Monomial representations for Gröbner bases computations

Bachmann, Olaf; Schönemann, Hans

doi:10.1145/281508.281657

Cited by 8 publications

(7 citation statements)

References 6 publications

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“…In our implementation we represent monomials as single machine integers (which allows us to compare and multiply monomials in one machine instruction). This representation, analyzed by Monagan and Pearce [6], is based on Bachmann and Schönemann's scheme [1]. The C-structure we are using to represent a lazy polynomial is given below.…”

Section: Methodsmentioning

confidence: 99%

Lazy and Forgetful Polynomial Arithmetic and Applications

Monagan

Vrbik

2009

Computer Algebra in Scientific Computing

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Abstract. We present lazy and forgetful algorithms for multiplying and dividing multivariate polynomials. The lazy property allows us to compute the i-th term of a polynomial without doing the work required to compute all the terms. The forgetful property allows us to forget earlier terms that have been computed to save space. For example, given polynomials A, B, C, D, E we can compute the exact quotient Q = A×B−C×D E without explicitly computing the numerator A × B − C × D which can be much larger than any of A, B, C, D, E and Q. As applications we apply our lazy and forgetful algorithms to reduce the maximum space needed by the Bareiss fraction-free algorithm for computing the determinant of a matrix of polynomials and the extended Subresultant algorithm for computing the inverse of an element in a polynomial quotient ring.

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Section: Methodsmentioning

confidence: 99%

Lazy and Forgetful Polynomial Arithmetic and Applications

Monagan

Vrbik

2009

Computer Algebra in Scientific Computing

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“…Some lessons to learn from are the data representation ( [5]) and fast additions ( [23], [22]). Furthermore, the memory management of Singular is optimized to handle many very small blocks of the same size (monomials) efficiently with a high locality of reference.…”

Section: Singular Frameworkmentioning

confidence: 99%

Plural

Levandovskyy

Schönemann

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

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Singular is a computer algebra system developed for efficient computations with polynomials. We describe Plural as an extension of Singular to noncommutative polynomial rings (G-/GR-algebras): to which structures does it apply, the prerequisites to monomial orderings, left-and two-sided Gröbner bases. The usual criteria to avoid "useless pairs" are revisited for their applicability in the case of G-/GRalgebras. Benchmark tests are used to evaluate the concepts compare them with other systems.

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“…The monomial overhead can be largely eliminated by packing multiple exponents into a machine word so that monomial comparisons and multiplications are reduced to machine integer comparisons and additions, which cost 1 cycle. This is done in the older ALTRAN system [9], and in Singular for Gröbner basis computations [1]. Our monomial packing is described in Section 5.1.…”

Section: Introductionmentioning

confidence: 99%

Sparse polynomial division using a heap

Monagan

Pearce

2011

Journal of Symbolic Computation

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In 1974, Johnson showed how to multiply and divide sparse polynomials using a binary heap. This paper introduces a new algorithm that uses a heap to divide with the same complexity as multiplication. It is a fraction-free method that also reduces the number of integer operations for divisions of polynomials with integer coefficients over the rationals. Heap-based algorithms use very little memory and do not generate garbage. They can run in the cpu cache and achieve high performance. We compare our C implementation of sparse polynomial multiplication and division with integer coefficients to the routines of the Magma, Maple, Pari, Singular and Trip computer algebra systems.

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Monomial representations for Gröbner bases computations

Cited by 8 publications

References 6 publications

Lazy and Forgetful Polynomial Arithmetic and Applications

Lazy and Forgetful Polynomial Arithmetic and Applications

Plural

Sparse polynomial division using a heap

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