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Sums of values represented by a quadratic form

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Abstract

Let q be a quadratic form over a field K of characteristic different from 2. We investigate the properties of the smallest positive integer n such that −1 is a sum of n values represented by q in several situations. We relate this invariant (which is called the q-level of K) to other invariants of K such as the level, the u-invariant and the Pythagoras number of K. The problem of determining the numbers which can be realized as a q-level for particular q or K is studied. We also observe that the q-level naturally emerges when one tries to obtain a lower bound for the index of the subgroup of non-zero values represented by a Pfister form q. We highlight necessary and/or sufficient conditions for the q-level to be finite. Throughout the paper, special emphasis is given to the case where q is a Pfister form.

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Authors and Affiliations

  1. Institut Fourier, 100 rue des Maths, BP 74, 38402, Saint Martin d’Héres Cedex, France

    G. Berhuy

  2. Laboratoire de Didactique André Revuz, LDAR (EA 1547), Université Paris Diderot, Paris, France

    N. Grenier-Boley

  3. Université de Rouen, Rouen, France

    N. Grenier-Boley

  4. Department of Mathematical Sciences, Sharif University of Technology, P.O.Box 11155-9415, Tehran, Iran

    M. G. Mahmoudi

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  1. G. Berhuy
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  2. N. Grenier-Boley
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  3. M. G. Mahmoudi
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Correspondence to N. Grenier-Boley.

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Berhuy, G., Grenier-Boley, N. & Mahmoudi, M.G. Sums of values represented by a quadratic form. manuscripta math. 140, 531–556 (2013). https://doi.org/10.1007/s00229-012-0551-4

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