论文标题
在有限估值环上的三变量扩展器上
On three-variable expanders over finite valuation rings
论文作者
论文摘要
令$ \ mathcal {r} $为订单$ q^r $的有限估值环。在本文中，我们证明，对于任何二次多项式$ f（x，x，y，z）\ in \ mathcal {r} [x，x，y，z] $，它是$ $ axy+axy+axy+r（x）+s（y）+s（y）+t（z）$，用于某些一个可差的polynomials $ r，s，s，s，s，s，t $对于任何$ a，b，c \ subset \ subset \ mathcal {r} $。我们还研究了有限估值环$ \ mathcal {r}的总和类型问题。 \ gg q^{r-1/3} $然后$$ \ max \ {| a \ cdot a |，| a^d + a^d | \}，\ max \ max \ {| a + a |，| a^2 + a^2 + a^2 | \ \} | a |^{2/3} q^{r/3}，$$和$ | f（a） + a | \ gg | a |^{2/3} q^{r/3} $对于任何一个变量二次多项式$ f $。
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 \[ |f(A,B,C)| \gg \min\left\{ q^r, \frac{|A||B||C|}{q^{2r-1}}\right\}\] 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 $|A| \gg q^{r-1/3}$ then $$\max\{ |A \cdot A|, |A^d + A^d|\},\max\{ |A + A|, |A^2 + A^2|\},\max\{|A-A|,|AA+AA|\} \gg |A|^{2/3}q^{r/3},$$ and $|f(A) + A| \gg |A|^{2/3}q^{r/3}$ for any one variable quadratic polynomial $f$.
