1,907 research outputs found
Factoring multivariate polynomials over algebraic number fields
We present an algorithm to factor multivariate polynomials over algebraic number fields that is polynomial-time in the degrees of the polynomial to be factored. The algorithm is an immediate generalization of the polynomial-time algorithm to factor univariate polynomials with rational coefficients
Factoring bivariate sparse (lacunary) polynomials
We present a deterministic algorithm for computing all irreducible factors of
degree of a given bivariate polynomial over an algebraic
number field and their multiplicities, whose running time is polynomial in
the bit length of the sparse encoding of the input and in . Moreover, we
show that the factors over \Qbarra of degree which are not binomials
can also be computed in time polynomial in the sparse length of the input and
in .Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a
multivariate version of Theorem 1 had independently been achieved by Erich
Kaltofen and Pascal Koira
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
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
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
Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach
We present a new algorithm for the computation of the irreducible factors of
degree at most , with multiplicity, of multivariate lacunary polynomials
over fields of characteristic zero. The algorithm reduces this computation to
the computation of irreducible factors of degree at most of univariate
lacunary polynomials and to the factorization of low-degree multivariate
polynomials. The reduction runs in time polynomial in the size of the input
polynomial and in . As a result, we obtain a new polynomial-time algorithm
for the computation of low-degree factors, with multiplicity, of multivariate
lacunary polynomials over number fields, but our method also gives partial
results for other fields, such as the fields of -adic numbers or for
absolute or approximate factorization for instance.
The core of our reduction uses the Newton polygon of the input polynomial,
and its validity is based on the Newton-Puiseux expansion of roots of bivariate
polynomials. In particular, we bound the valuation of where is
a lacunary polynomial and a Puiseux series whose vanishing polynomial
has low degree.Comment: 22 page
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