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187
Involutive Bases of Polynomial Ideals
, 1999
"... In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the ..."
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Cited by 54 (13 self)
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In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the selfconsistent separation of the whole set of variables into two disjoint subsets. They are called multiplicative and nonmultiplicative. Given an admissible ordering, this separation is applied to polynomials in terms of their leading monomials. As special cases of the separation we consider those introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis of partial differential equations. Given involutive division, we define an involutive reduction and an involutive normal form. Then we introduce, in terms of the latter, the concept of involutivity for polynomial systems. We prove that an involutive system is a special, generally redundant, form of a Gröbner basis. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain conditions, for example, those of Janet and Thomas, provide an algorithmic construction of an involutive basis for any polynomial ideal. Some optimization in computation of involutive bases is also analyzed. In particular, we incorporate Buchberger’s chain criterion to avoid unnecessary reductions. The implementation for Pommaret division has been done in Reduce.
On the optimum relaxation factor associated with pcyclic matrices, Linear Algebra Appl . 162/164
, 1992
"... Assume that the matrix coefficient of the nonsingular linear system Ax = b belongs to the class of the generalized consistently ordered ( p 9, 9) matrices, where p and 9 are relatively prime. It is well known that under the additional assumption that the pth powers of the eigenvalues of the Jacobi ..."
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Cited by 2 (2 self)
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, and by Nichols and Fox, the problem has been solved in the nonnegative case for any ( p, 9). In the nonpositive case, in view of the work by Kredell, by Niethammer, de Pillis, and Varga, by Galanis, Hadjidimos, and Noutsos, and by Wild and Niethammer, the corresponding problem seems to be more difficult; it has
Simplification of Quantifierfree Formulas over Ordered Fields
 Journal of Symbolic Computation
, 1995
"... this article is to provide a collection of practicable methods that have been implemented and extensively tested for their relevance. We further show how to combine different ideas for simplification in such a way that a formula is obtained which cannot be further simplified with any of the describe ..."
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Cited by 35 (16 self)
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this article is to provide a collection of practicable methods that have been implemented and extensively tested for their relevance. We further show how to combine different ideas for simplification in such a way that a formula is obtained which cannot be further simplified with any of the described methods. In other words, our simplifiers viewed as a function are idempotent. Achieving this is by no means trivial. On the algorithmic side, we introduce the concept of a background theory that is implicitly enlarged when entering a formula for simplification. Originally developed for detecting interactions between atomic formulas on different Boolean levels, it has turned out that this concept captures also other simplifiers that we had developed some time ago. These simplifiers, namely the Grobner simplifier and the Tableau simplifiers, could even be generalized due to this new viewpoint. 1.1. definitions Our formulas combine atomic formulas using the Boolean connectives "," "," "\Gamma!," "/\Gamma," "/!," and ":." Conjunction and disjunction are not binary but allow an arbitrary number of arguments. The atomic formulas are equations constructed with "=,"
Noncommutative Computer Algebra for polynomial algebras: Gröbner bases, applications and implementation
, 2005
"... ..."
Decoding affine variety codes using Gröbner bases
 CODES CRYPTOGR
, 1998
"... We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the Fqrational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance ..."
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Cited by 21 (0 self)
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We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the Fqrational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance by using the theory of Gröbner bases. We extend results of A. B. Cooper and of X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong.
Implicitization of rational Parametric Equations
 Journal of Symbolic Computation
, 1992
"... Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational parametric equations. (1) Find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) Decide whether the param ..."
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Cited by 21 (6 self)
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Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational parametric equations. (1) Find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) Decide whether the parameters of a set of rational parametric equations are independent. (3) If the parameters of a set of rational parametric equations are not independent, reparameterize the parametric equations so that the new parametric equations have independent parameters. (4) Compute the inversion maps of parametric equations, and as a consequence, give a method to decide whether a set of parametric equations is proper. (5) In the case of algebraic curves, find a proper reparameterization for a set of improper parametric equations. 1
The computation of Gröbner bases on a shared memory multiprocessor
 Proc. DISCO ‘90, Springer LNCS 429
, 1990
"... The computation of Gro╠êbner bases on a shared memory multiprocessor ..."
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Cited by 19 (0 self)
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The computation of Gro╠êbner bases on a shared memory multiprocessor
Results 1  10
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