多项式筛法的应用：超越变量分离
Application of a polynomial sieve: beyond separation of variables
AI总结：
本文首次应用多项式筛法解决非变量分离情形下的零点计数问题，为塞雷的问题提供了新结果。
摘要：
给定一个多项式$f \in \mathbb{Z}[X_1,\ldots,X_n]$，平方筛可以提供一个上界，用于计算积分$\mathbf{x} \in [-B,B]^n$使得$f(\mathbf{x})$为完全平方数的个数。最近这一方法已大幅推广：首先推广为幂筛，计数$\mathbf{x} \in [-B,B]^n$使得$f(\mathbf{x})=y^r$对$y \in \mathbb{Z}$可解；然后推广为多项式筛，计数$\mathbf{x} \in [-B,B]^n$使得$f(\mathbf{x})=g(y)$对给定多项式$g$可解。形式上，多项式筛引理可以涵盖更一般的问题，即计数$\mathbf{x} \in [-B,B]^n$使得$F(y,\mathbf{x})=0$可解，对于给定多项式$F$。然而，以前的应用仅在$F(y,\mathbf{x})$表现出变量分离的情况下成功，即$F(y,\mathbf{x})$形式为$f(\mathbf{x}) - g(y)$。在本文中，我们首次应用多项式筛法计数$\mathbf{x} \in [-B,B]^n$使得$F(y,\mathbf{x})=0$可解，且$F$不表现出变量分离。因此，我们获得了一个新的结果，回应塞雷关于计数薄集中的点的问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
