14,982 research outputs found
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
Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L\"uroth's Theorem
The extended L\"uroth's Theorem says that if the transcendence degree of
\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK is 1 then there exists f \in
\KK(\underline{X}) such that \KK(\mathsf{f}_1,\dots,\mathsf{f}_m) is equal
to \KK(f). In this paper we show how to compute with a probabilistic
algorithm. We also describe a probabilistic and a deterministic algorithm for
the decomposition of multivariate rational functions. The probabilistic
algorithms proposed in this paper are softly optimal when is fixed and
tends to infinity. We also give an indecomposability test based on gcd
computations and Newton's polytope. In the last section, we show that we get a
polynomial time algorithm, with a minor modification in the exponential time
decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001
Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution
In this paper, a new reduction based interpolation algorithm for black-box
multivariate polynomials over finite fields is given. The method is based on
two main ingredients. A new Monte Carlo method is given to reduce black-box
multivariate polynomial interpolation to black-box univariate polynomial
interpolation over any ring. The reduction algorithm leads to multivariate
interpolation algorithms with better or the same complexities most cases when
combining with various univariate interpolation algorithms. We also propose a
modified univariate Ben-or and Tiwarri algorithm over the finite field, which
has better total complexity than the Lagrange interpolation algorithm.
Combining our reduction method and the modified univariate Ben-or and Tiwarri
algorithm, we give a Monte Carlo multivariate interpolation algorithm, which
has better total complexity in most cases for sparse interpolation of black-box
polynomial over finite fields
On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization
In this paper, we give a theoretical analysis for the algorithms to compute
functional decomposition for multivariate polynomials based on differentiation
and homogenization which are proposed by Ye, Dai, Lam (1999) and Faugere,
Perret (2006, 2008, 2009). We show that a degree proper functional
decomposition for a set of randomly decomposable quartic homogenous polynomials
can be computed using the algorithm with high probability. This solves a
conjecture proposed by Ye, Dai, and Lam (1999). We also propose a conjecture
such that the decomposition for a set of polynomials can be computed from that
of its homogenization with high probability. Finally, we prove that the right
decomposition factors for a set of polynomials can be computed from its right
decomposition factor space. Combining these results together, we prove that the
algorithm can compute a degree proper decomposition for a set of randomly
decomposable quartic polynomials with probability one when the base field is of
characteristic zero, and with probability close to one when the base field is a
finite field with sufficiently large number under the assumption that the
conjeture is correct
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