Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting basic-irreducible factors mod $p^k$ in deterministic poly-time
  and $p$-adic applications   Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is
not known to be in deterministic polynomial time. Though there is such a
classical algorithm that {\em counts} the number of irreducible factors of
$f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The
irreducible factors of $f\bmod p^k$ blow up exponentially in number; making it
hard to describe them. Can we count those irreducible factors $\bmod~p^k$ that
remain irreducible mod $p$? These are called {\em basic-irreducible}. A simple
example is in $f=x^2+px \bmod p^2$; it has $p$ many basic-irreducible factors.
Also note that, $x^2+p \bmod p^2$ is irreducible but not basic-irreducible!
  We give an algorithm to count the number of basic-irreducible factors of
$f\bmod p^k$ in deterministic poly(deg$(f),k\log p$)-time. This solves the open
questions posed in (Cheng et al, ANTS'18 \& Kopp et al, Math.Comp.'19). In
particular, we are counting roots $\bmod\ p^k$; which gives the first
deterministic poly-time algorithm to compute Igusa zeta function of $f$. Also,
our algorithm efficiently partitions the set of all basic-irreducible factors
(possibly exponential) into merely deg$(f)$-many disjoint sets, using a compact
tree data structure and {\em split} ideals.

35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess