Becker, T., Weispfenning, V.: Grobner Bases -- A Computational Approach to Commutative Algebra. Springer-Verlag  Home/Search   Document Not in Database   Summary   Related Articles   Check

....order compatible with the monoid operation; in fact, it has infinitely many such total orders, which we call term orders. These total orders were studied already by Macaulay [53] and classified by Robbiano [71, 72] see also [1, Chapter II, section 8] Further references on term orderings are [9, 11, 26]. Three of the most commonly used term orders are the lexicographic, graded lexicographic and graded reverse lexicographic orders. On , they are defined as follows. For the lexicographic order, ff 1 ; ff n ) lex (fi 1 ; fi n ) iff the first non zero component of (ff 1 fi ....

....a Grobner basis for the ideal generated by a finite set of polynomials. We shall not dwell longer on this subject; there are several good texts on the subject, to which we refer the reader. Buchbergers papers on the subject are [19, 20, 23, 21, 18, 22] More recent introductory expositions are [11, 73, 60, 29]. 0.3 Initial ideals of generic ideals It is a well known fact (for a proof, see for instance [83] that although there exists infinitely many term orders on M , if we fix an ideal I ae K[x 1 ; x n ] and partition the term orders into equivalence classes, two term orders and ....

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Thomas Becker and Volker Weispfenning. Gr obner bases: a computational approach to commutative algebra. Graduate texts in mathematics. Springer Verlag, 1993.

....Bird s Eye View of Grobner Bases Andr e Heck CAN Expertise Center, email: heck can.nl October 11, 1996 Abstract In this expository paper we give a short introduction to Grobner basis theory and its main applications. Quite a few books ([1, 3, 11, 17]) and overview articles ( 2, 7, 8, 20, 27] on Grobner basis theory already exist. This one differs in style and in choice of examples. The style is concrete: examples illustrate the main techniques and the use of a computer algebra system, in our case Maple, is not shunned. Most examples are ....

....form with respect to each other already produces the requested Grobner basis fx Gamma 1 2 z 2 Gamma 1 2 z; y Gamma 1 2 z 2 Gamma 1 2 z; z 4 z 2 Gamma zg. More sophisticated improvements of the basic Buchberger algorithm can among others be found in [7] algorithm 6. 3) and [3] (algorithm GR OBNERNEW2) 3.6 Programs for Computing Grobner Bases Almost every modern computer algebra system contains an implementation of a Grobner basis algorithm, some more advanced than others. For example, the basic algorithm implemented in Mathematica (version 2.0 and later) is poor in ....

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T. Becker, V. Weispfenning. Grobner Bases: A Computational Approach to Commutative Algebra. Springer Verlag, 1993.

....a Gr obner basis for an algebraic set is another set of polynomials whose common zeroes form exactly that same algebraic set. But the polynomials in the Gr obner basis satisfy some additional technical properties which facilitate various computational algorithms. See for example [CLO97] or [BW93]. An additional piece of terminology will be required: the ideal generated by a set of polynomials consists of all sums of multiples of those polynomials (where the multipliers can be any other polynomials) Clearly all zeroes of a set of polynomials are also zeroes of the ideal generated by those ....

Thomas Becker and Volker Weispfenning, Gr obner Bases: A Computational Approach to Commutative Algebra, 1993.

....0, 1, 0, 0#, where 1 is in exactly the i th component. We maintain the invariant that [l i , r i ] e for all rules l i # r i # R and hence, we assume that [s i , t i ] e for every i = 1, 2, n. Let # be the lexicographic ordering, or the total degree lexicographic ordering [7]. ACC Chaining: II, E, R # s # t, u # v ) II, E, R # s # t, u # v, su [s,v] t,u] # tv [s,v] t,u] if [t, u] #= e and either (a) t # s and u # v; or (b) s # t, u # v, and s # t is not a C rule; or (c) t # s, v # u, and u # v is not a C rule. The new rule is ....

T. Becker and V. Weispfenning. Grobner bases: a computational approach to commutative algebra. Springer-Verlag, Berlin, 1993.

