The Euclidean distance degree of smooth complex projective varieties

Aluffi, Paolo; Harris, Corey

doi:10.2140/ant.2018.12.2005

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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

DOI: 10.2140/ant.2018.12.2005

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

Vol. 12, No. 8, 2018