Skip to main content
SpringerLink
Log in

Applications of the Newton Index to the Construction of Irreducible Polynomials

  • Conference paper
Computer Algebra in Scientific Computing (CASC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8660))

Included in the following conference series:

  • 965 Accesses

Abstract

We use properties of the Newton index associated to a polynomial with coefficients in a discrete valuation domain for generating classes of irreducible polynomials. We obtain factorization properties similar to the case of bivariate polynomials and we give new applications to the construction of families of irreducible polynomials over various discrete valuation domains. The examples are obtained using the package gp-pari.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Angermüller, G.: A generalization of Ehrenfeucht’s irreducibility criterion. J. Number Theory 36, 80–84 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bhatia, S., Khanduja, S.K.: Difference polynomials and their generalizations. Mathematika 48, 293–299 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bishnoi, A., Khanduja, S.K., Sudesh, K.: Some extensions and applications of the Eisenstein irreducibility criterion. Developments in Mathematics 18, 189–197 (2010)

    Article  Google Scholar 

  4. Bonciocat, A.I., Bonciocat, N.C.: Some classes of irreducible polynomials. Acta Arith. 123, 349–360 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bonciocat, N.C.: A Capelli type theorem for multiplicative convolutions of polynomials. Math. Nachr. 281, 1240–1253 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bonciocat, N.C.: On an irreducibility criterion of Perron for multivariate polynomials. Bull. Math. Soc. Sci. Math. Roumanie 53(101), 213–217 (2010)

    MathSciNet  Google Scholar 

  7. Bonciocat, N.C.: Schönemann-Eisenstein-Dumas-type irreducibility conditions that use arbitrarily many prime numbers. arXiv:1304.0874v1

    Google Scholar 

  8. Bonciocat, N.C., Bugeaud, Y., Cipu, M., Mignotte, M.: Irreducibility criteria for sums of two relatively prime polynomials. Int. J. Number Theory 9, 1529–1539 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bonciocat, N.C., Zaharescu, A.: Irreducible multivariate polynomials obtained from polynomials in fewer variables. J. Pure Appl. Algebra 212, 2338–2343 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dumas, G.: Sur quelques cas d’irréducibilité des polynômes à coefficients rationnels. Journal de Math. Pures et Appl. 12, 191–258 (1906)

    Google Scholar 

  11. Eisenstein, G.: Über die Irreductibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt. J. Reine Angew. Math. 39, 160–182 (1850)

    Article  MATH  Google Scholar 

  12. Lipkovski, A.: Newton Polyhedra and Irreducibility. Math. Z. 199, 119–128 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ore, O.: Zur Theorie der Eisensteinschen Gleichungen. Math. Z. 20, 267–279 (1924)

    Article  MATH  MathSciNet  Google Scholar 

  14. Panaitopol, L.D., Ştefănescu, D.: On the generalized difference polynomials. Pacific J. Math. 143, 341–348 (1990)

    Google Scholar 

  15. Rubel, L.A., Schinzel, A., Tverberg, H.: On difference polynomials and hereditary irreducible polynomials. J. Number Theory 12, 230–235 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  16. Schönemann, T.: Von denjenigen Moduln, welche Potenzen von Primzahlen sind. J. Reine Angew. Math. 32, 93–105 (1846)

    Google Scholar 

  17. Ştefănescu, D.: Construction of classes of irreducible bivariate polynomials. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 393–400. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  18. Ştefănescu, D.: On the irreducibility of bivariate polynomials. Bull. Math. Soc. Sci. Math. Roumanie 56(104), 377–384 (2013)

    Google Scholar 

  19. Zaharescu, A.: Residual transcendental extentions of valuations, irreducible polynomials and trace series over p–adic fields. Bull. Math. Soc. Sci. Math. Roumanie 56(104), 125–131 (2013)

    MathSciNet  Google Scholar 

  20. Weintraub, S.H.: A mild generalization of Eisenstein’s criterion. Proc. Amer. Math. Soc. 141, 1159–1160 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Bucharest, Romania

    Doru Ştefănescu

Authors
  1. Doru Ştefănescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research, 141980, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, 34132, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Ştefănescu, D. (2014). Applications of the Newton Index to the Construction of Irreducible Polynomials. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10515-4_33

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10514-7

  • Online ISBN: 978-3-319-10515-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation