Algorithms for the Quillen-Suslin theorem

Logar, Alessandro; Sturmfels, Bernd

doi:10.1016/0021-8693(92)90189-s

Cited by 76 publications

(56 citation statements)

References 4 publications

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“…In conclusion, the algorithm for checking local freeness of constant dimension is unsuitable for checking local freeness, or equivalently, it could not be used to check if a given module is projective. Compare with the comment at the end of [16], and see also the comment in [13] after Theorem 1.1.…”

Section: Projective Dimension Of a Modulementioning

confidence: 82%

Testing flatness and computing rank of a module using syzygies

Lezama

2009

Colloq. Math.

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Abstract. Using syzygies computed via Gröbner bases techniques, we present algorithms for testing some homological properties for submodules of the free module A m , where A = R[x1, . . . , xn] and R is a Noetherian commutative ring. We will test if a given submodule M of A m is flat. We will also check if M is locally free of constant dimension. Moreover, we present an algorithm that computes the rank of a flat submodule M of A m and also an algorithm that computes the projective dimension of an arbitrary submodule of A m . All algorithms are illustrated with examples.

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Section: Projective Dimension Of a Modulementioning

confidence: 82%

Testing flatness and computing rank of a module using syzygies

Lezama

2009

Colloq. Math.

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“…Conversely, if a~is a dual mask of a, setting It is well known that the invertible matrix (a~| = (z)) |, = # 0 M is used in deriving the associated dual wavelet masks from a and a~and such matrix can be constructed from the mask a by an algorithm proposed in [33]. With the help of Theorem 3.1 and Proposition 3.3, we demonstrate the following (obvious?)…”

Section: Bin Hanmentioning

confidence: 91%

“…Since a is a primal mask, by Quillen Suslin Theorem, there exist sequences a~| (| # 0 M ) in (l 0 (Z s )) r_r such that the conditions in Proposition 3.3 are satisfied. Such sequences a~| can be constructed from a by an algorithm proposed in [33]. Let a~be a sequence in (l 0 (Z s )) r_r given by a~|(z) := :…”

Section: Bin Hanmentioning

confidence: 99%

Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets

Han

2001

Journal of Approximation Theory

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Multiwavelets are generated from refinable function vectors by using multiresolution analysis. In this paper we investigate the approximation properties of a multivariate refinable function vector associated with a general dilation matrix in terms of both the subdivision operator and the order of sum rules satisfied by the matrix refinement mask. Based on a fact about the sum rules of biorthogonal multiwavelets, a coset by coset (CBC) algorithm is presented to construct biorthogonal multiwavelets with arbitrary order of vanishing moments. More precisely, to obtain biorthogonal multiwavelets, we have to construct primal and dual masks. Given any primal matrix mask a and a general dilation matrix M, the proposed CBC algorithm reduces the construction of all dual masks of a, which satisfy the sum rules of arbitrary order, to a problem of solving a well organized system of linear equations. We prove in a constructive way that for any given primal mask a with a dilation matrix M and for any positive integer k, we can always construct a dual mask aõ f a such that a~satisfies the sum rules of order k. In addition, we provide a general way for the construction of Hermite interpolatory matrix masks in the univariate setting with any dilation factors. From such Hermite interpolatory masks, smooth Hermite interpolants, including the well known cubic Hermite splines as a special case, are obtained and are used to construct biorthogonal multiwavelets. As an example, a C 3 Hermite interpolant with support [&3, 3] is presented. Then we shall apply the CBC algorithm to such Hermite interpolatory masks to construct biorthogonal multiwavelets. Several examples of biorthogonal multiwavelets are provided to illustrate the general theory. In particular, a C 1 dual function vector with support [&4, 4] of the cubic Hermite splines is given. Academic Press

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“…Serre's conjecture was proven independently by Quillen and Suslin in 1976, and is now known as the Quillen-Suslin Theorem. A number of algorithms for this theorem have been presented (Fitchas and Galligo, 1990;Logar and Sturmfels, 1992;Fitchas, 1993).…”

Section: Introductionmentioning

confidence: 96%

“…All algebras discussed are either polynomial rings over a field, or quotients of these polynomial rings, which allows us to apply the theory of Gröbner bases for all necessary calculations (Adams and Loustaunau, 1994;Cox et al, 1997). As a subalgorithm we use the determination of a free basis for projective modules over k [x], such as the one provided by Logar and Sturmfels (1992).…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

Laubenbacher

Schlauch

2000

Journal of Symbolic Computation

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This paper presents an algorithm for the Quillen-Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k[x 0 , . . . , xn]/I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P , described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis for P .

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Algorithms for the Quillen-Suslin theorem

Cited by 76 publications

References 4 publications

Testing flatness and computing rank of a module using syzygies

Testing flatness and computing rank of a module using syzygies

Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

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