Sums of squares of regular functions on real algebraic varieties

Abstract

Let V V be an affine algebraic variety over R \mathbb {R} (or any other real closed field R R ). We ask when it is true that every positive semidefinite (psd) polynomial function on V V is a sum of squares (sos). We show that for dim ⁡ V ≥ 3 \dim V\ge 3 the answer is always negative if V V has a real point. Also, if V 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 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 ≤ 1 \le 1 : If such a curve has a complex point at infinity, then every psd function is sos, provided the field R R is archimedean. If R R is not archimedean, there are counter-examples of genus  1 1 .