A removal lemma for systems of linear equations over finite fields

KRÁĽ, Daniel; O SERRA and L VENA. A removal lemma for systems of linear equations over finite fields. Israel Journal of Mathematics. JERUSALEM: HEBREW UNIV MAGNES PRESS, 2012, vol. 187, No 1, p. 193-207. ISSN 0021-2172. Available from: https://doi.org/10.1007/s11856-011-0080-y.

Basic information

Original name

A removal lemma for systems of linear equations over finite fields

Authors

KRÁĽ, Daniel; O SERRA and L VENA

Edition

Israel Journal of Mathematics, JERUSALEM, HEBREW UNIV MAGNES PRESS, 2012, 0021-2172

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Other information

Language

English

Type of outcome

Article in a journal

Confidentiality degree

is not subject to a state or trade secret

Impact factor

Impact factor: 0.646

DOI

https://doi.org/10.1007/s11856-011-0080-y

UT WoS

000300887800009

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Abstract

In the original language

We prove a removal lemma for systems of linear equations over finite fields: let X (1), aEuro broken vertical bar, X (m) be subsets of the finite field F (q) and let A be a (k x m) matrix with coefficients in F (q) ; if the linear system Ax = b has o(q (m-k) ) solutions with x (i) a 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.