Georgia Institute of Technology College of Sciences

The secondary polytope of a point configuration A is a polytope whose faces are in bijection with regular subdivions of A, e.g. the secondary polytope of the vertices of polygon is an associahedron. The resultant of a tuple of point configurations A_1, A_2, ..., A_k in Z^n is the set of coefficients for which the polynomials with supports A_1, A_2, ..., A_k have a common root with no zero coordinates over complex numbers, e.g. when each A_1 is a standard simplex and k = n+1, the resultant is defined by a determinant. The Newton polytope of a polynomial is the convex hull of the exponents, e.g. the Newton polytope of the determinant is the perfect matching polytope. In this talk, I will explain the close connection between secondary polytopes and Newton polytopes of resultants, using tropical geometry, based on joint work with Anders Jensen.