On the Removal Lemma for Linear Systems over Abelian Groups

Details On the Removal Lemma for Linear Systems over Abelian Groups On the Removal Lemma for Linear Systems over Abelian Groups

2011-06-21

arxiv.org/abs/1106.4243v2

Mathematics

Combinatorics

14 pages, fixed a problem with bibtex file

Scientific paper

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the $k$--determinantal of an integer $(k\times m)$ matrix $A$ is coprime with the order $n$ of a group $G$ and the number of solutions of the system $Ax=b$ with $x_1\in X_1,..., x_m\in X_m$ is $o(n^{m-k})$, then we can eliminate $o(n)$ elements in each set to remove all these solutions. This is a follow-up of our former paper 'A Removal Lemma for Systems of Linear Equations over Finite Fields' arXiv:0809.1846v1, which dealt with the case of finite fields.

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