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Croot, Lev and Pach used a new polynomial technique to give a new exponentialupper bound for the size of three-term progression-free subsets in the groups$(\mathbb Z _4)^n$.  The main tool in proving their striking result is a simple lemma aboutpolynomials, which gives interesting new bounds for the size of subsets of thevector space $({\mathbb Z _p})^n$.  Our main result is a generalization of this lemma. In the proof we combinedTao's slice rank bounding method with Gr\"obner basis technique.  As an application, we improve Green's results and present new upper boundsfor the size of difference sets in polynomial rings. We give a new, moreconcrete upper bound for the size of arithmetic progression-free subsets in$({\mathbb Z _p})^n$.

A generalization of Croot-Lev-Pach's Lemma and a new upper bound for the size of difference sets in polynomial rings