Results 1  10
of
22
Approximate Factorization of Multivariate Polynomials via Differential Equations
 Manuscript
, 2004
"... The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors ove ..."
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Cited by 37 (9 self)
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The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the perturbation in some selected distance measure, but no e#cient algorithm for that is known. We give a numerical multivariate greatest common divisor algorithm and use it on a numerical variant of algorithms by W. M. Ruppert and S. Gao. Our numerical factorizer makes repeated use of singular value decompositions. We demonstrate on a significant body of experimental data that our algorithm is practical and can find factorizable polynomials within a distance that is about the same in relative magnitude as the input error, even when the relative error in the input is substantial (10 3 ).
Factorization of Polynomials Given by StraightLine Programs
 Randomness and Computation
, 1989
"... An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and ..."
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Cited by 29 (9 self)
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An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and outputs a straightline program, which with controllably high probability correctly determines the irreducible factors. It also returns the probably correct multiplicities of each distinct factor. If th oefficient field has finite characteristic p and p divides the multiplicities of some irreducible factors our algorithm constructs straightline programs for the appropriate pth powers of such factors. Also a probabilistic algorithm is presented that allows to convert a polynomial given by a straightline program into its sparse representation. This conversion algorithm is in randompolynomial time in the previously cited parameters and in an upper bound for the number of nonzero...
Absolute Irreducibility Of Polynomials Via Newton Polytopes
, 1998
"... A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple ..."
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Cited by 24 (9 self)
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A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the wellknown Eisenstein's criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping certain collection of them nonzero.
Towards Factoring Bivariate Approximate Polynomials
"... A new algorithm is presented for factoring bivariate approximate polynomials over C [x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the de ..."
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Cited by 21 (0 self)
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A new algorithm is presented for factoring bivariate approximate polynomials over C [x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.
Factoring Rational Polynomials over the Complex Numbers
, 1989
"... eskeleton on the surface (P = 0) whose number of connected components is precisely the number of connected components of P =0minus its singular points. The connectivity of this curveskeleton is constructed symbolically using Sturm sequences associated with the various polynomials de ning these ..."
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Cited by 20 (2 self)
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eskeleton on the surface (P = 0) whose number of connected components is precisely the number of connected components of P =0minus its singular points. The connectivity of this curveskeleton is constructed symbolically using Sturm sequences associated with the various polynomials de ning these maps. Given the number of irreducible factors and their degree, the actual factors can be reconstructed using the recent result of Ne [22] on nding zeroes of one variable polynomials in NC. 1 Introduction Factoring polynomials is a basic problem in symbolic computation with applications as diverse as theorem proving and computeraided design. Our goal is to approximate the factors, irreducible over the complex numbers, of a multivariable polynomial with rational coecients in deterministic NC with respect to the polynomial's degree and coecient size, assuming that the number of variables is xed. Further if the number of variables is not xed, we will nd the number of irreducible facto
On Approximate Irreducibility of Polynomials in Several Variables
"... We study the problem of bounding a polynomial away from polynomials which are absolutely irreducible. Such separation bounds are useful for testing whether a numerical polynomial is absolutely irreducible, given a certain tolerance on its coefficients. Using an absolute irreducibility criterion due ..."
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Cited by 19 (7 self)
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We study the problem of bounding a polynomial away from polynomials which are absolutely irreducible. Such separation bounds are useful for testing whether a numerical polynomial is absolutely irreducible, given a certain tolerance on its coefficients. Using an absolute irreducibility criterion due to Ruppert, we are able to find useful separation bounds, in several norms, for bivariate polynomials. We also use Ruppert's criterion to derive new, more effective Noether forms for polynomials of arbitrarily many variables. These forms lead to small separation bounds for polynomials of arbitrarily many variables.
Improved dense multivariate polynomial factorization algorithms
 J. Symbolic Comput
, 2005
"... We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the numb ..."
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Cited by 15 (3 self)
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We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem. Key words: Polynomial factorization, Hensel lifting, Bertini’s irreducibility theorem.
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 14 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.