Abstract
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 ℓ be a power of a prime number p, and let F q be “the” finite field of order q.
For a 1,...,a n , N∈Z+, we denote by m(a 1,...,a n ;N)∈Z+ a certain combinatorial quantity defined and computed in Section 2.1.
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Clark, P.L., Forrow, A. & Schmitt, J.R. Warning’s Second Theorem with restricted variables. Combinatorica 37, 397–417 (2017). https://doi.org/10.1007/s00493-015-3267-8
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DOI: https://doi.org/10.1007/s00493-015-3267-8