20 research outputs found
Near NP-Completeness for Detecting p-adic Rational Roots in One Variable
We show that deciding whether a sparse univariate polynomial has a p-adic
rational root can be done in NP for most inputs. We also prove a
polynomial-time upper bound for trinomials with suitably generic p-adic Newton
polygon. We thus improve the best previous complexity upper bound of EXPTIME.
We also prove an unconditional complexity lower bound of NP-hardness with
respect to randomized reductions for general univariate polynomials. The best
previous lower bound assumed an unproved hypothesis on the distribution of
primes in arithmetic progression. We also discuss how our results complement
analogous results over the real numbers.Comment: 8 pages in 2 column format, 1 illustration. Submitted to a conferenc
Computing zeta functions of arithmetic schemes
We present new algorithms for computing zeta functions of algebraic varieties
over finite fields. In particular, let X be an arithmetic scheme (scheme of
finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of
its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a
single prime p in time p^(1/2+o(1)), and another algorithm that computes
zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise
previous results of the author from hyperelliptic curves to completely
arbitrary varieties.Comment: 23 pages, to appear in the Proceedings of the London Mathematical
Societ
Degenerations and limit Frobenius structures in rigid cohomology
We introduce a "limiting Frobenius structure" attached to any degeneration of
projective varieties over a finite field of characteristic p which satisfies a
p-adic lifting assumption. Our limiting Frobenius structure is shown to be
effectively computable in an appropriate sense for a degeneration of projective
hypersurfaces. We conjecture that the limiting Frobenius structure relates to
the rigid cohomology of a semistable limit of the degeneration through an
analogue of the Clemens-Schmidt exact sequence. Our construction is
illustrated, and conjecture supported, by a selection of explicit examples.Comment: 41 page