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 Cheval...
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2017
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| Online Access: | http://hdl.handle.net/1721.1/106843 https://orcid.org/0000-0001-8316-5369 |
| _version_ | 1826218130858311680 |
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| author | Clark, Pete L. Forrow, Aden Schmitt, John R. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Clark, Pete L. Forrow, Aden Schmitt, John R. |
| author_sort | Clark, Pete L. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T17:14:40Z |
| format | Article |
| id | mit-1721.1/106843 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T17:14:40Z |
| publishDate | 2017 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1068432022-10-03T11:22:12Z Warning’s Second Theorem with restricted variables Clark, Pete L. Forrow, Aden Schmitt, John R. Massachusetts Institute of Technology. Department of Mathematics Forrow, Aden 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. 2017-02-02T21:59:54Z 2017-03-01T16:14:47Z 2016-05 2014-05 2017-02-02T15:20:25Z Article http://purl.org/eprint/type/JournalArticle 0209-9683 1439-6912 http://hdl.handle.net/1721.1/106843 Clark, Pete L., Aden Forrow, and John R. Schmitt. “Warning’s Second Theorem with Restricted Variables.” Combinatorica (2016): n. pag. https://orcid.org/0000-0001-8316-5369 en http://dx.doi.org/10.1007/s00493-015-3267-8 Combinatorica 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. János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Clark, Pete L. Forrow, Aden Schmitt, John R. Warning’s Second Theorem with restricted variables |
| title | Warning’s Second Theorem with restricted variables |
| title_full | Warning’s Second Theorem with restricted variables |
| title_fullStr | Warning’s Second Theorem with restricted variables |
| title_full_unstemmed | Warning’s Second Theorem with restricted variables |
| title_short | Warning’s Second Theorem with restricted variables |
| title_sort | warning s second theorem with restricted variables |
| url | http://hdl.handle.net/1721.1/106843 https://orcid.org/0000-0001-8316-5369 |
| work_keys_str_mv | AT clarkpetel warningssecondtheoremwithrestrictedvariables AT forrowaden warningssecondtheoremwithrestrictedvariables AT schmittjohnr warningssecondtheoremwithrestrictedvariables |