Weighted sums of squares in local rings and their completions. I
Let \(\mathbb{R}[{\mathtt x}]=\mathbb{R}[x_1,\ldots, x_n],\) and let \(g_1,\ldots, g_k\in \mathbb{R}[{\mathtt x}]\) define the basic semialgebraic set \(K=\{x\in \mathbb{R}: g_i(x)\geq 0, i=1,\ldots,r\}.\) A natural and popular question is whether a polynomial \(f\in \mathbb{R}[{\mathtt x}]\) which is nonnegative on \(K\) lies in the preorder generated by \(g_1,\ldots, g_n,\) that is whether every such \(f\) has a representation \quad \((*): \; f=\sum_{i=1}^k a_i^2 h_i ,\) \quad where the \(h_i\) are products of some of the \(g_i,\) and the \(a_i\in \mathbb{R}[{\mathtt x}]\). Such representations are called nonnegativity certificates. \textit{C. Scheiderer} [Math. Z. 245, 725--760 (2003; Zbl 1056.14078)] has shown that, roughly speaking, an identity \((*)\) exists in \(\mathbb{R}[{\mathtt x}]\) iff it exists at sufficiently many localization \(\mathbb{R}[{\mathtt x}]_{\mathfrak{m}}\) by a maximal ideal \(\mathfrak{m}\). See [\textit{M. Marshall}, Positive Polynomials and Sums of Squares. chapter 9, Mathematical Surveys and Monographs 146. Providence, RI: American Mathematical Society (AMS). (2008; Zbl 1169.13001)] for a self-contained introduction to Scheiderer's local-global principles. In a more abstract setting, given a ring \(A\) and any subset \(M\subset A,\) define \(\mathcal{X}(M)=\{\alpha: \forall f\in M \; f(\alpha)\geq 0\},\) and the saturation of \(M\) by \(\text{Sat}(M)=\{f\in A: \forall \alpha\in \mathcal{X}(M) \; f(\alpha)\geq 0\}.\) Of course \(\text{Sat}(M)\supseteq M.\) So if \(T\) is a preorder generated by finitely many elements \(g_1,\dots,g_r\) in a local ring \(A\) the saturation Sat\((T)\) of \(T\) consists of all elements of \(A\) which are nonnegative where all \(g_i\) are nonnegative on the real spectrum \(\text{Sper}A\) of \(A.\) The existence of a nonnegativity certificate can now be seen to follow from answering for local rings \(A\) when is \(\text{Sat}(T)=T\)? \quad In this paper new light is shed onto the question by relating it to the same question in the completion \(\widehat{A}\) of \(A.\) In section 2 techniques from \textit{C. Scheiderer} [J. Reine Angew. Math. 540, 205--227 (2001; Zbl 0991.13014)] are extended from sums of squares to arbitrary preorderings. For example the basic fact that every preordering in a field is saturated is shown to generalize to a semilocal ring \(A\) in the form that Sat\((T)\cap A^* \subseteq T\) for \(T\) a preordering. For \(f\in A,\) let \(Z(f)=\{\alpha\in \text{Sper}(A): f(\alpha)=0\}.\) Theorem 2.6 shows for semilocal noetherian \(A,\) and \(f\in \text{Sat}(T)\) that if \(\text{supp}(\alpha)\in \text{Max}(A)\) for every \(\alpha \in Z(f)\cap \mathcal{X}(T),\) and if \(f\in T+ \text{Rad(A)}^n\) for all \(n\geq 1,\) then \(f\in T.\) In particular this is true if \(f\in \widehat{T}.\) In a geometric situation (\(A=R[{\mathtt x}]_{\mathfrak{m}_p}\) with \(p\) a point in \(K=\{h_1\geq 0,\dots,h_r \geq 0\}\)) the condition on the \(\alpha\) can be viewn as saying that \(p\) is the only zero of \(f\) in \(K\) in a Zariski neighbourhood of \(p.\) Section 3 (`from a preordering to its completion') starts by studying nonnegative extensions of functions defined on closed sets of the real spectrum. For example if \((A,\mathfrak{m})\) is a henselian noetherian local ring, \(X\subseteq \text{Sper}(A)\) closed and \(X'=\{\alpha \in X: \text{supp}(\alpha) \neq \mathfrak{m} \}, \) then for any \(f\in A\) with \(f>0\) on \(X'\) one can find an \(n\in \mathbb{N}\) so that every element of \(f+\mathfrak{m}^n\) is positive on \(X'.