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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sums of squares of regular functions
on real algebraic varieties


Author: Claus Scheiderer
Journal: Trans. Amer. Math. Soc. 352 (2000), 1039-1069
MSC (1991): Primary 14P99; Secondary 11E25, 12D15, 13H05, 14G30, 14H99, 14J99
Published electronically: September 8, 1999
MathSciNet review: 1675230
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Abstract: Let $V$ be an affine algebraic variety over $\mathbb{R}$ (or any other real closed field $R$). We ask when it is true that every positive semidefinite (psd) polynomial function on $V$ is a sum of squares (sos). We show that for $\dim V\ge 3$ the answer is always negative if $V$ has a real point. Also, if $V$ is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if $V$ is a smooth surface with only real divisors at infinity. The ``compact'' case is harder. We completely settle the case of smooth curves of genus $\le 1$: If such a curve has a complex point at infinity, then every psd function is sos, provided the field $R$ is archimedean. If $R$ is not archimedean, there are counter-examples of genus $1$.


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Additional Information

Claus Scheiderer
Affiliation: Fachbereich Mathematik, Universität Duisburg, 47048 Duisburg, Germany
Email: claus.@math.uni-duisburg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02522-2
PII: S 0002-9947(99)02522-2
Keywords: Sums of squares, positive semidefinite functions, preorders, real algebraic curves, Jacobians, real algebraic surfaces, real spectrum
Received by editor(s): October 5, 1997
Published electronically: September 8, 1999
Dedicated: Dedicated to Manfred Knebusch on the occasion of his 60th birthday
Article copyright: © Copyright 1999 American Mathematical Society