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For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma forany abelian group $G$. The best known bounds relating the parameters in thelemma for general $G$ are of tower-type. For $k>3$, even in the case$G=\mathbb{F}_2^n$ no better bounds were known prior to this paper. Thisspecial case has received considerable attention due to its close connection toproperty testing of boolean functions. For every $k\geq 3$, we prove apolynomial bound relating the parameters for $G=\mathbb{F}_p^n$, where $p$ isany fixed prime. This extends the result for $k=3$ by the first two authors.Due to substantial issues with generalizing the proof of the $k=3$ case, a newstrategy is developed in order to prove the result for $k>3$.

A polynomial bound for the arithmetic $k$-cycle removal lemma in vector spaces