  In this article, we consider the conjectured relationship between F-purity
and log canonicity for polynomials over the complex numbers. We associate to a
collection M of n monomials a rational polytope P contained in [0,1]^n. Using P
and the Newton polyhedron associated to M, we define a non-degeneracy condition
under which log canonicity and dense F-pure type are equivalent for all linear
combinations of the monomials in M. We also show that log canonicity
corresponds to F-purity for very general polynomials. Our methods rely on
showing that the F-pure and log canonical threshold agree for infinitely
primes, and we accomplish this by comparing these thresholds with the
thresholds associated to their monomial ideals.
