Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations

@article{Berthomieu2012ReductionOB,
  title={Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations},
  author={J{\'e}r{\'e}my Berthomieu and Gr{\'e}goire Lecerf},
  journal={Math. Comput.},
  year={2012},
  volume={81},
  pages={1799-1821}
}
In this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case. This reduction simply consists in computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial. This approach turns out to be very efficient in practice, as demonstrated with our implementation. 
Sparse bivariate polynomial factorization
TLDR
A generalized Hensel lifting, which takes advantage of sparsity, for factoring bivariate polynomial over the rational number field and a new recombination method, which can solve the extraneous factor problem before lifting based on numerical linear algebra are presented.
Multivariate sparse interpolation using randomized Kronecker substitutions
TLDR
A new algorithm for multivariate interpolation is given which uses these new techniques for reducing a multivariate sparse polynomial to a univariatePolynomial along with any existing univariate interpolations algorithm.
An efficient algorithm to factorize sparse bivariate polynomials over the rationals
TLDR
This poster gives a summary of an algorithm, recently presented in the authors’ manuscript, that uses both symbolic and numeric methods to exactly compute the irreducible factorization in Z, and therefore in Q, of any bivariate polynomial satisfying Hypothesis.
Bivariate Factorization Using a Critical Fiber
TLDR
It is shown that working along a critical fiber leads in some cases to a good theoretical complexity, due to the smaller number of analytic factors to recombine.
Exact bivariate polynomial factorization over ℚ by approximation of roots
TLDR
An algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients is presented and can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library.
Factoring bivariate polynomials using adjoints
A recombination algorithm for the decomposition of multivariate rational functions
TLDR
This paper shows how it can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy using the used of Darboux polynomials and deduces a decomposition algorithm in the sparse bivariate case.
Bounded-degree factors of lacunary multivariate polynomials
  • Bruno Grenet
  • Mathematics, Computer Science
    J. Symb. Comput.
  • 2016
A cache-oblivious engineering of the G2V algorithm for computing Gröbner bases
  • F. Salem
  • Computer Science, Mathematics
    ACCA
  • 2013
TLDR
A revised algorithm is presented that permits to preserve the beautiful relation between the classical Euclidean algorithm and the block diagonalization of Hankel matrices for two noncoprime polynomials.
...
...

References

SHOWING 1-10 OF 50 REFERENCES
Fast separable factorization and applications
  • Grégoire Lecerf
  • Mathematics, Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 2008
TLDR
It is shown that the separable decomposition of a univariate polynomial can be computed in softly optimal time, in terms of the number of arithmetic operations in the coefficient field, using the classical multi-modular strategy.
Improved dense multivariate polynomial factorization algorithms
A lifting and recombination algorithm for rational factorization of sparse polynomials
Complexity issues in bivariate polynomial factorization
TLDR
This work proposes an algorithm based on a faster multi-moduli computation for univariate polynomials and shows that it saves a constant factor compared to the classical multifactor lifting algorithm.
Sparse Hensel Lifting
  • E. Kaltofen
  • Computer Science
    European Conference on Computer Algebra
  • 1985
TLDR
It is shown how the content of the input polynomial in the main variable as a by-product can be taken advantage of when computing the GCD of multivariate polynomials by sparse Hensel lifting.
New recombination algorithms for bivariate polynomial factorization based on Hensel lifting
  • Grégoire Lecerf
  • Mathematics, Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 2010
We present new faster deterministic and probabilistic recombination algorithms to compute the irreducible decomposition of a bivariate polynomial via the classical Hensel lifting technique. For the
Sharp precision in Hensel lifting for bivariate polynomial factorization
TLDR
A new algorithm for the recombination stage that requires a lifting up to precision twice the total degree of the polynomial to be factored and it is shown that the bound on precision is asymptotically optimal.
Factoring polynomials via polytopes
TLDR
A new approach to multivariate polynomial factorisation is introduced which incorporates ideas from polyhedral geometry, and generalises Hensel lifting, and is able to exploit to some extent the sparsity of polynomials.
Factoring bivariate sparse (lacunary) polynomials
Probabilistic algorithms for sparse polynomials
TLDR
This work has tried to demonstrate how sparse techniques can be used to increase the effectiveness of the modular algorithms of Brown and Collins and believes this work has finally laid to rest the bad zero problem.
...
...