Collectanea Mathematica

, Volume 68, Issue 3, pp 419–432 | Cite as

Local Bézout theorem for Henselian rings

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Abstract

In this paper we prove what we call Local Bézout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and \(\mathfrak {m}\)-adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities. Our main tools are, first the border bases [11], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [7], for which there exists a constructive proof in [13]. Using these tools we find a very simple proof of our theorem, which seems new in the classical literature.

Keywords

Local Bézout Theorem Henselian rings Roots continuity Stable computations Constructive Algebra 

Mathematics Subject Classification

Primary 13J15 Secondary 13P10 13P15 14Q20 03F65 

References

  1. 1.
    Alonso, M. E., Brachat, J., Mourrain, B.: Stable Deformation of Zero-Dimensional Quotient Algebras, Technical report (2009)Google Scholar
  2. 2.
    Alonso, M.E., Coquand, T., Lombardi, H.: Revisiting Zariski main theorem from a constructive point of view. J. Algebra 406, 46–68 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alonso, M.E., Lombardi, H.: Local Bézout theorem. J. Symb. Comput. 45(10), 975–985 (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps, Volume 1. Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by V. I. Arnold. Boston, MA: Birkhäuser, reprint of the 1985 hardback edition (2012)Google Scholar
  5. 5.
    Bishop, Errett: Foundations of constructive analysis. McGraw-Hill Book Co, New York (1967)MATHGoogle Scholar
  6. 6.
    Brachat, J.: Schémas de Hilbert et Décomposition de tenseurs. PhD thesis, (2011)Google Scholar
  7. 7.
    de Smit, B., Lenstra, H.W.: Finite complete intersection algebras and the completeness radical. J. Algebra 196(2), 520–531 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Eisermann, M.: The fundamental theorem of algebra made effective: an elementary real-algebraic proof via Sturm chains. Am. Math. Monthly 119(9), 715–752 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Griffiths, Phillip, Harris, Joseph: Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York (1994). Reprint of the 1978 originalCrossRefMATHGoogle Scholar
  10. 10.
    Lombardi, H., Quitté, C.: Commutative Algebra. Constructive Methods. Algebra and Applications, Vol. 20. Springer, Berlin (2015)CrossRefMATHGoogle Scholar
  11. 11.
    Mourrain, B.: A new criterion for normal form algorithms. In: Applied algebra, algebraic algorithms and error correcting codes. 13th international symposium, AAECC-13, Honolulu, HI, USA, November 15–19, 1999. Proceedings, pp. 430–443. Berlin: Springer (1999)Google Scholar
  12. 12.
    Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, Third edition of ıt Solution of equations and systems of equations, Pure and Applied Mathematics, Vol. 9 (1973)Google Scholar
  13. 13.
    Quitté, C., and Lombardi, H.: Le théorème de de Smit et Lenstra, démonstration élémentaire. http://arxiv.org/abs/1508.05589 (2015)
  14. 14.
    Richman, F.: The fundamental theorem of algebra: a constructive development without choice. Pacific J. Math. 196(1), 213–230 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2016

Authors and Affiliations

  1. 1.Universidad ComplutenseMadridSpain
  2. 2.University de Franche-ComtéBesançon CedexFrance

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