....H. M. Moller and T. Sauer H bases for interpolation and system solving a problem in the infinite dimensional vector space of all polynomials in x 1 ; x n into a problem in one (or a series of) finite dimensional vector space(s) I P d . A similar concept is that of Grobner bases, see [1,3,7,11]. For these bases, one first has to order all power products (terms) x i 1 1 Delta Delta Delta x i n n linearly. Then, analogously to an H basis, a Grobner basis allows to find a linear generating system for the vector space I F t , where the linear space F t is generated by all terms ....

T. Becker and V. Weispfenning, Grobner Bases: A Computational Approach to Commutative Algebra, Springer Verlag, Berlin and New York, 1993.

....1=ff d g. The lemma now follows from the definition of I(V (I) 2 Lemma 5 J possesses a generating set contained in Q[x 1 ; x d ] and such a set can be computed from f 1 ; f d . Proof : By the Nullstellensatz, J is the radical of I over Q . There is an algorithm (see Chap. 8 of [1]) to compute a generating set for the radical of I over Q. We show that it is enough to work over Q. The radical of I is mapped to itself by every automorphism of Q since Q is fixed by any automorphism. By Lemma 2, p.19 of [13] this implies that the radical has a generating set with coefficients ....

T. Becker and V. Weispfenning. Grobner Bases: a Computational Approach to Commutative Algebra. Springer, 1993.

....uniquely and f 2 I if and only if r G (f) 0. An algorithm for computing Grobner bases (Buchberger s Algorithm) is described in the next subsection. Grobner bases together with Buchberger s Algorithm underlie many algorithms in algebraic geometry and commutative algebra, for details see [2], 7] 24] Buchberger s Algorithm To formulate a necessary and sufficient condition for a basis G of an ideal I ae k[x] to be a Grobner basis (with respect to a fixed monomial order OE) we introduce the notion of an S polynomial of two polynomials f 1 ; f 2 2 k[x] To avoid technicalities we ....

....division by G is zero (Theorem 6, 7] p. 84) This result is the key for constructing a Grobner basis from an arbitrary finite basis of an ideal. The method is due to Buchberger and will be listed here only in its basic form. Many improvements were worked out by Buchberger and other authors (see [2], 4] 7] and the references therein) Buchberger s Algorithm Let I = hf 1 ; f i. 16 INPUT: F = ff 1 ; f g G : F REPEAT G 0 : G FOR each pair fp; qg; p 6= q; in G 0 DO S : r G 0 (S(p; q) IF S 6= 0 THEN G : G [ fSg UNTIL G = G 0 . OUTPUT: A Grobner basis ....

T. Becker and V. Weispfennig. Grobner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, Berlin, 1993.

.... to check effectively whether the polynomial T (y 1 (f) y M (f) equals to identical zero, is reduced to testing the membership of the polynomial T (Y ) in the radical of the ideal generated by P 1 ; PM in the ring Q[f; Y ] 3 ; the latter problem is algorithmically solvable [13]. So, one can indeed construct all solutions to system (18) continuous in some neighborhood of zero, in the form of fractional power series in , keeping sufficiently many terms in these series to satisfy conditions (19) and then examine the asymptotic behavior of the Jacobian det D[ CK ....

....respect to I N prior to any forthcoming investigation. More precisely, for each polynomial C k , S k (belonging to the ring Q[ff; A; f ] although the dependence on f is not marked explicitly in the equations above) we will compute its fully reduced form with respect to some Grobner basis [13] associated with the ideal I N . Let us outline briefly the main idea of constructing and using Grobner bases. There must be introduced some admissible rule of ordering terms of multi variate polynomials in the ring under consideration. Hence, the generalization of the Euclidian algorithm which ....

Bekker T., Weispfenning V., and Kredel H. Grobner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, 1993. 33

....denominators of the last matrix don t vanish if u 1 , u 3 are non zero. In this situation, g 1 and g 2 are in hf 1 ; f 2 i in k(u 1 ; u 2 ) x 1 ; x 2 ] Thus, the geometric theorem holds if u 1 and u 3 are non zero. 7. The algorithms The source of the algorithms below is Becker Weispfenning [1]. The algorithm in Table 3 calculates the back and forth transformations. The algorithm listed in Table 4 computes reduced extended Grobner basis: 8 MAREK RYCHLIK Table 2. A calculation of back and forth transformations (C5) f1:determinant(matrix( 0,0,1] u1 u2,u3,1] x1,x2,1] D5) U2 ....

