In a former paper the first and third authors introduced the notion of direction set for a subset of R^n, and showed that the dimension of the common direction set of two subanalytic subsets, called directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we first prove our main theorem and discuss in detail directional properties in the case of an Archimedean real closed field, and then we give a proof in the case of a general real closed field. In addition, related to our main result, we show the existence of special polyhedra in some Euclidean space, illustrating that the bi-Lipschitz equivalence does not always imply the existence of a definable one.
This paper addresses to Nichtnegativstellensätze for definable functions in o-minimal structures on $(\mathbb{R}, +, \cdot).$ Namely, let $f, g_1, \ldots, g_l \colon \mathbb{R}^n \to \mathbb{R}$ be definable $C^p$-functions ($p \ge 2$) and assume that $f$ is non-negative on $S := \{x \in \mathbb{R}^n \ | \ g_1(x) \ge 0, \ldots, g_l(x) \ge 0 \}.$ Under some natural hypotheses on zeros of $f$ in $S,$ we show that $f$ is expressible in the form $f = ϕ_0 + \sum_{i = 1}^l ϕ_i g_i,$ where each $ϕ_i$ is a sum of squares of definable $C^{p - 2}$-functions. As a consequence, we derive global optimality…
We study the set of tangent limits at a given point to a set definable in any o-minimal structure by characterizing the set of exceptional rays in the tangent cone to the set at that point and investigating the set of tangent limits along these rays. Several criteria for determining exceptional rays will be given. The main results of the paper generalize, to the o-minimal setting and to arbitrary dimension, the main results of O'Shea--Wilson which deals with algebraic surfaces in $\mathbb R^3$.
In this mostly expository note, I give a very quick proof of the definable Chow theorem of Peterzil and Starchenko using the Bishop-Stoll theorem and a volume estimate for definable sets due to Nguyen and Valette. The volume estimate says that any $d$-dimensional definable subset of $S\subseteq\mathbb{R}^n$ in an o-minimal expansion of the ordered field of real numbers satisfies the inequality $\mathcal{H}^d(\{x\in S:\lVert x\rVert<r\})\leq Cr^d$, where $\mathcal{H}^d$ denotes the $d$-dimensional Hausdorff measure on $\mathbb{R}^n$ and $C$ is a constant depending on $S$. A closely related volu…
We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open interval is a finite union of points and open intervals. The latter is a variant of van den Dries and Miller's analytic geometric categories and Shiota's $\mathfrak X$-sets and $\mathfrak Y$-sets. In them, the family of definable sets are closed only under proper projections unlike first-order structures. We demonstrate that an $\mathfrak X$-expansion of an ord…
These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which is locally definably bi-Lipschitz trivial along the strata. This result is obtained as a byproduct of two foregoing results of the author. The first one asserts that, given a family definable in an o-minimal structure, there is a regular vector, up to a definable family of bi-Lipschitz homeomorphisms. The second one is a bi-Lipschitz version of the famous H…
Peterzil and Starchenko have proved the following surprising generalization of Chow's theorem: A closed analytic subset of a complex algebraic variety that is definable in an o-minimal structure, is in fact an algebraic subset. In this paper, we prove a non-archimedean analogue of this result.
We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period ma…
MPI of Molecular Cell Biology and Genetics & Center for Systems Biology, Dresden（德累斯顿分子细胞生物学与遗传学马普研究所及系统生物学中心）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
Instytut Matematyczny Uniwersytetu Wrocławskiego（弗罗茨瓦夫大学数学研究所）；Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf（杜塞尔多夫海因里希·海涅大学数学研究所）；School of Mathematics, University of Leeds（利兹大学数学学院）
