The Canny-Emiris conjecture for the sparse resultant
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We present a product formula for the initial parts of the sparse resultant associated to an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of the sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain a similar product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated to a mixed subdivision of a polytope. Applying these results, we prove that the sparse resultant can be computed as the quotient of the determinant of such a square matrix by a certain principal minor, under suitable hypothesis. This generalizes the classical Macaulay formula for the homogeneous resultant, and confirms a conjecture of Canny and Emiris.
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