Geodesic
PRIMES REPRESENTED BY INCOMPLETE NORM FORMS
Forum of Mathematics, Pi, Tome 8 (2020)

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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. 
@article{10_1017_fmp_2019_8,
     author = {JAMES MAYNARD},
     title = {PRIMES {REPRESENTED} {BY} {INCOMPLETE} {NORM} {FORMS}},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {8},
     year = {2020},
     doi = {10.1017/fmp.2019.8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2019.8/}
}
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JAMES MAYNARD. PRIMES REPRESENTED BY INCOMPLETE NORM FORMS. Forum of Mathematics, Pi, Tome 8 (2020). doi: 10.1017/fmp.2019.8

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