AbstractWe initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface $$f=0$$f=0 defined by a polynomialf that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of $$f=0$$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.
