Sums of squares in nonreal commutative rings

Detlev W. Hoffmann; David B. Leep

doi:10.1007/s11856-017-1575-y

Sums of squares in nonreal commutative rings

Detlev W. Hoffmann, David B. Leep

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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.

Original language	English
Pages (from-to)	803-835
Number of pages	33
Journal	Israel Journal of Mathematics
Volume	221
Issue number	2
DOIs	https://doi.org/10.1007/s11856-017-1575-y
State	Published - Sep 1 2017

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Publisher Copyright:
© 2017, Hebrew University of Jerusalem.

ASJC Scopus subject areas

General Mathematics

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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 sm(R) and the hyperlevel sh(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 sm(R) ≤ s(R) ≤ p(R) ≤ sh(R) + 1 ≤ sm(R) + 2. We also study generic rings that realize prescribed values for the various levels.",

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AB - 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 sm(R) and the hyperlevel sh(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 sm(R) ≤ s(R) ≤ p(R) ≤ sh(R) + 1 ≤ sm(R) + 2. We also study generic rings that realize prescribed values for the various levels.

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Sums of squares in nonreal commutative rings

Abstract

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ASJC Scopus subject areas

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