\) Using that every excellent henselian local ring has the strong Artin approximation property - the author refers to [\textit{D. Popescu}, Artin approximation. Handbook of algebra. Volume 2. Amsterdam: North-Holland. 321-356 (2000; Zbl 1005.13003)] and the definition of real dimension \(\dim_r(A)\) given by [\textit{J. F. Fernando, J. M. Ruiz} and \textit{C. Scheiderer} [Trans. Am. Math. Soc. 356, 2663--2684 (2004; Zbl 1080.14071)] - he shows that in an excellent henselian local ring \(A\) one has for the Pythagoras number \(p(A),\) that \(\dim_r(A) \leq 2\) usually implies finite Pythagoras number \(p(A),\) while \(\dim_r(A) \geq 3\) always implies \(p(A)=\infty\). Define a quadratic module \(M\) in \(A\) to be 0-saturated if every \(f\in \text{Sat}(M)\) for which \(\alpha\in Z(f)\cap \mathcal{X}(M)\) implies supp\((\alpha) \in \text{Max}(A)\) is in \(M\). Using further results on the Krull (or: \(\mathfrak{m}\)-adic) topology in a ring \(A\), theorem 3.20 presents one of the main results: let \(A\) be an excellent local ring whose completion has finite Pythagoras number. If \(M\) is a finitely generated quadratic module which is 0-saturated, then \(\widehat{M}\) (the corresponding module in the completion) is saturated. The converse is true if \(M\) is even a preordering. In section 4, `changing the real closed base field', let \(k\subset K\) be a field extension,and consider the following quotient rings \(A=k[[{\mathtt x}]]/I\) and \(B=K[[{\mathtt x}]]/I K[[{\mathtt x}]]\) of multivariate power series rings. Assuming \(T\) is a finitely generated preordering in \(A,\) write \(T_K\) for the preordering generated by \(T\) in \(B.\) If one of \(T\) or \(T_K\) is saturated, does this imply saturatedness of the other as well? The author shows a more general theorem yielding a positive answer in case that \(k\subset K\) is an extension of real closed field. His theorem implies that the `psd=sos' results of Fernando and Ruiz , see e.g. \textit{J. F. Fernando} [Math. Ann. 322, 49--67 (2002; Zbl 1006.32008)] and \textit{J. M. Ruiz} [Math. Z. 230, 317--328 (1999; Zbl 0930.32007)] extend from analytic local rings over \(\mathbb{R}\) to complete local rings over arbitrary real closed fields \(R.\) Section 5 presents selected applications of the results of this paper. If \(A\) is a ring, and \(X\subset \text{Sper}(A),\) then the set of all \(f\in A\) with \(f_{|X} \geq 0\) is a saturated preordering in \(A.\) Also write \(A_+\) for this set when \(X=\text{Sper}(A).\) For excellent local rings of dimension 1 for which \(A/\mathfrak{m}=R\) is a real closed field two results are shown guaranteeing finite generation of preorders \(\mathcal{P}(X).\) The results have direct implications for nonnegativity certificates for real polynomials which are nonnegative on suitable two-dimensional semi-algebraic sets. As is clear from the preceding review, the paper draws on a large number of concepts not found in all introductory texts to commutative algebra. Apart of the standard sources of real algebraic geometry the reviewer confesses to have used the books of [\textit{D. Eisenbud}, Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. 150. Berlin: Springer-Verlag (1995; Zbl 0819.13001)] and [\textit{H. Matsumura}, Commutative Ring theory. Cambridge Studies in Advanced Mathematics, 8. Cambridge etc.: Cambridge University Press. (1989; Zbl 0666.13002)] to learn about henselian rings, excellent rings, Krull topology and completions.
scientific article; zbMATH DE number 5782746
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