Warning’s Second Theorem with restricted variables
Author(s)Clark, Pete L.; Forrow, Aden; Schmitt, John R.
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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[superscript ℓ] be a power of a prime number p, and let F[subscript q] be “the” finite field of order q. For a[subscript 1],...,a[subscript n], N∈Z[superscript +], we denote by m(a[subscript 1],...,a[subscript n];N)∈Z[superscript +] a certain combinatorial quantity defined and computed in Section 2.1.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Springer Berlin Heidelberg
Clark, Pete L., Aden Forrow, and John R. Schmitt. “Warning’s Second Theorem with Restricted Variables.” Combinatorica (2016): n. pag.
Author's final manuscript