论文标题
多项式筛的应用：超出变量分离
Application of a polynomial sieve: beyond separation of variables
论文作者
论文摘要
在\ mathbb {z} [x_1，\ ldots，x_n] $中设置一个多项式$ f \。方形筛子可以为整体$ \ mathbf {x} \ in [-b，b]^n $的积分数量提供上限，以使$ f（\ mathbf {x}）$是一个完美的正方形。最近，这已经进行了总体的概括：首先是一个功率筛子，计数$ \ mathbf {x} \ in [-b，b]^n $ in $ f（\ mathbf {x}）= y^r $可解决$ y \ in \ y \ in \ mathbb {z} $;然后在多项式筛子上，计数$ \ mathbf {x} \在[-b，b]^n $中，$ f（\ mathbf {x}）= g（y）$是可以解决的，对于给定的多项$ g $。正式地，多项式筛引理可以包含[-b，b]^n $计数$ \ mathbf {x} \ in [-b，b]^n $的更普遍的问题，其中$ f（y，\ mathbf {x}）= 0 $可溶解，对于给定的多元元素$ f $。但是，以前的应用程序仅在$ f（y，\ mathbf {x}）$中表现出变量分离的情况下才成功，即$ f（y，\ mathbf {x}）$采用form $ f（\ mathbf {x}） - g（y） - g（y）$。在目前的工作中，我们介绍了多项式筛的第一个应用，以计数$ \ mathbf {x} \ in [-b，b]^n $ in [-b，b]^n $，以至于$ f（y，\ mathbf {x}）= 0 $是可解决的，在这种情况下，$ f $不显示变量分离的情况。因此，我们获得了一个新的结果，涉及一个与稀薄集中的计数点有关的问题。
Let a polynomial $f \in \mathbb{Z}[X_1,\ldots,X_n]$ be given. The square sieve can provide an upper bound for the number of integral $\mathbf{x} \in [-B,B]^n$ such that $f(\mathbf{x})$ is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=y^r$ is solvable for $y \in \mathbb{Z}$; then to a polynomial sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=g(y)$ is solvable, for a given polynomial $g$. Formally, a polynomial sieve lemma can encompass the more general problem of counting $\mathbf{x} \in [-B,B]^n$ for which $F(y,\mathbf{x})=0$ is solvable, for a given polynomial $F$. Previous applications, however, have only succeeded in the case that $F(y,\mathbf{x})$ exhibits separation of variables, that is, $F(y,\mathbf{x})$ takes the form $f(\mathbf{x}) - g(y)$. In the present work, we present the first application of a polynomial sieve to count $\mathbf{x} \in [-B,B]^n$ such that $F(y,\mathbf{x})=0$ is solvable, in a case for which $F$ does not exhibit separation of variables. Consequently, we obtain a new result toward a question of Serre, pertaining to counting points in thin sets.
