Sums of squares of regular functions on real algebraic varieties

Scheiderer, Claus

doi:10.1090/S0002-9947-99-02522-2

Sums of squares of regular functions on real algebraic varieties
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Trans. Amer. Math. Soc. 352 (2000), 1039-1069 Request permission

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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