No abstract

Let {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{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 have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

In this paper, we use a recent method given by Rudnev, Shakan, and Shkredov (2018) to improve results on sum-product type problems due to .

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 .