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Sums of squares in nonreal commutative rings

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Abstract

We investigate elements in a commutative ring that can be written as a sum of squares and we study invariants that measure how many squares are needed in such a representation. We focus on rings where −1 is a sum of squares. For such a ring R, we define the metalevel s m (R) and the hyperlevel s h (R), and we relate these to the classical level s(R) and the Pythagoras number p(R) of the ring. Among many results, we prove that s m (R) ≤ s(R) ≤ p(R) ≤ s h (R) + 1 ≤ s m (R) + 2. We also study generic rings that realize prescribed values for the various levels.

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Dortmund, D-44221, Dortmund, Germany

    Detlev W. Hoffmann

  2. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    David B. Leep

Authors
  1. Detlev W. Hoffmann
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  2. David B. Leep
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Correspondence to Detlev W. Hoffmann.

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Hoffmann, D.W., Leep, D.B. Sums of squares in nonreal commutative rings. Isr. J. Math. 221, 803–835 (2017). https://doi.org/10.1007/s11856-017-1575-y

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  • DOI: https://doi.org/10.1007/s11856-017-1575-y

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