Warning's Second Theorem with restricted variables

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 Schauz-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 Schauz-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. Let q = p be a power of a prime number p, and let Fq be “the” finite field of order q. For a1, . . . , an, N ∈ Z, we denote by m(a1, . . . , an;N) ∈ Z a certain combinatorial quantity defined and computed in §2.1.