论文标题
通过beta筛表示的多项式以规范形式表示
Polynomials represented by norm forms via the beta sieve
论文作者
论文摘要
算术几何形状中的一个核心问题是确定哪个多项式$ f \ in \ mathbb {z} [t] $以及哪些数字字段$ k $ hasse原理适用于仿射方程$ f（t）= n_ {k/\ mathbb {q}}（q}}}（\ boldsymbol）虽然在文献中进行了广泛的研究，但目前的结果在很大程度上仅限于多项式和低度的数字字段。在本文中，我们为广泛的多项式和数字领域的家庭建立了HASSE原理，包括多项式的多项式，这些多项式是任意的许多线性，二次或立方因素的产物。证明概括了欧文的论点，该论点利用了罗瑟（Rosser）和伊瓦尼克（Iwaniec）的beta筛子。作为我们的筛子结果的进一步应用，我们证明了Harpaz和Wittenberg的新案例在多项式上的局部拆分值对数字字段上的局部拆分值，并讨论了纤维化中理性点的后果。
A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasse principle holds for the affine equation $f(t) = N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving, which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.
