  We present a restricted variable generalization of Warning's Second Theorem
(a result giving a lower bound on the number of solutions of a low degree
polynomial system over a finite field, assuming one solution exists). This is
analogous to Brink's restricted variable generalization of Chevalley's Theorem
(a result giving conditions for a low degree polynomial system not to have
exactly one solution). Just as Warning's Second Theorem implies Chevalley's
Theorem, our result implies Brink's Theorem. We include several combinatorial
applications, enough to show that we have a general tool for obtaining
quantitative refinements of combinatorial existence theorems.
