On sums of squares in local rings
On sums of squares in local rings
复制标题
关于局部环中的平方和
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
C. Scheiderer
中科院分区：
文献类型：
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作者：
C. Scheiderer
Let A be a semilocal ring. We compare the set of positive semidefinite (psd) elements of A and the set of sums of squares in A. For psd f ∈ A, whether f is a sum of squares or not depends only on the behavior of f in an infinitesimal neighborhood of the real zeros of f in Spec A. We apply this observation, first to 1-dimensional local rings, then to 2-dimensional regular semilocal rings. For the latter, we show that every psd element is a sum of squares. On the quantitative side, we obtain explicit (finite) upper bounds for the Pythagoras number, for various classes of local rings for which finiteness of this invariant has been an open question so far. For example, a regular 2dimensional local ring has finite Pythagoras number if and only if its quotient field does.
