We present a restricted variable generalization of Warnings 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 Brinks restricted variable generalization of Chevalleys Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warnings Second Theorem implies Chevalleys Theorem, our result implies Brinks Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems.
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