  We give the first exact determinantal formula for the resultant of an unmixed
sparse system of four Laurent polynomials in three variables with arbitrary
support. This follows earlier work by the author on exact formulas for
bivariate systems and also uses the exterior algebra techniques of Eisenbud and
Schreyer. Along the way we will prove an interesting new vanishing theorem for
the sheaf cohomology of divisors on toric varieties. This will allow us to
describe some supports in four or more variables for which determinantal
formulas for the resultant exist.
