Smith normal form of a matrix of generalized polynomials with rational exponents

Miroslav Kureš; Ladislav Skula

Annales UMCS, Mathematica (2008)

  • Volume: 62, Issue: 1, page 81-90
  • ISSN: 2083-7402

Abstract

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It is proved that generalized polynomials with rational exponents over a commutative field form an elementary divisor ring; an algorithm for computing the Smith normal form is derived and implemented.

How to cite

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Miroslav Kureš, and Ladislav Skula. "Smith normal form of a matrix of generalized polynomials with rational exponents." Annales UMCS, Mathematica 62.1 (2008): 81-90. <http://eudml.org/doc/267665>.

@article{MiroslavKureš2008,
abstract = {It is proved that generalized polynomials with rational exponents over a commutative field form an elementary divisor ring; an algorithm for computing the Smith normal form is derived and implemented.},
author = {Miroslav Kureš, Ladislav Skula},
journal = {Annales UMCS, Mathematica},
keywords = {Smith normal form; generalized polynomials with rational exponents; elementary divisor ring; algorithm},
language = {eng},
number = {1},
pages = {81-90},
title = {Smith normal form of a matrix of generalized polynomials with rational exponents},
url = {http://eudml.org/doc/267665},
volume = {62},
year = {2008},
}

TY - JOUR
AU - Miroslav Kureš
AU - Ladislav Skula
TI - Smith normal form of a matrix of generalized polynomials with rational exponents
JO - Annales UMCS, Mathematica
PY - 2008
VL - 62
IS - 1
SP - 81
EP - 90
AB - It is proved that generalized polynomials with rational exponents over a commutative field form an elementary divisor ring; an algorithm for computing the Smith normal form is derived and implemented.
LA - eng
KW - Smith normal form; generalized polynomials with rational exponents; elementary divisor ring; algorithm
UR - http://eudml.org/doc/267665
ER -

References

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  1. Brown, W. C., Matrices over Commutative Rings, Marcel Dekker, 1993.[WoS] Zbl0782.15001
  2. Cohen, H., Hermite and Smith normal form algorithms over Dedekind domains, Math. Comp. 65 (1996), 1681-1699. Zbl0853.11100
  3. Cohn, P. M., On the structure of GL2 of a ring, Publications Mathématiques de l'I. H.É. S. 30 (1966), 5-53. 
  4. Delzell, C. N., Extension of Pólya's theorems to signomials with rational exponents, Preprint available on 
  5. Gantmacher, F. R., The Theory of Matrices, Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. Zbl0927.15002
  6. Gillman, L., Henriksen, M., Some remarks about elementary divisor rings, Trans. Am. Math. Soc. 82 (1956), 362-365. Zbl0073.02203
  7. Helmer, O., The elementary divisor theorem for certain rings without chain condition, Bull. Am. Math. Soc. 49 (1943), 225-236. Zbl0060.07606
  8. Kannan, A., Bachem, A., Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix, SIAM J. Comput. 8 (1979), 499-507. Zbl0446.65015
  9. Kaplansky, I., Elementary divisors and modules, Trans. Am. Math. Soc. 66 (1949), 464-491. Zbl0036.01903
  10. Karásek, J., Šlapal, J., Polynomials and Generalized Polynomials in Control Theory, (Czech), CERM Academic Publishing Brno, 2007. 
  11. Kureš, M., On the computation of Smith normal form of a matrix of generalized polynomials with rational exponents, in: Aplimat 2007, Proceedings of the International Conference Aplimat, Bratislava, Slovakia, February 6-9, 2007, Slovak University of Technology, 2007, pp. 103-108. 
  12. Pascolletti, A.: SmithForm.m, version 1.0 (2005). The Mathematica package available on 
  13. Suslin, A. A., On the structure of the special linear group over polynomial rings, Math. USSR, Izv. 11 (1977), 221-238. Zbl0378.13002
  14. Villard, G., Computation of the Smith normal form of polynomial matrices, in: ISSAC '93, Proceedings of the 1993 international symposium on Symbolic and algebraic computation, Kiev, Ukraine, July 6-8, 1993, ACM Press Baltimore, 1993, pp. 209-217. Zbl0964.65507

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