B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Philosophische Fakultat an der LeopoldFranzens -Universitat, Innsbruck, Austria, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... ; g r g is a basis of I; ii) fLT (g 1 ) LT (g r )g is a basis of the leading term ideal of I, which is the smallest ideal containing the leading terms of all f 2 I; or, equivalently: if f 2 I, then LT (f) 2 (LT (g 1 ) LT (g r ) Grobner bases have been introduced in [8]. For an excellent exposition of their numerous useful properties, see e.g. 9] A basis is called minimal if it does not strictly contain some other basis of the same ideal. A Grobner basis is called reduced if no term in any one of its polynomials is divisible by the leading term of some other ....

Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.d. thesis, Department of Mathematics, University of Innsbruck, 1965.

.... introduction to commutative and non commutative Grobner Bases Teo Mora Introduction In 1965, Buchberger introduced the notion of Grobner bases for a polynomial ideal and an algorithm (Buchberger Algorithm) for their computation ([B1], B2] Since the end of the Seventies, Grobner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in Algebraic Geometry and Commutative Algebra. Bergman ( Ber] ....

B.Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph. D. Thesis, Innsbruck, (1965)

....of mathematics, and they also have a number of applications in various areas of computer science, like language generating and term rewriting systems, tiling problems, algebraic manifolds, motion planing, and several models for parallel systems. Through the introduction of Grobner bases (see [Buc65], also [Hi64] many of the mentioned problems become easily expressible and algorithmically solvable. For practical applications, in particular, the implementation in computer algebra systems, it is important to establish upper complexity bounds for the normal form algorithms which transform a ....

....Notations The polynomial ideals which we get by using the relationship of finitely presented commutative semigroups and polynomial ideals are pure difference binomial ideals, i.e. each polynomial in the basis of such an ideal is the difference of two terms. By looking at Buchberger s algorithm [Buc65], it is not hard to see that the reduced Grobner basis of a pure difference binomial ideal still consists only of pure difference binomials. Let X denote the finite set fx 1 ; x k g, and Q[X] the (commutative) ring of polynomials with indeterminates x 1 ; x k and rational ....

B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.d. thesis, Department of Mathematics, University of Innsbruck, 1965.

....B. Buchberger, in 1965, was the first to show that for polynomials over a field it is possible to construct, from an arbitrary given basis, a detaching basis, the so called Grobner basis, such that many of the problems mentioned above become easily expressible and algorithmicly solvable (see [Buc65], also [Hi64] Although versions of Buchberger s algorithm have been somewhat successful in practice, the complexity of the algorithm is not well understood. First steps towards an upper bound were obtained in [Bay82] and [MoMo84] where upper bounds for the degrees in a minimal Grobner basis ....

....denotes the Q[X] ideal generated by fl 1 Gamma r 1 ; l h Gamma r h g, i.e. I(P) p i (l i Gamma r i ) p i 2 Q[X] for i 2 I h We call such an ideal a binomial ideal , i.e. each polynomial in the basis is the difference of two terms. By looking at Buchberger s algorithm [Buc65] it is not hard to see that the reduced Grobner bases of binomial ideals still consist only of binomials. The following proposition shows the connection between the uniform word problem for commutative semigroups and the membership problem for ideals in Q[X] 5 The uniform word problem for ....

Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.d. thesis, Department of Mathematics, University of Innsbruck, 1965.

.... a basis of I; G2) fLT (g 1 ) LT (g r )g is a basis of the leading term ideal of I, which is the smallest ideal containing the leading terms of all f 2 I, or equivalently: if f 2 I, then LT (f) 2 hLT (g 1 ) LT (g r )i : Grobner bases have been introduced in [Hi64, Hi64a] and [Buc65] A basis is called minimal if it does not strictly contain some other basis of the same ideal. A Grobner basis is called reduced if no monomial in any one of its polynomials is divisible by the leading term of some other polynomial in the basis. 5 Now let P = fl i j r i ; i 2 I h g be any ....

