Euclidean Distance Degree via Mixed Volume

The Euclidean distance degree (EDD) of a variety X in R^n measures
the algebraic complexity of computing the point of X closest to a
general point u in R^n.  It is the number of critical points of the
complexified distance function from u to X.  Known formulas involve
polar classes of the conormal variety to X or Chern classes of X.

In this talk, I will discuss formulas of a different character, when
X is a hypersurface whose defining equation is general given its
Newton polytope.  In this case, the EDD is shown to be the mixed
volume of the critical point equations.  This uses Bernstein's Other
Theorem, which is of independent interest.  We give an interesting
closed formula for the EDD when the Newton polytope is a rectangular
parallelepiped.  This is joint work with Paul Breiding and James
Woodcock.