Applicable Algebra in Engineering, Communication and Computing

, Volume 8, Issue 2, pp 125–134

On Intrinsic Bounds in the Nullstellensatz

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

DOI: 10.1007/s002000050057

Cite this article as:
Krick, T., Sabia, J. & Solernó, P. AAECC (1997) 8: 125. doi:10.1007/s002000050057

Abstract.

 Let k be a field and  f1, . . . ,  fs be non constant polynomials in k[X1, . . . , Xn] which generate the trivial ideal. In this paper we define an invariant associated to the sequence  f1, . . . ,  fs: 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, . . . ,  fs and d :=maxj deg( fj), then there exist polynomials p1, . . . , psk[X1, . . . , Xn] such that 1=∑jpjfj and deg pj  fj≦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 idealstrace theoryeffective Nullstellensatzgeometric degree.

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