....denotes the Q[X] ideal generated by fl 1 Gamma r 1 ; l h Gamma r h g, i.e. I(P) p i (l i Gamma r i ) p i 2 Q[X] for i 2 I h We call such an ideal a binomial ideal , i.e. each polynomial in the basis is the difference of two terms. By looking at Buchberger s algorithm [Buc65] it is not hard to see that the reduced Grobner basis of a binomial ideal still consists only of binomials. The following proposition shows the connection between the uniform word problem for commutative semigroups and the membership problem for ideals in Q[X] The uniform word problem for ....

Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.d. thesis, Department of Mathematics, University of Innsbruck, 1965. 19

....sets of points, e.g. to Cayley Bacharach schemes. A preliminar version of the first part of this paper appeared in the Proceedings of the ISSAC 91 Meeting [20] A more general interpolation problem with symbolic techniques using Gr6bner bases, but with no control complexity, is discussed in [4]. 1 FUNCTIONALS OVER THE POLYNOMIAL RING 12:56 4 Let K be a field and P : K[X 1 . Xn] Considering P as a K vector space, a K functional L over P is a linear morphism L: P K, i.e an element of the K vector space pa = HOmK(P K) If V C pa is a K vector subspace, then Z(V) f G P: ....

.... t G T: 3 f G F s.t. T(f) divides t . Notice that if I is an ideal, then T(I) T I . We will freely use the notation N(F) N(T(F) G(F) G(T(F) B(F) B(T(F) I(F) I(T(F) Definition 3.2 Let I C P be an ideal. A finite set F C I 0 is called a Gr6bner basis of I if T(F) T(I) [4, 5] Theorem 3.3 Let I C P be an ideal and let : P P I be the canonical projection. If F C I 0 , then the following conditions are equivalent: 1) F is a Gr6bner basis of I. 2) 0: t G N(F) is a K basis of P I. 3) V f P, f I if and only if f = h i fi, hi P, fi F, T(hi) T(f i) T(f) 6 6 96 ....

[Article contains additional citation context not shown here]

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, Innsbruck, (1965)

....are available for some algebraic objects. Naturally, they are limited to particular classes of rings, e.g. GCD and factorization of polynomials. Operations on ideals play a central role in the system. They are based to a large extent on Buchberger s algorithm for Grobner Basis computation [6]. That algorithm and its various generalizations to non polynomial rings (e.g. 11, 9, 12, 5, 1] is used to compute elimination ideals (which relates to solving systems of non linear algebraic equations) modules and chains of syzygies, as well as intersections and quotients of ideals. A ....

B. Buchberger. Ein Algorithmus zum Auf finden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Dissertation, Mathematisches Institut der Universitat Innsbruck, Innsbruck, Osterreich, 1965.

....Nfg = GammaNf g 0 , i.e. the normal form of g does not depend on the second part of the reduction strategy. There still remains an open gap, namely the reduction of convergent power series. This gap will be closed in section 3. Now we give a short introduction to the theory of Grobner bases ([BB65]) which turned out to be a very powerful tool in constructive commutative algebra (cf. BB84, G T Z] Definition 2.3: A subset F ae I of the ideal I ae R such that inFR = inI R is called a Grobner basis of I (with respect to OE) This is the most common Grobner basis definition which can be ....

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. Thesis, Univ. Innsbruck, 1965.

....Maple in the moregroebner package [17] The input consists of a finite set of monomials and a list of gradings forming a weight system. It is an important and well known fact about Grobner bases that they enable the computation of Hilbert series of homogeneous ideals as was already pointed out in [8]. Lemma 2.14 [35] Let (W 1 ; W r ) be a weight system for K[x] and I a W homogeneous ideal. Let LT (I) be the monomial ideal generated by all leading terms ht(f) of f 2 I with respect to a term order of K[x] Then the Hilbert series of I and LT (I) are equal. Since the leading terms ....

B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, (An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal). PhD thesis, Math. Inst. University of Innsbruck,Austria, 1965.

