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Israel Journal of Mathematics

, Volume 72, Issue 1–2, pp 118–132 | Cite as

The algebraic structure of linearly recursive sequences under hadamard product

  • Richard G. Larson
  • Earl J. Taft
Article

Abstract

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field.

Keywords

Hopf Algebra Algebraic Structure Algebra Homomorphism Zero Divisor Hadamard Product 
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References

  1. 1.
    M. F. Atyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969.Google Scholar
  2. 2.
    B. Benzaghou,Algèbres de Hadamard, Bull. Soc. Math. France98 (1970), 209–252.zbMATHMathSciNetGoogle Scholar
  3. 3.
    C. W. Curtis and I. Reiner,Representation Theory of Finite Groups and Associative Algebras, Wiley-Interscience, New York, 1962.zbMATHGoogle Scholar
  4. 4.
    D. S. Passman,Infinite Group Rings, Marcel Dekker, New York, 1971.zbMATHGoogle Scholar
  5. 5.
    B. Peterson and E. J. Taft,The Hopf algebra of linearly recursive sequences, Æquationes Math.20 (1980), 1–17.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. J. van der Poorten,Some facts that should be better known, especially about rational functions, inNumber Theory and Applications (R. A. Mollin, ed.), Kluwer Acad. Publ., Dordrecht, 1989.Google Scholar
  7. 7.
    C. Reutenauer,Sur les éléments inversibles de l’algèbre de Hadamard des séries rationelles, Bull. Soc. Math. France110 (1982), 225–232.zbMATHMathSciNetGoogle Scholar
  8. 8.
    C. Ronse,Feedback Shift Registers, Springer-Verlag, Berlin, 1984.zbMATHGoogle Scholar
  9. 9.
    M. Sweedler,Hopf Algebras, Benjamin, New York, 1969.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1990

Authors and Affiliations

  • Richard G. Larson
    • 1
  • Earl J. Taft
    • 2
  1. 1.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA

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