claus@math.uni-duisburg.de

Submission: 2001, Jan 2

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 $f\in A$, 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 $2$-dimensional local ring has finite Pythagoras number if and only if its quotient field does.

1991 Mathematics Subject Classification: Primary 11E25; Secondary 13H, 14P

Keywords and Phrases: Sums of squares, semilocal rings, Pythagoras number, quadratic forms, orderings, real spectrum

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