Euclidean distance degree and mixed volume
Algebraic Geometry
2021-06-01 v2
Abstract
We initiate a study of the Euclidean Distance Degree in the context of sparse polynomials. Specifically, we consider a hypersurface f=0 defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean Distance Degree of f=0 equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped.
Cite
@article{arxiv.2012.06350,
title = {Euclidean distance degree and mixed volume},
author = {Paul Breiding and Frank Sottile and James Woodcock},
journal= {arXiv preprint arXiv:2012.06350},
year = {2021}
}
Comments
20 pages, four figures