  Croot, Lev and Pach used a new polynomial technique to give a new exponential
upper 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 about
polynomials, which gives interesting new bounds for the size of subsets of the
vector space $({\mathbb Z _p})^n$.
  Our main result is a generalization of this lemma. In the proof we combined
Tao's slice rank bounding method with Gr\"obner basis technique.
  As an application, we improve Green's results and present new upper bounds
for the size of difference sets in polynomial rings. We give a new, more
concrete upper bound for the size of arithmetic progression-free subsets in
$({\mathbb Z _p})^n$.
