Emergent Mind
Counting Roots of Polynomials Over Prime Power Rings
(1711.01355)
Published Nov 3, 2017
in
math.NT
,
cs.CC
,
and
cs.SC
Abstract
Suppose $p$ is a prime, $t$ is a positive integer, and $f!\in!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<!pt$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}/(pt)$ of $f$ in deterministic time $(d+\log p){O(1)}$. This fixed parameter tractability appears to be new for $t!\geq!3$. A consequence for arithmetic geometry is that we can efficiently compute Igusa zeta functions $Z$, for univariate polynomials, assuming the degree of $Z$ is fixed.
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