## Abstract

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart–Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.

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## Acknowledgments

Jan Draisma was supported by a Vidi Grant from the Netherlands Organisation for Scientific Research (NWO), and Emil Horobeţ by the NWO Free Competition Grant *Tensors of bounded rank*. Giorgio Ottaviani is member of GNSAGA-INDAM. Bernd Sturmfels was supported by the NSF (DMS-0968882), DARPA (HR0011-12-1-0011), and the Max-Planck Institute für Mathematik in Bonn, Germany. Rekha Thomas was supported by the NSF (DMS-1115293).

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Communicated by James Renegar.

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Draisma, J., Horobeţ, E., Ottaviani, G. *et al.* The Euclidean Distance Degree of an Algebraic Variety.
*Found Comput Math* **16**, 99–149 (2016). https://doi.org/10.1007/s10208-014-9240-x

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DOI: https://doi.org/10.1007/s10208-014-9240-x