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58
Converting bases with the Gröbner walk
 Journal of Symbolic Computation
, 1997
"... We present an algorithm which converts a given Gröbner basis of a polynomial ideal I to a Gröbner basis of I with respect to another term order. The conversion is done in several steps following a path in the Gröbner fan of I. Each conversion step is based on the computation of a Gröbner basis of a ..."
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Cited by 35 (1 self)
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We present an algorithm which converts a given Gröbner basis of a polynomial ideal I to a Gröbner basis of I with respect to another term order. The conversion is done in several steps following a path in the Gröbner fan of I. Each conversion step is based on the computation of a Gröbner basis of a toric degeneration of I. c ○ 1997 Academic Press Limited 1.
Topology on the spaces of orderings of groups
, 2003
"... A natural topology on the space of left orderings of an arbitrary semigroup is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases. ..."
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A natural topology on the space of left orderings of an arbitrary semigroup is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases.
On the structure of order domains
 Finite Fields Appl
, 2002
"... Finite Fields and their Applications, vol. 8, pp. 369396, 2002. The notion of an order domain is generalized. The behaviour of an order domain by taking a sub algebra, the extension of scalars and the tensor product is studied. The relation of an order domain with valuation theory, Gröbner algebr ..."
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Cited by 21 (6 self)
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Finite Fields and their Applications, vol. 8, pp. 369396, 2002. The notion of an order domain is generalized. The behaviour of an order domain by taking a sub algebra, the extension of scalars and the tensor product is studied. The relation of an order domain with valuation theory, Gröbner algebras and graded structures is given. The theory of Gröbner bases for order domains is developed and used to show that the factor ring theorem and its converse, the presentation theorem hold. The dimension of an order domain is related to the rank of its value semigroup.
Some Complexity Results for Polynomial Ideals
, 1997
"... In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,.. ..."
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Cited by 16 (0 self)
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In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert’s Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.
Monomial Representations for Gröbner Bases Computations
 PROCEEDINGS OF ISSAC 1998, ACM PRESS
, 1998
"... Monomial representations and operations for Gröbner bases computations are investigated from an implementation point of view. The technique of vectorized monomial operations is introduced and it is shown how it expedites computations of Gröbner bases. Furthermore, a rankbased monomial representatio ..."
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Cited by 15 (1 self)
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Monomial representations and operations for Gröbner bases computations are investigated from an implementation point of view. The technique of vectorized monomial operations is introduced and it is shown how it expedites computations of Gröbner bases. Furthermore, a rankbased monomial representation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorized comparisons. Extensive benchmark tests with the Computer Algebra System Singular are used to evaluate these concepts.
Rankings of Partial Derivatives
 in: W. Kuchlin, Proc. ISSAC '97 (ACM
, 1998
"... Let m be a nonnegative integer, n a positive integer, N = f0; 1; 2; :::g and Nn = f1; : : : ; ng. A ranking is a total order of N m Nn such that (a; i) (b; j) implies (a + c; i) (b + c; j) for a, b, c 2 N m and i; j 2 Nn . We describe an approach to such rankings and a theorem which gives a ..."
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Let m be a nonnegative integer, n a positive integer, N = f0; 1; 2; :::g and Nn = f1; : : : ; ng. A ranking is a total order of N m Nn such that (a; i) (b; j) implies (a + c; i) (b + c; j) for a, b, c 2 N m and i; j 2 Nn . We describe an approach to such rankings and a theorem which gives an explicit construction of an arbitrary ranking using nite real data. The case n = 1 corresponds to termorderings of monomials which are crucial inputs for Buchberger's Grobner Basis algorithm for polynomial rings. The case n > 1 corresponds to rankings of partial derivatives which are inputs in algorithms in dierential algebra and Buchberger's algorithm for free modules over polynomial rings. A subclass of such rankings determined by nite integer data is given which is suÆcient for eective implementation of such rankings. This has been implemented in the symbolic language Maple. The rankings considered by Riquier are a special case of those considered here. Examples including appl...
A Gröbner Approach to Involutive Bases
 J.SYMB.COMP
, 1995
"... Recently, Zharkov and Blinkov introduced the notion of involutive bases of polynomial ideals. This involutive approach has its origin in the theory of partial differential equations and is a translation of results of Janet and Pommaret. In this paper we present a pure algebraic foundation of involut ..."
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Recently, Zharkov and Blinkov introduced the notion of involutive bases of polynomial ideals. This involutive approach has its origin in the theory of partial differential equations and is a translation of results of Janet and Pommaret. In this paper we present a pure algebraic foundation of involutive bases of Pommaret type. In fact, they turn out to be generalized left Gröbner bases of ideals in the commutative polynomial ring with respect to a noncommutative grading. The introduced theory will allow not only the verification of the results of Zharkov and Blinkov but it will also provide some new facts.
Gröbner Bases for Binomials with Parametric Exponents
 TECHNISCHE UNIVERSITÄT MÜNCHEN
, 2004
"... We study the uniformity of Buchberger algorithms for computing Gröbner bases with respect to a natural number parameter k in the exponents of the input polynomials. The problem is motivated by positive results of T. Takahashi on special exponential parameter series of polynomial sets in singular ..."
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We study the uniformity of Buchberger algorithms for computing Gröbner bases with respect to a natural number parameter k in the exponents of the input polynomials. The problem is motivated by positive results of T. Takahashi on special exponential parameter series of polynomial sets in singularity theory. For arbitrary input sets uniformity is in general impossible. By way of contrast we show that the Buchberger algorithm is indeed uniform up to a finite case distinction on the exponential parameter k for inputs consisting of monomials and binomials only. Under this hypothesis the case distinction is algorithmic and partitions the parameter range into Presburger sets. In each case the Buchberger algorithm is uniform and can be described explicitly and algorithmically. In the course of the algorithm the exponential parameter k enters also the coefficients as exponent. Thus the uniformity in k is established with respect to parametric exponents in both terms and coefficients. These results are obtained as a consequence of a much more general theorem concerning Buchberger algorithms for sets of monomials and binomials with arbitrary parametric coefficients and exponents, generalizing the construction of Gröbner systems.
Algorithms and Orders for Finding Noncommutative Gröbner Bases
, 1997
"... The problem of choosing efficient algorithms and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Grobner bases is expe ..."
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Cited by 9 (1 self)
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The problem of choosing efficient algorithms and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Grobner bases is expensive, and in noncommutative algebras is not guaranteed to terminate. The algorithm, together with the order used to determine the leading term of each polynomial, are known to affect the cost of the computation, and are the focus of this thesis. A Gröbner basis is a set of polynomials computed, using Buchberger's algorithm, from another set of polynomials. The noncommutative form of Buchberger's algorithm repeatedly constructs a new polynomial from a triple, which is a pair of polynomials whose leading terms overlap and form a nontrivial common multiple. The algorithm leaves a number of details underspecified, and can be altered to improve its behavior. A significant improvement is the devel...