Vol. 12, No. 8, 2018

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The Euclidean distance degree of smooth complex projective varieties

Paolo Aluffi and Corey Harris

Vol. 12 (2018), No. 8, 2005–2032
Abstract

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern–Schwartz–MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X.

Keywords
algebraic optimization, intersection theory, characteristic classes, Chern–Schwartz–MacPherson classes
Mathematical Subject Classification 2010
Primary: 14C17
Secondary: 14N10, 57R20
Milestones
Received: 3 November 2017
Revised: 3 May 2018
Accepted: 19 June 2018
Published: 4 December 2018
Authors
Paolo Aluffi
Department of Mathematics
Florida State University
Tallahassee, FL
United States
Corey Harris
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany