Recent zbMATH articles in MSC 32Bhttps://zbmath.org/atom/cc/32B2022-07-25T18:03:43.254055ZWerkzeugRepresentation of positive semidefinite elements as sum of squares in 2-dimensional local ringshttps://zbmath.org/1487.141252022-07-25T18:03:43.254055Z"Fernando, José F."https://zbmath.org/authors/?q=ai:fernando.jose-fFor 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 \(\kappa[[x,\,y]]\). In Appendix A are given two additional examples with \(\mathcal{P}(A)=\Sigma A^2\).
Reviewer: Yong Hu (Guangdong)Stabilisation of geometric directional bundle for a subanalytic sethttps://zbmath.org/1487.141272022-07-25T18:03:43.254055Z"Koike, Satoshi"https://zbmath.org/authors/?q=ai:koike.satoshi"Paunescu, Laurentiu"https://zbmath.org/authors/?q=ai:paunescu.laurentiuSummary: In the previous paper [the authors, Rev. Roum. Math. Pures Appl. 64, No. 4, 479--504 (2019; Zbl 1463.14007)] we have introduced the notion of geometric directional bundle of a singular space, in order to introduce global bi-Lipschitz invariants. Then we have posed the question of whether or not the geometric directional bundle is stabilised as an operation acting on singular spaces. In this paper we give a positive answer in the case where the singular spaces are subanalytic sets, thus providing a new invariant associated with the subanalytic sets.Relative subanalytic sheaves. IIhttps://zbmath.org/1487.180072022-07-25T18:03:43.254055Z"Monteiro Fernandes, Teresa"https://zbmath.org/authors/?q=ai:monteiro-fernandes.teresa"Prelli, Luca"https://zbmath.org/authors/?q=ai:prelli.lucaSummary: We give a new construction of sheaves on a relative site associated to a product \(X \times S\) where \(S\) plays the role of a parameter space, expanding the previous construction by the same authors, where the subanalytic structure on \(S\) was required. Here we let this last condition fall. In this way the construction becomes much easier to apply when the dimension of \(S\) is bigger than one. We also study the functorial properties of base change with respect to the parameter space.
For Part I, see [the authors, Fundam. Math. 226, No. 1, 79--99 (2014; Zbl 1305.18056)].Topology of 1-parameter deformations of non-isolated real singularitieshttps://zbmath.org/1487.320332022-07-25T18:03:43.254055Z"Dutertre, Nicolas"https://zbmath.org/authors/?q=ai:dutertre.nicolas"Moya-Pérez, Juan Antonio"https://zbmath.org/authors/?q=ai:moya-perez.juan-antonioSummary: Let \(f : (\mathbb{R}^n,0) \to (\mathbb{R},0)\) be an analytic function germ with non-isolated singularities and let \(F : (\mathbb{R}^{n+1},0) \to (\mathbb{R},0)\) be a 1-parameter deformation of \(f\). Let \(f_t^{-1}(0) \cap B_\epsilon^n\), \(0 < | t| \ll \epsilon \), be the ``generalized'' Milnor fiber of the deformation \(F\). Under some conditions on \(F\), we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.Bounding the length of gradient trajectorieshttps://zbmath.org/1487.320342022-07-25T18:03:43.254055Z"D'Acunto, Didier"https://zbmath.org/authors/?q=ai:dacunto.didier"Kurdyka, Krzysztof"https://zbmath.org/authors/?q=ai:kurdyka.krzysztofSummary: We propose a method to bound the length of gradient trajectories by comparison with the length of corresponding talwegs (ridge or valley lines), and we obtain several applications. We show that gradient trajectories of a definable (in an o-minimal structure) family of functions are of uniformly bounded length. We prove that the length of a trajectory of the gradient of a polynomial in \(n\) variables of degree \(d\) in a ball of radius \(r\) is bounded by \(rA(n,d)\), where \(A(n,d)=\nu (n)((3d-4)^{n-1} + 2(3d-3)^{n-2})\) and \(\nu (n)\) is an explicit constant. We give explicit bounds for the length of gradient trajectories of quasipolynomials and trigonometric quasipolynomials. As an application we give a construction of a curve (piecewise gradient trajectory of a polynomial) joining two points in an open connected semialgebraic set. We give an explicit bound for its length. We also obtain an explicit and quite sharp bound in Yomdin's version of the quantitative Morse-Sard theorem.Fibration theorems for subanalytic mapshttps://zbmath.org/1487.320352022-07-25T18:03:43.254055Z"Martins, Rafaella de Souza"https://zbmath.org/authors/?q=ai:martins.rafaella-de-souza"Menegon, Aurélio"https://zbmath.org/authors/?q=ai:menegon-neto.aurelioSummary: We study subanalytic maps \(f: X \rightarrow Y\) between subanalytic sets \(X \subset{\mathbb{R}}^m\) and \(Y \subset{\mathbb{R}}^n\). In the case when \(f\) extends to an analytic map \({\mathbb{R}}^m \rightarrow{\mathbb{R}}^n\), we define the singular set and the discriminant set of \(f\), in a stratified sense, and we give a fibration theorem for \(f\).Segre-degenerate points form a semianalytic sethttps://zbmath.org/1487.320372022-07-25T18:03:43.254055Z"Lebl, Jiří"https://zbmath.org/authors/?q=ai:lebl.jiriSummary: We prove that the set of Segre-degenerate points of a real-analytic subvariety \(X\) in \({\mathbb{C}}^n\) is a closed semianalytic set. It is a subvariety if \(X\) is coherent. More precisely, the set of points where the germ of the Segre variety is of dimension \(k\) or greater is a closed semianalytic set in general, and for a coherent \(X\), it is a real-analytic subvariety of \(X\). For a hypersurface \(X\) in \(\mathbb{C}^n\), the set of Segre-degenerate points, \(X_{[n]}\), is a semianalytic set of dimension at most \(2n-4\). If \(X\) is coherent, then \(X_{[n]}\) is a complex subvariety of (complex) dimension \(n-2\). Example hypersurfaces are given showing that \(X_{[n]}\) need not be a subvariety and that it also need not be complex; \(X_{[n]}\) can, for instance, be a real line.Conservative and semismooth derivatives are equivalent for semialgebraic mapshttps://zbmath.org/1487.490142022-07-25T18:03:43.254055Z"Davis, Damek"https://zbmath.org/authors/?q=ai:davis.damek"Drusvyatskiy, Dmitriy"https://zbmath.org/authors/?q=ai:drusvyatskiy.dmitriySummary: Subgradient and Newton algorithms for nonsmooth optimization require generalized derivatives to satisfy subtle approximation properties: conservativity for the former and semismoothness for the latter. Though these two properties originate in entirely different contexts, we show that in the semi-algebraic setting they are equivalent. Both properties for a generalized derivative simply require it to coincide with the standard directional derivative on the tangent spaces of some partition of the domain into smooth manifolds. An appealing byproduct is a new short proof that semi-algebraic maps are semismooth relative to the Clarke Jacobian.