A semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities having real coefficients and is a union of finitely many maximally connected components. We consider the problem of deciding whether two given points in a semi-algebraic set are connected; that is, whether the two points lie in the same connected component. In particular, we consider the semi-algebraic set defined by f <> 0 where f is a given polynomial with integer coefficients. The motivation comes from the observation that many important or non-trivial problems in science and engineering can be often reduced to that of connectivity. Due to its importance, there has been intense research effort on the problem. We will describe a symbolic-numeric method based on gradient ascent. The method will be described in two papers. The first paper (the present one) will describe the symbolic part and the forthcoming second paper will describe the numeric part. In the present paper, we give proofs of correctness and termination for the symbolic part and illustrate the efficacy of the method using several non-trivial examples.
In this paper we show that the $d$-dimensional algebraic connectivity of an arbitrary graph $G$ is bounded above by its $1$-dimensional algebraic connectivity, i.e., $a_d(G) \leq a_1(G)$, where $a_1(G)$ corresponds the well-studied second smallest eigenvalue of the graph Laplacian.
Let $X \subset \mathbb{P}^{n}$ be a non-empty closed subscheme over an algebraically closed field $k$, and $\mathrm{J}^{[p]}(X) = \mathrm{J}(X,\mathrm{J}(X,\cdots,\mathrm{J}(X,X)\cdots)$ denote the $p$-fold iterated join of $X$ with itself. In this article, we prove that the restriction homomorphism on cohomology $\mathrm{H}^{i}(\mathbb{P}^{N}) \rightarrow \mathrm{H}^{i}(\mathrm{J}^{[p]}(X))$, with $N = (p+1)(n+1)-1$, is an isomorphism for $0 \leq i < p$, and injective for $i=p$, for any good cohomology theory. We also prove this result in the more general setting of relative joins for $X$ ove…
We show that the Fargues--Fontaine curve associated to an algebraically closed field of characteristic p is geometrically simply connected; that is, its base extension from Q_p to any complete algebraically closed overfield admits no nontrivial connected finite etale covering. We then deduce from this an analogue for perfectoid spaces (and some related objects) of Drinfeld's lemma on the fundamental group of a product of schemes in characteristic p.
Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degree $d$ on the variety $X$. An effective $r$-cycle of minimal degree on $X$ gives rise to a chain of embeddings of $C_{r,d}(X)$ into $C_{r,d+1}(X)$, whose colimit is the connective Chow monoid $C_r^{\infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{\infty }(X)$ be the motivic classifying space of this monoid. In the paper we establish an isomorphism bet…
We define a category of motives for semi-algebraic spaces and show that it is trivial. This implies that there is no good extension of algebraic de Rham cohomology to semi-algebraic spaces compatible with the period isomorphism.
Given a closed semi-algebraic set $X \subset \mathbb{R}^n$ and a continuous semi-algebraic mapping $G \colon X \to \mathbb{R}^m,$ it will be shown that there exists an open dense semi-algebraic subset $\mathscr{U}$ of $L(\mathbb{R}^n, \mathbb{R}^m),$ the space of all linear mappings from $\mathbb{R}^n$ to $\mathbb{R}^m,$ such that for all $F \in \mathscr{U},$ the image $(F + G)(X)$ is a closed (semi-algebraic) set in $\mathbb{R}^m.$ To do this, we study the tangent cone at infinity $C_\infty X$ and the set $E_\infty X \subset C_\infty X$ of (unit) exceptional directions at infinity of $X.$ Spe…
A semigroup $S$ is called an equational domain if any finite union of algebraic sets over $S$ is algebraic. We prove some necessary and sufficient conditions for a completely simple semigroup to be an equational domain.
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.
We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a "patch" -- a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.
