Hausdorff approximations and volume of tubes of singular algebraic sets

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 (Proc...

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Vydané v:Mathematische annalen Ročník 387; číslo 1-2; s. 79 - 109
Hlavní autori: Basu, Saugata, Lerario, Antonio
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2023
Springer Nature B.V
Predmet:
Algebra
Algorithms
Euclidean geometry
Geometry
Hyperspaces
Mathematical analysis
Mathematics
Mathematics and Statistics
Numerical analysis
Polynomials
Smoothness
Tubes
ISSN:0025-5831, 1432-1807
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Popis
Shrnutí: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 (Proc Am Math Soc 143(5):1875–1889, 2015) on smooth complete intersections in the euclidean space and of Bürgisser et al. (Math Comp 77(263):1559–1583, 2008) on hypersurfaces in the sphere, and gives a complete solution to Bürgisser and Cucker (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 349, Springer, Heidelberg, 2013, Problem 17).
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-022-02458-w