Algebra & Number Theory

The Euclidean distance degree of smooth complex projective varieties

Paolo Aluffi and Corey Harris

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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.

Article information

Algebra Number Theory, Volume 12, Number 8 (2018), 2005-2032.

Received: 3 November 2017
Revised: 3 May 2018
Accepted: 19 June 2018
First available in Project Euclid: 21 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14N10: Enumerative problems (combinatorial problems) 57R20: Characteristic classes and numbers

algebraic optimization intersection theory characteristic classes Chern–Schwartz–MacPherson classes


Aluffi, Paolo; Harris, Corey. The Euclidean distance degree of smooth complex projective varieties. Algebra Number Theory 12 (2018), no. 8, 2005--2032. doi:10.2140/ant.2018.12.2005.

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