A removal lemma for systems of linear equations over finite fields

作者：Kral' Daniel^*; Serra Oriol; Vena Lluis

来源：Israel Journal of Mathematics, 2012, 187(1): 193-207.

DOI：10.1007/s11856-011-0080-y

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.

• 出版日期2012-1