ECCC-Report TR19-033https://eccc.weizmann.ac.il/report/2019/033Comments and Revisions published for TR19-033en-usTue, 05 Mar 2019 10:51:33 +0200
Paper TR19-033
| Counting basic-irreducible factors mod $p^k$ in deterministic poly-time and $p$-adic applications |
Ashish Dwivedi,
Rajat Mittal,
Nitin Saxena
https://eccc.weizmann.ac.il/report/2019/033Finding 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.Tue, 05 Mar 2019 10:51:33 +0200https://eccc.weizmann.ac.il/report/2019/033