8,280 research outputs found
Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree
In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if is a
polynomial with monomials, with individual degrees of its variables bounded
by , then can be deterministically factored in time . Prior to our work, the only efficient factoring algorithms known for
this class of polynomials were randomized, and other than for the cases of
and , only exponential time deterministic factoring algorithms were
known.
A crucial ingredient in our proof is a quasi-polynomial sparsity bound for
factors of sparse polynomials of bounded individual degree. In particular we
show if is an -sparse polynomial in variables, with individual
degrees of its variables bounded by , then the sparsity of each factor of
is bounded by . This is the first nontrivial bound on
factor sparsity for . Our sparsity bound uses techniques from convex
geometry, such as the theory of Newton polytopes and an approximate version of
the classical Carath\'eodory's Theorem.
Our work addresses and partially answers a question of von zur Gathen and
Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the
sparsity of factors of sparse polynomials
Gradual sub-lattice reduction and a new complexity for factoring polynomials
We present a lattice algorithm specifically designed for some classical
applications of lattice reduction. The applications are for lattice bases with
a generalized knapsack-type structure, where the target vectors are boundably
short. For such applications, the complexity of the algorithm improves
traditional lattice reduction by replacing some dependence on the bit-length of
the input vectors by some dependence on the bound for the output vectors. If
the bit-length of the target vectors is unrelated to the bit-length of the
input, then our algorithm is only linear in the bit-length of the input
entries, which is an improvement over the quadratic complexity floating-point
LLL algorithms. To illustrate the usefulness of this algorithm we show that a
direct application to factoring univariate polynomials over the integers leads
to the first complexity bound improvement since 1984. A second application is
algebraic number reconstruction, where a new complexity bound is obtained as
well
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
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
- …