Warning’s Second Theorem with restricted variables
Name
493_2015_Article_3267.pdf
Size
474.42 KB
Format
Adobe PDF
Checksum (MD5)
66f8098d9fb2524a1e83566ac33a6c47
Author(s)
Clark, Pete L.
Forrow, Aden
Schmitt, John R.
Date Issued
May 2016
Journal
Combinatorica
Publisher
Springer Berlin Heidelberg
Citation
Clark, Pete L., Aden Forrow, and John R. Schmitt. “Warning’s Second Theorem with Restricted Variables.” Combinatorica (2016): n. pag.
Version
Author's final manuscript
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[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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00493-015-3267-8
