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Square-free criteria for polynomials using no derivatives


Authors: E. Alkan, A. I. Bonciocat, N. C. Bonciocat and A. Zaharescu
Journal: Proc. Amer. Math. Soc. 135 (2007), 677-687
MSC (2000): Primary 11C08, 11C20
DOI: https://doi.org/10.1090/S0002-9939-06-08526-1
Published electronically: September 11, 2006
MathSciNet review: 2262863
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Abstract | References | Similar Articles | Additional Information

Abstract: We provide some square-free criteria for primitive polynomials over unique factorization domains, which do not make use of derivatives or discriminants. Using some ideas of Ostrowski we establish nonvanishing conditions for determinants of matrices with polynomial entries and deduce square-free criteria for polynomials in several variables.


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  • 1. F. DeMeyer, Separable polynomials over a commutative ring, Rocky Mountain J. Math. 2 (1972), no. 2, 299-310. MR 0294321 (45:3390)
  • 2. F. DeMeyer, Twisted forms of finite étale extensions and separable polynomials, Int. J. Math. Math. Sci. 26 (2001), no. 8, 467-473. MR 1851100 (2002f:13022)
  • 3. J. Hadamard, Leçons sur la propagation des ondes, Paris, 1903, 13-14.
  • 4. D. K. Harrison and T. Mckenzie, Towards an arithmetic of polynomials, Aequationes Math. 43 (1992), no. 1, 21-37. MR 1144586 (92m:13008)
  • 5. G. J. Janusz, Separable algebras over commutative rings, Trans. Amer. Math. Soc. 122 (1966), 461-479. MR 0210699 (35:1585)
  • 6. A. I. Kostrikin, Introduction to Algebra (translated from Russian) Springer-Verlag, 1982. MR 0661256 (83f:00003)
  • 7. T. McKenzie, The separable closure of a local ring, J. Algebra 207 (1998), no. 2, 657-663. MR 1644231 (99g:13006)
  • 8. T. Nagahara, On separable polynomials over a commutative ring, Math. J. Okayama Univ. 14 (1969/70), 175-181. MR 0289495 (44:6684)
  • 9. T. Nagahara, Characterization of separable polynomials over a commutative ring, Proc. Japan Acad. 46 (1970), 1011-1015. MR 0284432 (44:1659)
  • 10. A. M. Ostrowski, Mathematische Miszellen. XXIV. Zur relativen Stetigkeit von Wurzeln algebraischer Gleichungen, Jahresber. Deutsch. Math.-Verein. 58 (1956), Abt.1, 98-102. MR 0078327 (17:1175g)
  • 11. A. M. Ostrowski, On some conditions for nonvanishing of determinants, Proc. Amer. Math. Soc. 12 (1961), 268-273. MR 0137719 (25:1168)
  • 12. A. M. Ostrowski, Sur les conditions générales pour la régularité des matrices, Rend. Mat. e Appl. ser. V, vol. X (1951), 156-168. MR 0049151 (14:125g)
  • 13. A. M. Ostrowski, Ueber das Nichtverschwinden einer Klasse von Determinanten und die Lokalisierung der characteristischen Wurzeln von Matrizen, Compositio Math. 9 (1951), 209-226. MR 0045081 (13:524b)
  • 14. A. M. Ostrowski, Ueber Determinanten mit überwiegender Hauptdiagonale, Comm. Math. Helv., Bd. 10 (1937), 69-96. MR 1509568
  • 15. O. Tausski-Todd, A recurring theorem on determinants, Amer. Math. Monthly 56 (1949), 672-676. MR 0032557 (11:307b)
  • 16. R. J. Walker, Algebraic Curves, Princeton University Press, 1950. MR 0033083 (11:387e)

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Additional Information

E. Alkan
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, Illinois 61801
Email: alkan@math.uiuc.edu

A. I. Bonciocat
Affiliation: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 014700, Romania
Email: Anca.Bonciocat@imar.ro

N. C. Bonciocat
Affiliation: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 014700, Romania
Email: Nicolae.Bonciocat@imar.ro

A. Zaharescu
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, Illinois 61801
Email: zaharesc@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08526-1
Keywords: Square-free criteria, primitive polynomials, resultants, Frobenius map.
Received by editor(s): September 12, 2005
Received by editor(s) in revised form: October 10, 2005
Published electronically: September 11, 2006
Additional Notes: This research was partially supported by the CERES Programs 4-147 and 4-187/2004 of the Romanian Ministry of Education and Research.
Communicated by: Ken Ono
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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