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ISSN 1088-6850(e) ISSN 0002-9947(p)

Sums of squares of regular functions on real algebraic varieties

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

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