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Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, isnot known to be in deterministic polynomial time. Though there is such aclassical algorithm that {\em counts} the number of irreducible factors of$f\bmod p$. We can ask the same question modulo prime-powers $p^k$. Theirreducible factors of $f\bmod p^k$ blow up exponentially in number; making ithard to describe them. Can we count those irreducible factors $\bmod~p^k$ thatremain irreducible mod $p$? These are called {\em basic-irreducible}. A simpleexample 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 openquestions posed in (Cheng et al, ANTS'18 \& Kopp et al, Math.Comp.'19). Inparticular, we are counting roots $\bmod\ p^k$; which gives the firstdeterministic 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 compacttree data structure and {\em split} ideals.

Counting basic-irreducible factors mod $p^k$ in deterministic poly-time and $p$-adic applications