Counting basic-irreducible factors mod pk in deterministic poly-time and p-adic applications

Abstract

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 counts the number of irreducible factors of f mod p. We can ask the same question modulo prime-powers pk. The irreducible factors of f mod pk blow up exponentially in number; making it hard to describe them. Can we count those irreducible factors mod pk that remain irreducible mod p? These are called basic-irreducible. A simple example is in f = x2 + px mod p2; it has p many basic-irreducible factors. Also note that, x2 + p mod p2 is irreducible but not basic-irreducible! We give an algorithm to count the number of basic-irreducible factors of f mod pk 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 mod pk; 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 split ideals.