745 research outputs found
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
Value sets of sparse polynomials
We obtain a new lower bound on the size of value set f(F_p) of a sparse
polynomial f in F_p[X] over a finite field of p elements when p is prime. This
bound is uniform with respect of the degree and depends on some natural
arithmetic properties of the degrees of the monomial terms of f and the number
of these terms. Our result is stronger than those which canted be extracted
from the bounds on multiplicities of individual values in f(F_p)
Bounded-degree factors of lacunary multivariate polynomials
In this paper, we present a new method for computing bounded-degree factors
of lacunary multivariate polynomials. In particular for polynomials over number
fields, we give a new algorithm that takes as input a multivariate polynomial f
in lacunary representation and a degree bound d and computes the irreducible
factors of degree at most d of f in time polynomial in the lacunary size of f
and in d. Our algorithm, which is valid for any field of zero characteristic,
is based on a new gap theorem that enables reducing the problem to several
instances of (a) the univariate case and (b) low-degree multivariate
factorization.
The reduction algorithms we propose are elementary in that they only
manipulate the exponent vectors of the input polynomial. The proof of
correctness and the complexity bounds rely on the Newton polytope of the
polynomial, where the underlying valued field consists of Puiseux series in a
single variable.Comment: 31 pages; Long version of arXiv:1401.4720 with simplified proof
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
- …