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Register a new userHausdorff approximations and volume of tubes of singular algebraic sets
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Saugata Basu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This generalizes previous work of Lotz on smooth complete intersections in the euclidean space and of Burgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled Condition by Burgisser and Cucker.
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In this note, we propose a method to under-approximate finite-time reachable sets and tubes for a class of continuous-time linear uncertain systems. The class under consideration is the linear time-varying (LTV) class with integrable time-varying system matrices and uncertain initial and input values belonging to known convex compact sets. The proposed method depends upon the iterative use of constant-input reachable sets which results in convergent under-approximations in the sense of Hausdorff distance. We illustrate our approach through two numerical examples.
Let $Ssubset R^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining $S$ and an integer $pgeq 0$ and returns the $n$-dimensional volume of $S$ at absolute precision $2^{-p}$.Our algorithm relies on the relationship between volumes of semi-algebraic sets and periods of rational integrals. It makes use of algorithms computing the Picard-Fuchs differential equation of appropriate periods, properties of critical points, and high-precision numerical integration of differential equations.The algorithm runs in essentially linear time with respect to~$p$. This improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods. Assuming a conjecture of Dimca, the arithmetic cost of the algebraic subroutines for computing Picard-Fuchs equations and critical points is singly exponential in $n$ and polynomial in the maximum degree of the input.
Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdans motivic integration and Ayoubs equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.
Given any arbitrary semi-algebraic set $X$, any two points in $X$ may be joined by a piecewise $C^2$ path $gamma$ of shortest length. Suppose $mathcal{A}$ is a semi-algebraic stratification of $X$ such that each component of $gamma cap mathcal{A}$ is either a singleton or a real analytic geodesic segment in $mathcal{A}$, the question is whether $gamma cap mathcal{A}$ 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 insights for high dimensional stratifications of semi-algebraic sets in connection with geodesics.
Let $ Y subseteq Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and assume that $dim Y=n+h$ and $ dim Y_{sing} le min{ d+h-1 , n-1 } $. Let $Z$ be an algebraic cycle on $Y$ of dimension $d+h$, whose homology class in $H_{2(d+h)}(Y; Bbb Q)$ is non-zero. In the present paper we prove that the restriction of $Z$ to $X$ is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case $Y$ is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.
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