Abstract
Let $K=\mathbb{Q}(\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D714}$ the root of a degree $n$ monic irreducible polynomial $f\in \mathbb{Z}[X]$ . We show that the degree $n$ polynomial $N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$ in $n-k$ variables takes the expected asymptotic number of prime values if $n\geqslant 4k$ . In the special case $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$ , we show that $N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$ takes infinitely many prime values, provided $n\geqslant 22k/7$ . 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.
Highlights
It is believed that any integer polynomial satisfying some simple necessary conditions should represent infinitely many primes
= Q( n θ ), takes the we show expected that N (
The paper of Heath-Brown [13] suggested that one might hope to utilize similar techniques when considering higher degree norm forms with appropriate variables set equal to zero. We address this problem in this paper, thereby giving further examples of thin polynomials which represent infinitely many primes
Summary
It is believed that any integer polynomial satisfying some simple necessary conditions should represent infinitely many primes. The paper of Heath-Brown [13] suggested that one might hope to utilize similar techniques when considering higher degree norm forms with appropriate variables set equal to zero We address this problem in this paper, thereby giving further examples of thin polynomials which represent infinitely many primes. (The fact that n > 3k means that characters of large conductor do not play a role, and so we do not even require a large sieve type estimate as in [13].) On the other hand, the fact that we consider polynomials in an arbitrary number of variables and with multiple coordinates of the norm form set to 0 introduces different complications of a geometric nature It is handling such issues, which is the key innovation of this paper. One could give a quantitative bound to the o(1) error term appearing in Theorem 1.1
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