minimality of γ, we get a contradiction. Suppose that there exists i 0 such that Lc(g i0 ) / J ε then from the previous lemma (*) become: ε Lt(h i )g i = Lt(h i )(εg i )+ Lt(h i )(εg i ) mdeg(h i (εg i ))=γ mdeg(h i (εg i ))<γ m i (X α i (εg i )) = w i,j S(X α i g j ). i,j Lt(h i )(εg i )= mdeg(h i (εg i ))=γ i From the previous item we have seen that mdeg(s(x α i g j )) < γ and we can easily see that mdeg(m i (X α i εg i )) = mdeg(m i (X α i g i )), we get a contradiction. Suppose that Lc(g i ) / J ε i then from the previous lemma (*) become (ε m i (X α i g i )) = m i (X αi εg i )= mdeg(h i (εg i ))=γ i,j where t i,j V[ε]. From the previous item we have seen that mdeg(s(x α i g j )) <γthus we get a contradiction. Since mdeg(f) = γ, then there exists j 0 such that mdeg(f) = mdeg(h j0 g j0 )=γ. We have Lm(f) =Lm(h j0 )Lm(g j0 )=. Put = {i/lm(h i )Lm(g i )=Lm(h j0 )Lm(g j0 )} then Lc(f) = Lc(h i )Lc(g i ). i Therefore Lt(f) = Lt(h i )Lt(g i ) hence Lt(f) Lt(g i ) i i Lt(G). The previous theorem guaranties that we can compute a Gröbner basis of an ideal of R after a finite number of step. We are now ready to give the algorithm for computing a Gröbner basis. Buchberger s algorithm. Input: g 1,...,g s R and < a monomial order. Output: A Gröbner basis G for I = g 1,...,g s with {g 1,...,g s } G G := {g 1,...,g s } REPEAT G := G t i,j S(X α i g j )

9 Gröbner bases over a dual Noetherian valuation domain 547 For each pair g i,g j in G DO S := S(g i,g j ) G If S 0 THEN G := G {S} UNTIL G = G Example 3.5. Let R = Z 3Z[x, y] and let us fix a monomial order x< lex y. Consider the set F = {f 1 =(3+5ε)xy 2 +3εy, f 2 =5εx (1 + ε)y 2 }. We want to construct a Gröbner basis for I = F in R with respect to x< lex y. Set G = {g 1 := f 1,g 2 := f 2 } S(g 1,g 2 )=(εg 1 ) 3 5 y2 f 2 = g 3, G = {g 1,g 2,g 3 }. 3(1 + ε) y 4 and S(g 1,g 2 ) G = 5 S(g 2,g 2 )=εg 2 = εy 2 = S(g 2,g 2 ) G = g 4, G = {g 1,g 2,g 3,g 4 }. 3(1 + ε) y 4 = 5 Notice that S(g 1,g 4 )=S(g 1,g 3 )=S(g 4,g 4 )=S(g 3,g 4 )=0and S(g 2,g 3 ) G = S(g 2,g 4 ) G =0. Thus G = {(3 + 5ε)xy 2 +3εy, 5εx (1 + ε)y 2 3(1 + ε), y 4, εy 2 } is a 5 Gr bner basis for I = (3 + 5ε)xy 2 +3εy, 5εx (1 + ε)y 2. References [1] B. Buchberger (1965), An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, Ph.D. Thesis, Univ. of Innsbruck, Austria, Math., Inst. [2] B. Buchberger, Introduction to Gröbner bases. (B. Buchberger, F. Winkler, eds.) Gröbner bases and Applications, London Mathematical Society Lecture Note Series 251, Cambridge University Press (1998). [3] D. Cox, J. Little and D. O Shea, Ideals, Varieties and Algorithms, 3rd ed., Springer-Verlag, New York, (2007). [4] Deepak Kapur and Yongyang Cai, An Algorithm for Computing a Gröbner Basis of a Polynomial Ideal over a Ring with Zero Divisors. Math.comput.sci. 2 (2009), pp [5] E. Green, Noncommutative Gröbner bases, and projective resolutions. Computational methods for representations of groups and algebras. Papers from the First Euroconference held at the University of Essen. Basel, (1999), P. Dräxler, G. O. Michler, and C. M. Ringel, Eds., no. 173 in Progress in Math., Birkhäuser Verlag, pp

10 548 André Saint Eudes Mialébama Bouesso [6] A. Hadj Kacem and I. Yengui, Dynamical Gröbner bases over Dedekind rings, J. Algebra 324 (2010) pp [7] D. Kapur and K. Madlener, Construction of Gröbner bases in special rings, in Manuscript presented at the Gröbner bases workshop (1988), Cornell. [8] A.S.E.Mialébama Bouesso and D. Sow, Noncommutative Gröbner bases over rings, (preprint), to be published in communications in algebra (2013). [9] T. Mora, Groebner bases for non-commutative polynomial rings, Proc. AAECC3, LNCS 229 (1986) [10] Yengui I., Dynamical Gröbner bases. J. Algebra 301 (2006) pp [11] Yengui I., Corrigendum to Dynamical Gröbner bases [J. Algebra 301 (2) (2006) pp ] &to Dynamical Gröbner bases over Dedekind rings [J. Algebra 324 (1) (2010) pp ]. J. Algebra 339 (2011) pp Received: May 10, 2013