In this article, we address the classification of smooth projective algebraic surfaces over complex numbers admitting algebraic semigroup structures. We give a full description of those surfaces $S$, which has at least one non-trivial algebraic semigroup structure, when the Kodaira dimension of $S$ is $ -\infty$ and $ 0$. For the case "$ κ(S)=1$", we give a description of one special type of elliptic surfaces which admit non-trivial algebraic semigroup laws. \\ For a given surface $S$, it is an interesting problem to describe all algebraic semigroup structures on it and determine the dimension…
We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalize the correspondence of facets of a polytope to the vertices of the dual polytope to general semi-algebraic convex bodies. In the general setup, exceptional families of extreme points might exist and we characterize them semi-algebraically. We also give an algorithm to compute a complete list of exceptional families, given the algebraic boundary of the dual convex set.
Let k be an algebraically closed field of characteristic 0, and let f be a morphism of smooth projective varieties from X to Y over the ring k((t)) of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the Berkovich analytifications of X and Y are homotopy equivalent. Two important consequences of this result are that the homotopy type of the Berkovich analytification of any smooth projective variety X over k((t)) is a birational invariant of X, and that the Berkovich analytification of a rationally connected smooth projective variety over k((t…
This paper concerns an analytical stratification question of real algebraic and semi-algebraic sets. For Whitney's stratification in 1957, it partitions a real algebraic set into partial algebraic manifolds\cite{W}. In 1975 Hironaka reproved that a real algebraic set is triangulable and also generalized it to semi-algebraic sets, following the idea of Lojasiewicz's triangulation of semi-analytic sets in 1964. Following their examples and wondering how geometry looks like locally. this paper tries to come up with a stratification, in particular a cell decomposition, such that it satisfies the f…
Given any arbitrary semi-algebraic set $X$, any two points in $X$ may be joined by a piecewise $C^2$ path $γ$ of shortest length. Suppose $\mathcal{A}$ is a semi-algebraic stratification of $X$ such that each component of $γ\cap \mathcal{A}$ is either a singleton or a real analytic geodesic segment in $\mathcal{A}$, the question is whether $γ\cap \mathcal{A}$ has at most finitely many such components. This paper gives a semi-algebraic stratification, in particular a cell decomposition, of a real semi-algebraic set in the plane whose open cells have this finiteness property. This provides insig…
We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions on real algebraic varieties. We provide in particular several characterizations (algebraic or geometric) of these varieties, and provide a full description of centrally seminormal curves.
Let $\mathrm{R}$ be a real closed field and $\mathrm{C}$ the algebraic closure of $\mathrm{R}$. We give an algorithm for computing a semi-algebraic basis for the first homology group, $\mathrm{H}_1(S,\mathbb{F})$, with coefficients in a field $\mathbb{F}$, of any given semi-algebraic set $S \subset \mathrm{R}^k$ defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity. As an intermediate step in our algorithm we construct a semi-algebraic subset $Γ$…
Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolic algorithm to find the equations and inequalities defining a semi-algebraic set $Q$ which is identical to the closure of the image of $S$ under $f$, i.e., \begin{equation} Q=\overline{f(S)}\,. \end{equation} Consequently, every polynomial optimization problem whose optimum value is finite has an equivalent form with attained optimum value, i.e., \begin{equation} \min \limits_{t\in Q} t =\inf\limits_{x\in S} f(x) \end{equation} whenever the right-hand side is finite. Given $d$ as the upper bound on the degrees of $f$ and …
We prove that the wreath product $C=A wr B$ of a semigroup $A$ with zero and an infinite cyclic semigroup $B$ is $q_ω$-compact (logically Noetherian). Our result partially solves the Plotkin`s problem about wreath products
We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex algebraic curves. We also address the minimal surfaces with finite total curvature.
In this paper consisting of two parts, we study the integral of a logarithmic differential form on a compact semi-algebraic set in R^n or C^n. In Part I, we prove the convergence of the integral when the semi-algebraic set satisfies allowability (or admissibility), a condition on the dimension of the intersection of the set and the pole divisor of the differential form.
