  We present a product formula for the initial parts of the sparse resultant
associated to an arbitrary family of supports, generalising 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
generalises the classical Macaulay formula for the homogeneous resultant, and
confirms a conjecture of Canny and Emiris.