....algorithms below is Becker Weispfenning [1] The algorithm in Table 3 calculates the back and forth transformations. The algorithm listed in Table 4 computes reduced extended Grobner basis: 8 MAREK RYCHLIK Table 2. A calculation of back and forth transformations (C5) f1:determinant(matrix([0,0,1], u1 u2,u3,1] x1,x2,1] D5) U2 U1) X2 U3 X1 (C6) f2:determinant(matrix( u1,0,1] u2,u3,1] x1,x2,1] D6) U2 X2 U1 (U3 X2) U3 X1 (C7) xgextredgrobner( f1,f2] x1,x2] U3 U2 U1 (D7) R [ X2 , X1 ] U2 U1) X2 U3 X1, 2 2 2 [ 1 1 ] 2 U1 2 U1 ] U2 ....

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T. Becker and V. Weispfenning. Grobner bases---A Computational approach to Commutative Algebra. Springer-Verlag, New York, Berlin, Heidelberg, 1993.

....f 1 ; f n Gamma1 dans les calculs des r esultants successifs. 5. R eductions avec des bases de Gr obner 5.1. Bases de Grobner. L id eal I est engendr e par les n polynomes e 1 Gamma e 1 (ff) e n Gamma e n (ff) Mais ces polynomes n en forment pas une base de Grobner (voir [6]) Rappelons le th eor eme historique de Cauchy (voir [8] dans lequel il enonce comment les utiliser pour evaluer les polynomes sym etriques : 8] Soit F (x 1 ; x n ) un polynome a coefficients dans K et sym etrique en les variables x 1 ; x n . Pour eliminer x n ; x ....

....Notation 5.1. Pour i 2 [1; n] la notation I i d esigne l id eal engendr e dans K[x i ; x n ] par les modules de Cauchy f i ; f n . Il est bien connu qu alors les polynomes f i ; f n forment une base de Grobner r eduite pour l ordre lexicographique de l id eal I i (voir [6]) Ainsi, en consid erant K[x i ; x n ] comme un sous anneau de K[x 1 ; x n ] I i = I K[x i ; x n ] fP 2 K[x i ; x n ] j (8oe 2 S n ) oe:P ) ff) 0g : En particulier I 1 = I. Le th eor eme de Cauchy se g en eralise facilement ainsi: Th eor eme 5.2. ....

Becker (Thomas), Weispfenning (Volker) et Kredel (Heinz). -- Grobner bases : a computational approach to commutative algebra. -- Springer-Verlag, 1993.

....Reinterpreting the steps of this algorithm as operations on lattice vectors yields a combinatorial algorithm for computing test sets, see [92] 95] We consider here a geometric interpretation of Grobner bases for integer programs. We refer to Cox, Little O Shea [25] and Becker Weispfenning [6] for basics on Grobner bases and on Buchberger s algorithm for polynomial ideals that motivated these constructions. As in the previous section let L denote the lattice x # Z n : Ax = 0 . In order to avoid technical di#culties we make the following two assumptions: Assumption 3.1. c is ....

T. Becker, V. Weispfenning (1993), Grobner bases: A computational approach to commutative algebra, Springer Verlag, New York.

....Suppose g Gamma h and g Gamma k. Then there exists l with h Gamma l and k Gamma l. In particular, g has a unique complete reduction. This lemma follows from the previous lemma. The proof [18] is the virtually the same as in the algebraic case (for example see Theorem 4.75, pp. 176 7 of [1]) The analog of the Grobner basis in the differential context is given by the following Definition 1 We say that M is a Riquier basis if for all ff; ff 0 2 N m and f; f 0 2 M with hdD ff f = hdD ff 0 f 0 , the integrability condition D ff f Gamma D ff 0 f 0 can be reduced to 0. A ....

....known [14] This result gives a very natural finite set of integrability conditions that it is sufficient to check reduce to 0. However, this set of integrability conditions is in general far from minimal. As shown in [18] a well known criterion due to Buchberger in the algebraic case (see e.g. [1], p. 223) goes through in the differential case. Also see the preprint [2] for a related result in the case of differential polynomials. 5 Leading Nonlinear Theory The Janet al..gorithm, and consequently the Riquier Existence Theorem, can only be successfully applied to systems that are linear or ....

T. Becker, V. Weispfenning, and H. Kredel, Grobner Bases: A Computational Approach to Commutative Algebra, vol. 141 of Graduate Texts in Mathematics, Springer, 1993.

