Algebraically constructible functions: real algebra and topology

Ultrafilter and Constructible topologies on spaces of valuation domains

Let $K$ be a field and let $A$ be a subring of $K$. We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on the space Zar$(K|A)$ of all valuation domains having $K$ as quotient field and containing $A$. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar$(K|A)$. We extend results regarding distinguished spectral topologies on spaces of valuation domains.

The constructible topology on spaces of valuation domains

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an {\sl arbitrary spectral space} and we observe that this topology coincides with the constructible topology. If $K$ is a field and $A$ a subring of $K$, we show that the space Zar$(K|A)$ of all valuation domains, having $K$ as quotient field and containing $A$, (endowed with the Zariski topology) is a spectral space by giving in this general setting the explicit construction of a ring whose Zariski spectrum is homeomorphic to Zar$(K|A)$. We extend results regarding spectral t…

Variants of the Kakeya problem over an algebraically closed field

First, we study constructible subsets of $\A^n_k$ which contain a line in any direction. We classify the smallest such subsets in $\A^3$ of the type $R\cup\{g\neq 0\},$ where $g\in k[x_1,...,x_n]$ is irreducible of degree $d$, and $R\subset V(g)$ is closed. Next, we study subvarieties $X\subset\A^N$ for which the set of directions of lines contined in $X$ has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.

The Riemann-Hilbert Correspondence for Algebraic Stacks

Using the theory infinity-categories we construct derived (dg-)categories of regular, holonomic D-modules and algebraically constructible sheaves on a complex smooth algebraic stack. We construct a natural infinity-categorical equivalence between these two categories generalising the classical Riemann-Hilbert correspondence.

Semi-algebraic geometry with rational continuous functions

Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.

Topology of real algebraic sets of dimension 4: necessary conditions

Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield $2^{43}-43$ independent characteristic numbers mod 2, which generalize the Akbulut-King numbers in dimension three.

Stratifications of real vector spaces from constructible sheaves with conical microsupport

Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of real vector spaces that play well with sheaves having microsupport in a given cone or, equivalently, sheaves in the corresponding conic topology.

On some finiteness results in real étale cohomology

We show that for quasi-compact quasi-separated schemes of finite dimension, the constructibility condition in real étale cohomology agrees with a notion of constructibility arising naturally from topology. As application we prove that the derived direct image functor preserves constructibility under some assumptions, and compute the Grothendieck group of the constructible rational stable motivic homotopy category for reasonable schemes. We prove the generic base change property for constructible real étale sheaves, and deduce the same property for rational motivic spectra and $b$-sheaves.

Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

Let $V$ be a valuation domain of rank one with quotient field $K$. We study the set of extensions of $V$ to the field of rational functions $K(X)$ induced by pseudo-convergent sequences of $K$ from a topological point of view, endowing this set either with the Zariski or with the constructible topology. In particular, we consider the two subspaces induced by sequences with a prescribed breadth or with a prescribed pseudo-limit. We give some necessary conditions for the Zariski space to be metrizable (under the constructible topology) in terms of the value group and the residue field of $V$.

Erzeugungsgrad、VC维与具有有理激活函数的神经网络

Erzeugunsgrad, VC-Dimension and Neural Networks with rational activation function

该研究扩展了Erzeugungsgrad概念及其组合界，证明其是连接代数闭域仿射相交理论与计算学习VC理论的关键，还揭示VC维与Krull维的线性关系，并将成果应用于有理激活函数神经网络分析。

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in th…

fppf拓扑中的点

Points in the fppf topology

本研究利用交换代数与拓扑斯理论构造概形fppf拓扑的拓扑斯点，定义fppf-局部环并证明其素局部化性质，还给出两拓扑位点空间sobrification相同的抽象准则，适用于多种经典Grothendieck拓扑。

Éz fields

Let $K$ be a field. The étale open topology on the $K$-points $V(K)$ of a $K$-variety $V$ was introduced in our previous work. The étale open topology is non-discrete if and only if $K$ is large. If $K$ is separably, real, $p$-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large field…

Topology and Geometry of the Berkovich Ramification Locus for Rational Functions

Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula. Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curv…

Topology of real multi-affine hypersurfaces and a homological stability property

Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is sharp and is independent of $n$ (as opposed to the classical bound of $d(2d -1)^{n-1}$ on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree $d$ in $\mathrm{R}^n$ due to Petrovski{\uı} and Ole{\uı}nik, Thom and Milnor). Moreover, we show there exists $c > 1$, such that given a sequence $(B_n)_{n >0}$ where $B_n$ is a closed ball in…