arXiv 1507.050802015-07math.NT

不完全范数形式表示的素数

Primes represented by incomplete norm forms

James Maynard

备注：: 103 pages; v2 is significant rewrite of v1, main results unchanged

摘要：

设$K=\mathbb{Q}(ω)$为一阶$n$一元不可约多项式$f\in\mathbb{Z}[X]$的根，其中$ω$为该多项式的根.我们显示的程度$n$多项式$N(\sum_{i=1}^{n-k}x_iω^{i-1})$在$n-k$变量通过设置最终的$k$系数为0采取期望的渐近数的素数，如果$n\ge 4k$。在$K=\mathbb{Q}(\sqrt[n]θ)$的特殊情况下，我们证明了$N(\sum_{i=1}^{n-k}x_i\sqrt[n]{θ^{i-1}})$取无穷多个素数，条件是$n\ge 22k/7$ .我们的证明依赖于使用合适的“I型”和“II型”估计在哈曼筛，这是建立在一个类似的整体方式，以前的工作Friedlander和Iwaniec对素数值的$X^2+Y^4$和希斯布朗对$X^3+2Y^3$。我们的证明最终依赖于采用明确的初等估计，从几何的数字和代数几何控制的高度倾斜的格子出现在我们的最终估计的数量。

英文摘要：

Let $K=\mathbb{Q}(ω)$ with $ω$ the root of a degree $n$ monic irreducible polynomial $f\in\mathbb{Z}[X]$. We show the degree $n$ polynomial $N(\sum_{i=1}^{n-k}x_iω^{i-1})$ in $n-k$ variables formed by setting the final $k$ coefficients to 0 takes the expected asymptotic number of prime values if $n\ge 4k$. In the special case $K=\mathbb{Q}(\sqrt[n]θ)$, we show $N(\sum_{i=1}^{n-k}x_i\sqrt[n]{θ^{i-1}})$ takes infinitely many prime values provided $n\ge 22k/7$. Our proof relies on using suitable `Type I' and `Type II' estimates in Harman's sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^2+Y^4$ and of Heath-Brown on $X^3+2Y^3$. Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.

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。

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英文摘要：

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