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Results in Mathematics

, 55:383 | Cite as

Polynomial Parametrization of the Solutions of Certain Systems of Diophantine Equations

  • Franz Halter-KochEmail author
  • Günter Lettl
Article
  • 44 Downloads

Abstract.

Let \(f_1, f_2, \ldots , f_k \in {\mathbb {Z}}[X_0, X_1, \ldots , X_N]\) be non-constant homogeneous polynomials which define a projective variety V over \(\mathbb {Q}\). Under the hypothesis that, for some \(n \in \mathbb {N}\), there is a surjective morphism \(\varphi: \mathbb {P}^n_\mathbb {Q} \rightarrow V\), we show that all integral solutions of the system of Diophantine equations f1 = 0, . . . , f k  = 0 (outside some exceptional set) can be parametrized by a single k-tuple of integer-valued polynomials. This result only depends on φ, but not on the embedding given by f1, f2, . . . , f k . If, in particular, φ is a normalization of V, then the exceptional set is really small.

Mathematics Subject Classification (2000).

Primary 11D85 secondary 14G05, 13F20, 14M20, 11D41 

Keywords.

Integer-valued polynomials rational variety 

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Institut für Mathematik und wissenschaftliches RechnenKarl-Franzens-UniversitätGrazAustria

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