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Discrete & Computational Geometry

, Volume 62, Issue 4, pp 990–991 | Cite as

Correction to: Metric Properties of Semialgebraic Mappings

  • Krzysztof KurdykaEmail author
  • Stanisław Spodzieja
  • Anna Szlachcińska
Correction
  • 212 Downloads

1 Correction to: Discrete & Computational Geometry (2016) 55:786–800  https://doi.org/10.1007/s00454-016-9776-4

In some formulas in the original article we have omitted in the exponent the number of functions (or equivalently the dimension of the target) which is denoted by m. This shortcoming could lead to misunderstandings in applications.

1. In Corollary 2.2, inequality (2.6), the exponent \(N+r-1\) should be replaced by \(N+m+r-1\). Namely it should be
$$\begin{aligned} {\mathcal {L}}^\mathbf {\mathbb {R}}_0(F|X)\le d(6d-3)^{N+m+r-1} \end{aligned}$$
(2.6)
instead of
$$\begin{aligned} {\mathcal {L}}^\mathbf {\mathbb {R}}_0(F|X)\le d(6d-3)^{N+r-1}. \end{aligned}$$
(2.6)
Indeed, let \(F:X\rightarrow \mathbf {\mathbb {R}}^m\) be a continuous semialgebraic mapping, where \(X\subset \mathbf {\mathbb {R}}^N\) is a closed semialgebraic set. Assume that \(0\in X\) and \(F(0)=0\). Set \(r=r(X)+r(\,\mathrm{graph}\,F)\) and \(d=\max {\{\kappa (X),\kappa (\,\mathrm{graph}\,F)\}}\). Clearly
$$\begin{aligned} \,\mathrm{graph}\,F\subset \mathbf {\mathbb {R}}^{N+m}. \end{aligned}$$
Therefore, it follows from (2.4) in Theorem 2.1 that
$$\begin{aligned} |F(x)| \ge {\text {dist}}((x,0),\,\mathrm{graph}\,F)\ge C{\text {dist}}((x,0),[X\cap F^{-1}(0)]\times \{0\})^{d(6d-3)^{N+m+r-1}} \end{aligned}$$
for \((x,0)\in U\cap [X\times \{0\}]\), where U is a neighborhood \(0 \in \mathbf {\mathbb {R}}^{N+m}\). This gives the correct version of inequality (2.6).
2. Analogously, in Corollary 2.2, inequality (2.7), the exponent \(N+r\) should be replaced by \(N+m+r\). Namely it should be
$$\begin{aligned} {\mathcal {L}}^\mathbf {\mathbb {R}}_0(F|X)\le & {} \frac{(2d-1)^{N+m+r}+1}{2} \end{aligned}$$
(2.7)
instead of
$$\begin{aligned} {\mathcal {L}}^\mathbf {\mathbb {R}}_0(F|X)\le & {} \frac{(2d-1)^{N+r}+1}{2}. \end{aligned}$$
(2.7)
3. Since formula (2.6) was corrected, in the last paragraph of Remark 2.4 the statement “... with our estimate \({\mathcal {L}}^\mathbf {\mathbb {R}}_0(F|X)\le d(6d-3)^{N+r-1}\) ...” should be replaced by “... with our estimate \({\mathcal {L}}^\mathbf {\mathbb {R}}_0(F|X)\le d(6d-3)^{N+r+m-1}\) ...”.
4. In Corollary 3.3 (the first centered inequality), m should be added in the exponent. Namely it should be
$$\begin{aligned} |F(x)|\ge C\left( \frac{{\text {dist}}(x,F^{-1}(0)\cap X)}{1+|x|^{d}}\right) ^{{d(6d-3)^{N+m+r-1}}} \quad \hbox {for}\ x\in X \end{aligned}$$
instead of
$$\begin{aligned} |F(x)|\ge C\left( \frac{{\text {dist}}(x,F^{-1}(0)\cap X)}{1+|x|^{d}}\right) ^{{d(6d-3)^{N+r-1}}} \quad \hbox {for}\ x\in X. \end{aligned}$$
5. Finally, in Corollary 3.3 (the second centered inequality), it should be
$$\begin{aligned} {\mathcal {L}}^\mathbf {\mathbb {R}}_\infty (F|X)\ge (1-d)d(6d-3)^{N+m+r-1} \end{aligned}$$
instead of
$$\begin{aligned} {\mathcal {L}}^\mathbf {\mathbb {R}}_\infty (F|X)\ge (1-d)d(6d-3)^{N+r-1}. \end{aligned}$$
These corrections do not affect the asymptotics of the Łojasiewicz exponent.

6. The correct number of the grant from the Polish National Science Centre is 2012/07/B/ST1/03293.

Notes

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathematiques (LAMA)Universié Savoie Mont Blanc, UMR-5127 de CNRSLe Bourget-du-Lac CedexFrance
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of ŁódźŁódźPoland

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