On sums of squares in local rings (Q2754273)

Let \(A\) be a semilocal ring. An element \(f \in A\) is called totally positive or positive semidefinite (pds) if \(f\) has positive or non-negative sign with respect to any ordering of any residue field of \(A\). One may raise the natural question when an element is a sum of squares (sos). This is known to be the case if \(A\) is a field or a valuation ring. In an earlier paper [\textit{C. Schneiderer}, Trans. Am. Math. Soc. 352, No. 3, 1039-1069 (1999; Zbl 0941.14024)], the author showed the existence of elements which are not sos for certain rings \(A\) with \(\dim A \geq 3\). This paper studies the above question for the case \(\dim A = 1,2\). The starting point is the observation that \(f\) is sos in \(A\) if \(f\) is sos in a suitable infinitesimal neighborhood of its real zero locus. It is shown that pds = sos holds in a Nagata local ring \(A\) with \(\dim A =1\) iff it holds in the completion \(\widehat A\). If the residue field \(k\) of \(A\) is moreover real closed, then pds = sos iff \(\widehat A \cong k[[x_1,\ldots,x_n]]/(x_ix_j\mid i < j)\). The main result in the case \(\dim A = 2\) states that if \(A\) is a regular local domain, then \(f\) is sos in \(A\) iff \(f\) is sos in the quotient field \(K\) of \(A\). NEWLINENEWLINENEWLINEThe paper also proves the following results on the Pythagoras number \(p^+(A)\) or \(p(A)\), that is the least number \(n\) such that every totally positive sos or every sos in \(A\) is a sum of \(n\) squares: \(p^+(A) \leq 2^{d+1}\) if \(A\) is a semilocal algebra of finite transcendence degree \(d\) over a real closed field; \(p(A) \leq 2^d\) if \(A\) is a two-dimensional local ring of an algebraic variety \(V\) over a real closed field with \(\dim V = d\) (which is the best possible bound); \(p(A) < 4p(K)\) if \(A\) is a two-dimensional regular local ring.