The Euclidean Distance Degree of Fermat Hypersurfaces
Hwangrae Lee
TL;DR
This paper derives a piecewise polynomial formula for the Euclidean distance degree of Fermat hypersurfaces, revealing intricate congruence conditions that influence the optimization problem's complexity.
Contribution
It provides the first explicit formula for the Euclidean distance degree of Fermat hypersurfaces, highlighting the role of congruence conditions.
Findings
The Euclidean distance degree is a piecewise polynomial.
Congruence conditions determine the polynomial pieces.
The formula advances understanding of algebraic optimization problems.
Abstract
Finding the point in an algebraic variety that is closest to a given point is an optimization problem with many applications. We study the case when the variety is a Fermat hypersurface. Our formula for its Euclidean distance degree is a piecewise polynomial whose pieces are defined by subtle congruence conditons.
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