  We consider arbitrary algebraic families of lower order deformations of
nondegenerate toric exponential sums over a finite field. We construct a
relative polytope with the aid of which we define a ring of coefficients
consisting of p-adic analytic functions with polyhedral growth prescribed by
the relative polytope. Using this we compute relative cohomology for such
families and calculate sharp estimates for the relative Frobenius map. In
applications one is interested in L-functions associated with linear algebra
operations (symmetric powers, tensor powers, exterior powers and combinations
thereof) applied to the relative Frobenius. Using methods pioneered by Ax, Katz
and Bombieri we prove estimates for the degree and total degree of the
associated L-function and p-divisibility of the reciprocal zeros and poles.
Similar estimates are then established for affine families and pure Archimedean
weight families (in the simplicial case).
