On Intrinsic Bounds in the Nullstellensatz

  • T. Krick
  • J. Sabia
  • P. Solernó

Abstract.

 Let k be a field and  f1, . . . ,  f s be non constant polynomials in k[X1, . . . , X n ] which generate the trivial ideal. In this paper we define an invariant associated to the sequence  f1, . . . ,  f s : the geometric degree of the system. With this notion we can show the following effective Nullstellensatz: if δ denotes the geometric degree of the trivial system  f1, . . . ,  f s and d :=max j  deg( f j ), then there exist polynomials p1, . . . , p s k[X1, . . . , X n ] such that 1=∑ j p j f j and deg p j   f j ≦3n2δd. Since the number δ is always bounded by (d+1) n-1 , one deduces a classical single exponential upper bound in terms of d and n, but in some cases our new bound improves the known ones.

Keywords: complete intersection polynomial ideals trace theory effective Nullstellensatz geometric degree. 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • T. Krick
    • 1
  • J. Sabia
    • 1
  • P. Solernó
    • 2
  1. 1.Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, -1428- Buenos Aires, Argentina (e-mail: krick@dm.uba.ar/jsabia@dm.uba.ar)AR
  2. 2.Departamento de Economía y Matemática, Universidad de San Andrés, Vito Dumas 284, -1644- Victoria, Buenos Aires, Argentina (e-mail: psolerno@udesa.edu.ar)AR

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