....in [9] to show that the latter question is decidable. Of course the ideal membership problem is also solvable using Buchberger s method of Grobner bases, which is based on a special reduction system associated to finite sets of polynomials which represent ideal congruences in polynomial rings [6]. It was observed independently in [19, 22, 16] that similar results hold for congruences on arbitrary finitely generated monoids and groups. Here we want to develop these ideas for the free group case in order to give a coset enumerating procedure using Grobner techniques for free group rings: ....

B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, Universit at Innsbruck, 1965.

.... algorithms to compute the function ; these algorithms are now obsolete but have been the first ones to have polynomial complexity in the number of generators of M (6) The reduction to the monomial case via Grobner bases and Macaulay theorem and the use of Moebius function is already in [4]. Bayer Stillman algorithm Let M be generated by f 1 ; r g, M 0 be generated by f 1 ; r Gamma1 g and let M 00 = M 0 : r . Bayer Stillman algorithm ( 1] computes PM (t) by the formula: PM (t) PM 0 (t) Gamma t deg( r ) PM 00 (t) using induction on the number ....

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, Innsbruck, (1965).

....zeros of F over an algebraic closure of k) Such a description is usually given by a finite family fT 1 ; T r g of polynomial sets with particular properties, a link between the T i and F , and an algorithm to compute the T i from F . A well developed strategy since Buchberger s work ([Buc65]) is the following : given an ordering on the monomials, to choose for T 1 the Grobner basis of the ideal generated by F and compute it by the Buchberger s algorithm. Wu Wen Tsun in [Wu87] introduced another way of solving algebraic systems which is the one we are concerned with in this paper. In ....

B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Innsbruck, 1965.

....students (T. Bayer, M. Schulze, M. Wenk) respectively from joint research projects together with C. Lossen and E. Shustin. All algorithms are implemented in the computer algebra system Singular [GPS] They are mainly based on Grobner basis methods which were foundationally developed by Buchberger [Bu1], Bu2] for polynomial rings. Subsequently, they have been extended to local and mixed rings in [GP2] for use in singularity theory. Grobner basis computations are, nowadays, implemented in all major general purpose computer algebra systems such as the big M systems (Magma, Maple, ....

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, PhD Thesis, University of Innsbruck, Austria, 1965.

....in this case, r is independent of the choices so it is a canonical form (aka normal form) for f and is denoted N(f,G) The main theoretical results in the commutative case were obtained in the mid 1960 s. Hironaka ( Hiro] 1964) proved the existence of a Grbner Basis for any ideal. Buchberger ([Buch1], 1965) showed this independently. He also provided an algorithm for obtaining a Grbner Basis from an arbitrary basis. By 1985, Buchberger [Buch2] substantially improved the efficiency of the algorithm. He and others also explored a variety of applications. In the non commutative case, ideals may ....

B. Buchberger, "Ein Algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal", Doctoral Dissertation Math Inst University of Innsbruck, Austria.

....rings of polynomials, germs, and entire functions. Due to the lack of completeness and for some rings also semi locality we can not simply extend the ideas of Krull and Chevalley. However, at least in the case of polynomial functions we can give a satisfactory answer by means of Grobner bases (cf. [B] and [BWK] Theorem 1 Let I 1 I 2 : be a descending sequence of ideals of the polynomial ring R = K [X ] Furthermore, for each = 1; 2; let G be the reduced Grobner basis of I with respect to a fixed order OE of order type . Then the set G = T 1 = G is the reduced Grobner ....

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. Thesis, Univ. Innsbruck, 1965.

.... a confluent prefix rewriting system for the subgroup [KuMaOt94] The application of reduction techniques in rings for solving membership problems of ideals and subalgebras also has a long tradition and has produced multiple results beginning with Buchberger s fundamental work on Grobner bases [Bu65]. The main purpose of this paper is to relate the reduction techniques used for monoids, groups and rings by explicitly relating decision problems in appropriate related structures. Using reductions, e.g. from the word problem for finitely presented monoids or groups to the ideal membership ....

B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Dissertation. Universitat Innsbruck. 1965.

