Computing Valuation Popov Forms

Giesbrecht, Mark; Labahn, George; Zhang, Yang

doi:10.1007/11428862_84

Cited by 3 publications

(1 citation statement)

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“…The problem of defining and computing Popov forms over non-commutative valuation domains such as rings of Ore polynomials was considered in Ref. 9, but efficient computation of Popov forms is not considered. In practice, row reductions can introduce significant coefficient growth which must be controlled.…”

Section: Introductionmentioning

confidence: 99%

Modular Computation for Matrices of Ore Polynomials

Cheng

Labahn²,

Cheriton³

2007

Computer Algebra 2006

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We give a modular algorithm to perform row reduction of a matrix of Ore polynomials with coefficients in Z [t]. Both the transformation matrix and the transformed matrix are computed. The algorithm can be used for finding the rank and left nullspace of such matrices. In the special case of shift polynomials, we obtain algorithms for computing a weak Popov form and for computing a greatest common right divisor (GCRD) and a least common left multiple (LCLM) of matrices of shift polynomials. Our algorithms improve on existing fraction-free algorithms and can be viewed as generalizations of the work of Li and Nemes on GCRDs and LCLMs of Ore polynomials. We define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. Our algorithm is output-sensitive, such that the number of homomorphic images required depends on the size of the output. Furthermore, there is no need to verify the result by trial division or multiplication. When our algorithm is used to compute a GCRD and a LCLM of shift polynomials, we obtain a new output-sensitive modular algorithm.

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Section: Introductionmentioning

confidence: 99%