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A removal lemma for systems of linear equations over finite fields
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Králʼ, Daniel, Serra, Oriol and Vena, Lluís (2012) A removal lemma for systems of linear equations over finite fields. Israel Journal of Mathematics, Vol.187 (No.1). pp. 193-207. doi:10.1007/s11856-011-0080-y ISSN 0021-2172.
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Official URL: http://dx.doi.org/10.1007/s11856-011-0080-y
Abstract
We prove a removal lemma for systems of linear equations over finite fields: let X 1, …, X m be subsets of the finite field F q and let A be a (k × m) matrix with coefficients in F q ; if the linear system Ax = b has o(q m−k ) solutions with x i ∈ X i , then we can eliminate all these solutions by deleting o(q) elements from each X i . This extends a result of Green [Geometric and Functional Analysis 15 (2) (2005), 340–376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Israel Journal of Mathematics | ||||
Publisher: | Magnes Press | ||||
ISSN: | 0021-2172 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Vol.187 | ||||
Number: | No.1 | ||||
Page Range: | pp. 193-207 | ||||
DOI: | 10.1007/s11856-011-0080-y | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
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