Sums of squares of regular functions on real algebraic varieties
Sums of squares of regular functions on real algebraic varieties
复制标题
实代数簇上正则函数的平方和
DOI：
--
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
C. Scheiderer
中科院分区：
文献类型：
--
作者：
C. Scheiderer
. Let V be an a(cid:14)ne algebraic variety over R (or any other real closed (cid:12)eld R ). We ask when it is true that every positive semide(cid:12)nite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 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 in(cid:12)nity are real, the answer is again negative. The same holds if V is a smooth surface with only real divisors at in(cid:12)nity. The compact" case is harder. We completely settle the case of smooth curves of genus ≤ 1: If such a curve has a complex point at in(cid:12)nity, then every psd function is sos, provided the (cid:12)eld R is archimedean. If R is not archimedean, there are counter-examples of genus 1.
