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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pfister's theorem fails in the Hermitian case


Authors: John P. D’Angelo and Jiří Lebl
Journal: Proc. Amer. Math. Soc. 140 (2012), 1151-1157
MSC (2010): Primary 12D15, 14P05, 15B57, 32V15
Published electronically: April 1, 2011
MathSciNet review: 2869101
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Abstract: We show that the Hermitian analogue of a famous result of Pfister fails. To do so we provide a Hermitian symmetric polynomial $ r$ of total degree $ 2d$ such that any nonzero multiple of it cannot be written as a Hermitian sum of squares with fewer than $ d+1$ squares.


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Additional Information

John P. D’Angelo
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Email: jpda@math.uiuc.edu

Jiří Lebl
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112
Email: jlebl@math.uiuc.edu, jlebl@math.ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10841-4
PII: S 0002-9939(2011)10841-4
Keywords: Hilbert’s $17$-th problem, Hermitian forms, sums of squares, Hermitian length, Huang lemma.
Received by editor(s): July 6, 2010
Received by editor(s) in revised form: October 8, 2010, and December 22, 2010
Published electronically: April 1, 2011
Communicated by: Franc Forstneric
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.