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Hausdorff approximations and volume of tubes of singular algebraic sets

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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 (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).

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Notes

  1. Here we turn the ball \(B(p, \sigma )\) into a probability space using the Lebesgue measure normalized by the volume of the ball itself.

  2. For the rest of the paper, given a family \({\mathcal {P}}\) of polynomials, we denote by \(Z({\mathcal {P}}, X)\) their common zero set, where X will be \(\mathbb {R}^n\) or \(S^n\).

  3. The Curve Selection Lemma is used to construct a sequence \(a_{t_n}\in A_{t_n}\) converging to \(a_0\). Without this we would only be able to construct a sequence \(a_{t'_n}\) still converging to \(a_0\) but with \(t'_n\le t_n\) (remember that in the quantifiers we have “...for every \(n>0\) there exists \(t_n>0\)...”) For instance, if one takes as A the graph of the function \(t\mapsto \sin \left( \frac{1}{t}\right) \), we see that the first inclusion is still true, what fails is the second one.

  4. These coordinates are called “good for Q” in [1].

  5. The reader unfamiliar with the notion of density can assume that M is orientable and read this paragraph simply substituting the word “density” with the word “form”. We refer to [4, Chapter 1, §7 ] for more details.

  6. Given a map \(\gamma {:}\,A\rightarrow B \) between smooth manifolds and a submanifold \(S\hookrightarrow B\), the symbol “\(\gamma \pitchfork S\)” stands for “\(\gamma \) is transversal to S”, i.e. \(\mathrm {im}(d_x\gamma )+T_{\gamma (x)}S=T_{\gamma (x)}B\) for every \(x\in A\) such that \(\gamma (x)\in S\).

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Authors and Affiliations

  1. Purdue University, West Lafayette, IN, USA

    Saugata Basu & Antonio Lerario

  2. SISSA, Trieste, Italy

    Antonio Lerario

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  1. Saugata Basu
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Basu was supported in part by the NSF Grant CCF-1910441.

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Basu, S., Lerario, A. Hausdorff approximations and volume of tubes of singular algebraic sets. Math. Ann. 387, 79–109 (2023). https://doi.org/10.1007/s00208-022-02458-w

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