4 research outputs found
Randomized Polynomial-Time Root Counting in Prime Power Rings
Suppose with prime and is
a univariate polynomial with degree and all coefficients having absolute
value less than . We give a Las Vegas randomized algorithm that computes
the number of roots of in within time
. (We in fact prove a more intricate complexity bound
that is slightly better.) The best previous general algorithm had
(deterministic) complexity exponential in . We also present some
experimental data evincing the potential practicality of our algorithm.Comment: 11 pages, 3 figures. Qi Cheng just pointed out that [3, Cor. 4, Pg.
16] proves a generalization of the main result (Theorem 1.1), and gives a
sharper complexity bound. Nevertheless, the underlying algorithms are
approached differently, so the development of our paper (the recursion tree
structure, in particular) may still be of valu
Noisy polynomial interpolation modulo prime powers
We consider the {\it noisy polynomial interpolation problem\/} of recovering
an unknown -sparse polynomial over the ring of
residues modulo , where is a small prime and is a large integer
parameter, from approximate values of the residues of . Similar results are known for residues modulo a large prime ,
however the case of prime power modulus , with small and large , is
new and requires different techniques. We give a deterministic polynomial time
algorithm, which for almost given more than a half bits of for
sufficiently many randomly chosen points , recovers
Sub-Linear Point Counting for Variable Separated Curves over Prime Power Rings
Let with prime and let be a
bivariate polynomial with degree and all coefficients of absolute value at
most . Suppose also that is variable separated, i.e., for
. We give the first algorithm, with complexity
sub-linear in , to count the number of roots of over mod
for arbitrary : Our Las Vegas randomized algorithm works in time
, and admits a quantum version for smooth curves
working in time . Save for some subtleties concerning
non-isolated singularities, our techniques generalize to counting roots of
polynomials in over mod .
Our techniques are a first step toward efficient point counting for varieties
over Galois rings (which is relevant to error correcting codes over
higher-dimensional varieties), and also imply new speed-ups for computing Igusa
zeta functions of curves. The latter zeta functions are fundamental in
arithmetic geometry.Comment: 18 pages, no figures. Submitted to a conference. Comments and
questions welcome
A complexity chasm for solving univariate sparse polynomial equations over -adic fields
We reveal a complexity chasm, separating the trinomial and tetranomial cases,
for solving univariate sparse polynomial equations over certain local fields.
First, for any fixed field
, we prove that any
polynomial with exactly monomial terms, degree , and
all coefficients having absolute value at most , can be solved over in
deterministic time in the classical Turing model. (The
best previous algorithms were of complexity exponential in , even for
just counting roots in .) In particular, our algorithm generates
approximations in with bit-length to all the
roots of in , and these approximations converge quadratically under
Newton iteration. On the other hand, we give a unified family of tetranomials
requiring digits to distinguish the base- expansions of
their roots in .Comment: 19 pages, 3 figures. This version contains an Appendix missing from
the ISSAC 2021 conference version, as well as some corrections and
improvement