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A functional equation related to the product in a quadratic number field

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Abstract

The functional equation

$$f(x_{1},y_{1})f(x_{2},y_{2})=f(x_{1}x_{2}+\alpha y_{1}y_{2},x_{1}y_{2}+x_{2}y_{1}),\ (x_{1},y_{1}),\,(x_{2},y_{2})\in \mathbb{ R}^{2}$$

arises from the formula for the product of two numbers in the quadratic field \({\mathbb{Q}(\sqrt{\alpha})}\). The general solution \({f:\mathbb{R}\rightarrow \mathbb{R}}\) to this equation is determined. Moreover, it is shown that no more general equations arise from a change of basis in the field.

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References

  1. Aczél, J., Dhombres, J.: Functional Equations in Several Variables (with applications to mathematics, information theory and to the natural and social sciences). In: Encyclopaedia of Mathematics and its Applications, vol. 31, Cambridge Univ. Press, Cambridge (1989)

  2. Leonardo of Pisa, The Book of Squares. An annotated translation of the Liber Quadratorum into modern English by L. E. Siegler. Academic Press, London (1987)

  3. Jung S.-m., Bae J.-H.: Some functional equations originating from number theory. Proc. Indian Acad. Sci. (Math. Sci.) 113(2), 91–98 (2003)

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Authors and Affiliations

  1. Laboratorio de Acústica y Electroacústica, Facultad de Ciencias Exactas, Ing. y Agrim., Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Rosario, Riobamba 245 bis, 2000, Rosario, Argentina

    Lucio R. Berrone

  2. Departament d’Algebra i Geometria, Universitat de Barcelona, G. V. de les Corts Catalanes 585, 08007, Barcelona, Spain

    Luis V. Dieulefait

Authors
  1. Lucio R. Berrone
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  2. Luis V. Dieulefait
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Correspondence to Luis V. Dieulefait.

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Berrone, L.R., Dieulefait, L.V. A functional equation related to the product in a quadratic number field. Aequat. Math. 81, 167–175 (2011). https://doi.org/10.1007/s00010-010-0049-7

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  • DOI: https://doi.org/10.1007/s00010-010-0049-7

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