Hausdorff approximations and volume of tubes of singular algebraic sets

TL;DR

This paper establishes bounds on the volume of neighborhoods around algebraic sets in Euclidean space and spheres, based on polynomial degree, variables, and dimension, extending previous smoothness-dependent results.

Contribution

It generalizes existing bounds to include singular algebraic sets without smoothness assumptions, solving a key open problem in the field.

Findings

Derived volume bounds for neighborhoods of algebraic sets

Extended results to singular sets in Euclidean space and spheres

Provided a complete solution to a known open problem

Abstract

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 B\"urgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled "Condition" by B\"urgisser and Cucker.