Euclidean distance degree and mixed volume

TL;DR

This paper explores the Euclidean Distance Degree for sparse polynomial hypersurfaces, establishing its relation to mixed volume of Newton polytopes and providing formulas for specific cases, impacting computational complexity understanding.

Contribution

It introduces a novel connection between Euclidean Distance Degree and mixed volume for sparse polynomials, with explicit formulas for certain Newton polytope shapes.

Findings

Euclidean Distance Degree equals the mixed volume of Newton polytopes.

Provides a formula for the Euclidean Distance Degree for rectangular parallelepiped Newton polytopes.

Discusses implications for computational complexity.

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.