....Let P = P 1 ; P ) be a system of points (nodes) P i 2 K n and D = D 1 ; D ) a system of Ferrers diagrams. Then we require jffj F X ff (P j ) b j;ff (2) for ff 2 D j , j = 1; and given data b j;ff 2 K . There can be applied Grobner basis techniques (c.f. [Bu65], BWK] CLO] in order to solve this problem for arbitrary given data P, D and b j;ff (c.f. BW91] To each pair (P j ; D j ) we assign the ideal I j = I(P j ; D j ) F 2 R fi fi fi fi fi jffj F X ff (P j ) 0 for all ff 2 D j ) 3 which is generated by the set of all (X ....

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. Thesis, Univ. Innsbruck, 1965.

....So, the algorithmic computation of and division modulo Grobner bases can be considered as the fundamental problems of computational ideal theory. During the last more than three decades Buchberger s algorithm became a central tool in constructive commutative algebra and algebraic geometry (cf. 1 [1, 8, 9, 11]) and motivated by the achievements in polynomial rings many efforts have been spent in generalizations to other types of rings. The concept of graded structures due to Robbiano [26] and Mora [22] provides an excellent frame for investigating Grobner bases in very general situations. What remains ....

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, University of Innsbruck, 1965.

....of this paper is the presentation of some classes of effective graded structures. 1 Introduction Buchberger s algorithm for the computation of Grobner bases has become a central tool in the field of constructive commutative algebra and algebraic geometry during the last three decades (c.f. [Bu65], Bu85] BW93] AL94] Motivated by the achievements in polynomial rings many efforts have put towards generalisations of the Grobner theory to other types of rings. The concept of graded structures due to Robbiano (see [Ro86] and Mora (see [Mo88a] provides an excellent frame for investigating ....

....of effective Grobner structures, i.e. such graded structures which allow the algorithmic computation of Grobner bases. Important examples for classes of rings covered by Theorem 1 and Theorem 2 are, for instance, the polynomial rings in finitely many indeterminates over effective fields (see [Bu65]) or over effective principal ideal domains (see [KK84] Pa85] or, even more general, over commutative rings in which linear equations are solvable (see [Tr78] Za78] Sch79] Mo88] BW93] AL94] the algebras of solvable type (see [KW90] the G algebras (see [Ap92] or the solvable 1 ....

Buchberger, B., Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhDThesis, Univ. Innsbruck, 1965.

....we describe the various access methods we provided and their technical realization. In the concluding Section 4 we discuss various aspects of the use of Java for our project that will apply to similar projects. 2 Background 2. 1 Gr obner Bases We will recall some basic facts about Grobner bases [7, 8, 6] which will be needed in our system description below. A Grobner basis is computed for a set of polynomials with respect to a term ordering induced by an ordering on the indeterminates of the polynomials. For many applications a Grobner basis with respect to the so called lexicographic term ....

BUCHBERGER, B. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Universitat Innsbruck, 1965.

....of P and Q is defined by SP ol(P; Q) C i (Q)H i (Q) P Gamma C i (P )H i (P ) Q: This definition enables us to assert that H(SP ol(P; Q) lcm(H(P ) H(Q) Note also that Spol(P; Q) 2 (P; Q) and a fortiori Spol(P; Q) 2 (P; Q) A=S . We now apply Buchberger s algorithm (see [CLO 92] Buch 65] with our alternative definitions of heads, leading coefficients, reduction, and S polynomials. We recall hereafter a compact but non optimized version of this algorithm. Algorithm Pseudo Groebner basis Input: a finite set E of non zero polynomials in A. Output: a pseudo Groebner basis G of ....

B. Buchberger. Ein Algorithmus zum auffinden der Basiselementedes Restklassenringes nach einem null-dimensionalen Polynomideal. PhD. thesis, University of Innsbruck.

No context found.

B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Philosophische Fakultat an der LeopoldFranzens -Universitat, Innsbruck, Austria, 1965.

No context found.

Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Universitat Innsbruck, 1965.

No context found.

Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.d. thesis, Department of Mathematics, University of Innsbruck, 1965.

No context found.

Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Thesis, Univ. Innsbruck, 1965.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC