Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The primes contain arbitrarily long polynomial progressions   We establish the existence of infinitely many \emph{polynomial} progressions
in the primes; more precisely, given any integer-valued polynomials $P_1, >...,
P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any
$\eps > 0$, we show that there are infinitely many integers $x,m$ with $1 \leq
m \leq x^\eps$ such that $x+P_1(m), ..., x+P_k(m)$ are simultaneously prime.
The arguments are based on those in Green and Tao, which treated the linear
case $P_i = (i-1)\m$ and $\eps=1$; the main new features are a localization of
the shift parameters (and the attendant Gowers norm objects) to both coarse and
fine scales, the use of PET induction to linearize the polynomial averaging,
and some elementary estimates for the number of points over finite fields in
certain algebraic varieties.

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The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
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The categories of this paper are: eess