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On the Definition of Quadratic Mappings

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Abstract

In 1966, Gleason published some interesting results about quadratic mappings. He considered a mapping Q : V → W such that Q(u + v) + Q(u − v) = 2Q(u) + 2Q(v), and he proved that it was \({\mathbb{Z}}\) -quadratic, provided that the multiplication by 2 in W was injective. Then he answered this question negatively: if V and W are vector spaces over a field K, and if Q is \({\mathbb{Z}}\)-quadratic and K-homogeneous of order 2, is it always true that Q is K-quadratic? These considerations and other related topics are here revisited, without the systematic hypothesis about the multiplication by 2.

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Authors and Affiliations

  1. Université de Grenoble I, Institut Fourier, B.P. 74, 38400, Saint-Martin d’Hères, France

    Jacques Helmstetter

  2. Université Montpellier II, 34095, Montpellier, France

    Philippe Revoy

Authors
  1. Jacques Helmstetter
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  2. Artibano Micali
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  3. Philippe Revoy
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Correspondence to Jacques Helmstetter.

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died in Montpellier on December 26, 2011

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Helmstetter, J., Micali, A. & Revoy, P. On the Definition of Quadratic Mappings. Adv. Appl. Clifford Algebras 24, 125–140 (2014). https://doi.org/10.1007/s00006-013-0426-0

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  • DOI: https://doi.org/10.1007/s00006-013-0426-0

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