Abstract
Let X be a closed semialgebraic set of dimension k. If \(n\ge 2k+1\), then there is a bi-Lipschitz and semialgebraic embedding of X into \(\mathbb {R}^n\). Moreover, if \(n \ge 2k+2\), then this embedding is unique (up to a bi-Lipschitz and semialgebraic homeomorphism of \(\mathbb {R}^n\)). We also give local and complex algebraic counterparts of these results.
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Acknowledgements
The authors are grateful to the anonymous referee for his helpful remarks and for Edson Sampaio for a nice and useful example (Example 3.4) and interesting discussions.
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Lev Birbrair was partially supported by CNPq grant 302655/2014-0. Alexander Fernandes and Zbigniew Jelonek are partially supported by the grant of Narodowe Centrum Nauki number 2019/33/B/ST1/00755.
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Birbrair, L., Fernandes, A. & Jelonek, Z. On the extension of bi-Lipschitz mappings. Sel. Math. New Ser. 27, 15 (2021). https://doi.org/10.1007/s00029-021-00625-6
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DOI: https://doi.org/10.1007/s00029-021-00625-6