Euclidean Distance Degree and Mixed Volume

@article{Breiding2021EuclideanDD,
  title={Euclidean Distance Degree and Mixed Volume},
  author={Paul Breiding and Frank Sottile and James D. Woodcock},
  journal={Foundations of Computational Mathematics},
  year={2021}
}
We 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 polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of $$f=0$$ 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… 
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