Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings
For an arbitrary commutative ring \(A\), the set \(\mathcal{P}(A)\) of positive semidefinite elements in \(A\) can be defined by using the real spectrum of \(A\). Let \(\Sigma A^2\) denote the set of finite sums of squares in \(A\). Comparing the two sets \(\mathcal{P}(A)\) and \(\Sigma A^2\) is a classical problem both in the theory of quadratic forms and in real geometry. When \(A\) is a henselian excellent local ring, it is known that the equality \(\mathcal{P}(A)=\Sigma A^2\) implies \(\dim A\le 2\). A natural question is to know which rings \(A\) of dimension \(\le 2\) have the property \(\mathcal{P}(A)=\Sigma A^2\). The 1-dimensional case was completely solved by \textit{C. Scheiderer} [J. Reine Angew. Math. 540, 205--227 (2001; Zbl 0991.13014)]. The paper under review considers the case where \(A\) is further assumed to have dimension 2 and embedding dimension 3. When the residue field \(\kappa\) of \(A\) is real closed, the author has determined in previous works (see [\textit{J. F. Fernando}, Math. Ann. 322, No. 1, 49--67 (2002; Zbl 1006.32008); Math. Z. 244, No. 4, 725--752 (2003; Zbl 1052.14069)]) a necessary and sufficient condition for the equality \(\mathcal{P}(A)=\Sigma A^2\) in terms of the analytic equivalence class (or equivalently, the structure of the completion \(\hat{A}\)) of \(A\). In the present paper there are two main theorems. In the first one (Theorem 1.5), the residue field \(\kappa\) is only assumed to be formally real. The result is that if \(\mathcal{P}(A)=\Sigma A^2\), then \(\hat{A}\) must be one of the candidates listed in the theorem. In the second main result (Theorem 1.8), one assumes that \(\kappa\) is formally real and that \(\tau(\kappa)<+\infty\). (The latter condition means that the level \(s(F)\) of every finite non-real extension \(F/\kappa\) is uniformly bounded from the above.) In this case, it is proved that when \(\hat{A}\) belongs to a given list the equality \(\mathcal{P}(A)=\Sigma A^2\) holds. As a corollary, when \(\kappa\) has a unique ordering and \(\tau(\kappa)<+\infty\) (with \(A\) of dimension 2 and embedding dimension 3), the structure of \(\hat{A}\) is completely determined for all \(A\) with \(\mathcal{P}(A)=\Sigma A^2\). The paper also discusses some applications to principal preorderings of the two-variable Laurent series ring \(\kappax,\,y\). In Appendix A are given two additional examples with \(\mathcal{P}(A)=\Sigma A^2\).
scientific article; zbMATH DE number 7462213
scientific article