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Mathematische Zeitschrift

, Volume 231, Issue 4, pp 679–706 | Cite as

On the degrees of bases of free modulesover a polynomial ring

  • Marcela Almeida
  • Lisi D'Alfonso
  • Pablo Solernó

Abstract.

Let k be an infinite field, A the polynomial ring \(k[x_1,...,x_n]\) and \(F\in A^{N\times M}\) a matrix such that \({\rm Im}\,F\subset A^N\) is A-free (in particular, Quillen-Suslin Theorem implies that \({\rm Ker}\,F\) is also free). Let D be the maximum of the degrees of the entries of F and s the rank of F. We show that there exists a basis \(\{ v_1,\ldots,v_{M} \}\) of \(A ^M\) such that \(\{ v_1,\ldots ,v_{M-s} \}\) is a basis of \({\rm Ker}\,F\), \(\{ F(v_{M-s+1}), \ldots , F(v_M) \}\) is a basis of \({\rm Im}\,F\) and the degrees of their coordinates are of order \(((M-s)sD)^{O(n^4)}\).

This result allows to obtain a single exponential degree upper bound for a basis of the coordinate ring of a reduced complete intersection variety in Noether position.

Keywords

Complete Intersection Intersection Variety Polynomial Ring Coordinate Ring Infinite Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Marcela Almeida
    • 1
  • Lisi D'Alfonso
    • 1
  • Pablo Solernó
    • 2
  1. 1. Departamento de Matemática. Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina (e-mails: malmeida@dm.uba.ar, lisi@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@dm.uba.ar) AR

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