On three-variable expanders over finite valuation rings

Abstract

Let R be a finite valuation ring of order q r . In this paper, we prove that for any quadratic polynomial f (x, y, z) ∈ R[x, y, z] that is of the form axy +R(x)+S(y)+T (z) for some one-variable polynomials R, S, T , we havefor any A, B, C ⊂ R. We also study the sum-product type problems over finite valuation ring R. More precisely, we show that for any A ⊂ R with |A| ≫ q r−1/3 then max{|A•A|, |A d +A d |}, max{|A+A|, |A 2 +A 2 |}, max{|A−A|, |AA+AA|} ≫ |A| 2/3 q r/3 , and |f (A) + A| ≫ |A| 2/3 q r/3 for any one variable quadratic polynomial f .