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The Ramanujan Journal

, Volume 48, Issue 2, pp 351–355 | Cite as

Holzer’s theorem in k[t]

  • José Luis Leal-RupertoEmail author
  • David B. Leep
Article
  • 63 Downloads

Abstract

Let abc be nonzero polynomials in k[t] where k[t] is the ring of polynomials with coefficients in k. We prove that if \(ax^2 \,{+}\, by^2 \,{+}\, cz^2 = 0\) has a nonzero solution in k[t], then there exist \(x_0,y_0,z_0 \in k[t]\), not all zero, such that \(ax_0^2 \,{+}\, by_0^2 \,{+}\, cz_0^2 = 0\) and \(\deg x_0 \le \frac{1}{2}(\deg b \,{+}\, \deg c)\), \(\deg y_0 \le \frac{1}{2}(\deg a \,{+}\, \deg c)\), and \(\deg z_0 \le \frac{1}{2}(\deg a \,{+}\, \deg b)\). This is the polynomial analogue of Holzer’s theorem for \(ax^2 \,{+}\, by^2 \,{+}\, cz^2 = 0\) when abc are integers.

Keywords

Legendre equation Rational polynomial ring Holzer’s theorem 

Mathematics Subject Classification

11D 11G 

References

  1. 1.
    Holzer, L.: Minimal solutions of Diophantine equations. Can. J. Math. 2, 238–244 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Mordell, L.J.: On the magnitude of the integer solutions of the equation \(ax^2 +by^2 +cz^2 =0\). J. Number Theory 1, 1–3 (1969)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Prestel, A.: On the size of zeros of quadratic forms over rational function fields. J. Reine Angew Math. 378, 101–112 (1987)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, Escuela Politécnica SuperiorUniversidad de MálagaMálagaSpain
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA

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