....iff 8p 2 O(t) p; t) 2 F b , p) w( p; t) in . An enabled transition can fire in the respective mode, creating a new marking in the accordingly. 3 Review of the Theory of Grobner bases In this section we give a self contained review of the main results of the theory of Grobner bases, see [5, 3] for a detailed exposition . We assume that the reader is familiar with the concept of polynomial over a field. Let T Theta T be an equivalence relation on a decidable set T . Definition 2 (Canonical Simplifier) An algorithm S, with inputs and outputs in T , is a canonical simplifier for on T ....

T. Becker and V. Weispfenning. Grobner Bases: a Computational Approach to Commutative Algebra. Graduate Texts in Mathematics. Springer--Verlag, 1993.

.... Buchberger s algorithm for constructing a Grobner basis for polynomial ideals over a field is one of the central methods in computational algebra [9, 11] It has been extended in two directions: to polynomials over coefficient domains other than fields and to non commutative polynomial rings, cf. [8]. For instance, Kandri Rody and Kapur [13] and Pan [17] considered polynomials over Euclidian rings and principal ideal domains, respectively. A Grobner basis for an ideal I induces a terminating rewrite relation on polynomials, such that two polynomials have the same normal form if, and only if, ....

....left hand sides, for n 2 N, where xn is constrained by an ideal membership constraint xn 2 I n . Since the divisibility relation on power products is a Dickson partial order, there exists an infinite subsequence t n1 ; t n2 ; such that t n i properly divides t n j for all i j (cf. [8], pp. 162 189) Since N1 is left reduced, we have I n i 1 6 I n i , for all i. We next claim that necessarily I n j ae I n j 1 for all j; but this contradicts the assumption that the coefficient domain is Noetherian. In order to prove the claim, suppose we have I n j 6 I n j 1 , for some j. ....

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T. Becker and V. Weispfenning. Grobner bases: a computational approach to commutat ive algebra. Springer-Verlag, Berlin, 1993.

....and such that [A] S 1 = A] S 1 . Delta Delta Delta u x u y u Delta Delta Delta v xx v xy v yy v x v y v: Running Rosenfeld Grobner directly over Sigma and R makes the memory of the computer explode. 1. 2 Grobner bases Grobner bases are presented in [8, 2]. Let R = K[X] be a polynomial ring. A term over X is a power product of elements of X. If B is a Grobner basis then Gamma Gamma Gamma B denotes the reduction by the basis B, using the classical reduction algorithm of the Grobner basis theory, which rewrites a term as a polynomial. Let S = ....

Thomas Becker and Volker Weispfenning. Grobner Bases: a computational approach to commutative algebra, volume 141 of Graduate Texts in Mathematics. Springer Verlag, 1991.

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Becker, T., Weispfenning, V.: Grobner Bases -- A Computational Approach to Commutative Algebra. Springer-Verlag

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T. Becker and V. Weispfenning. Grobner bases: a computational approach to commutative algebra. Springer-Verlag, Berlin, 1993. 13

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T. Becker and V. Weispfenning. Grobner bases: a computational approach to commutative algebra. Springer-Verlag, Berlin, 1993.

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T. Becker and V. Weispfenning. Grobner bases: a computational approach to commutative algebra. Springer-Verlag, Berlin, 1993.

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T. Becker and V. Weispfenning. Grobner bases: a computational approach to commutative algebra. Springer-Verlag, Berlin, 1993.

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T. Becker and V. Weispfenning, Grobner Bases: A Computational Approach to Commutative Algebra, Graduate Texts in Mathematics, Springer-Verlag, New York, 1993. 20

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T.Becker, V.Weispfenning, "Grobner Bases, a Computational Approach to Commutative Algebra", Graduate Texts in Mathematics, Springer, Berlin, 1993.

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T. Becker and V. Weispfenning, "Grobner Bases, A Computational Approach to Commutative Algebra", Springer-Verlag, New York, 1993.

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Thomas Becker and Volker Weispfenning, Grobner Bases: A Computational Approach to Commutative Algebra, Springer-Verlag, Heidelberg, 1993.

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T. Becker and V. Weispfenning, "Grobner Bases, A Computational Approach to Commutative Algebra," Graduate texts in Mathematics, Springer Verlag (1993).

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