Multivariate Polynomial Factorization

Musser, David R.

doi:10.1145/321879.321890

Cited by 88 publications

(37 citation statements)

References 5 publications

(11 reference statements)

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“…Musser [6] or Wang and Rothschild [7] give a detailed exposition of the considerations involved in applying Hensel's lemma to the factoring process.…”

Section: On the Efficiency Of Algorithms For Polynomial Factoringmentioning

confidence: 99%

“…Note that if two (or more) factors give the same residue of V(x): (2)(3)(4)(5)(6) V(x)-S = fl-a=f2-b=fl-f2-c, then the GCD operation of (2.3) will yield their product and not the individual factors.…”

Section: On the Efficiency Of Algorithms For Polynomial Factoringmentioning

confidence: 99%

“…The SAC-1 implementation of the Berlekamp-Hensel algorithms [6] used 15 minutes to factor the polynomial on a Honeywell 6050. However, a special purpose SAC-1 program, on a Honeywell 6050, using the Roots modification, working modulo 1790967809 = 427-222 + 1, was able to discover the factors in under three minutes.…”

Section: =0mentioning

confidence: 99%

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On the Efficiency of Algorithms for Polynomial Factoring

Moenck¹

1977

Mathematics of Computation

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Abstract.Algorithms for factoring polynomials over finite fields are discussed. A construction is shown which reduces the final step of Berlekamp's algorithm to the problem of finding the roots of a polynomial in a finite field Z_. p i It is shown that if the characteristic of the field is of the form p = L ■ 2 +1, where / = L, then the roots of a polynomial of degree n may be found in 2 2 0(n log p + n log p) steps.As a result, a modification of Berlekamp's method can be performed in 3 2 2 0(n + n log p + n log p) steps. If n is very large then an alternative method finds ,22 2 the factors of the polynomial in 0{n log n + n lognlogp). Some consequences and empirical evidence are discussed.

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“…Musser [6] or Wang and Rothschild [7] give a detailed exposition of the considerations involved in applying Hensel's lemma to the factoring process.…”

Section: On the Efficiency Of Algorithms For Polynomial Factoringmentioning

confidence: 99%

Section: On the Efficiency Of Algorithms For Polynomial Factoringmentioning

confidence: 99%

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On the Efficiency of Algorithms for Polynomial Factoring

Moenck¹

1977

Mathematics of Computation

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“…The idea of using lifting for K[x, y] is pioneered by [10,19,18]. For a long time the recombination step was performed by an exhaustive search, which means the computation of all the possible recombinations: true factors are recognized by means of Euclidean divisions.…”

Section: Introductionmentioning

confidence: 99%

Sharp precision in Hensel lifting for bivariate polynomial factorization

Lecerf

2006

Math. Comp.

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Abstract. Popularized by Zassenhaus in the seventies, several algorithms for factoring polynomials use a so-called lifting and recombination scheme. Concerning bivariate polynomials, we present a new algorithm for the recombination stage that requires a lifting up to precision twice the total degree of the polynomial to be factored. Its cost is dominated by the computation of reduced echelon solution bases of linear systems. We show that our bound on precision is asymptotically optimal.

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“…Although to date the short vector algorithm provides the only worst-case polynomial-time factoring procedure for univariate integer polynomials, older algorithms, based on Berlekamp [1970] and Zassenhaus [1969], perform well in practice. For multivariate polynomials, the competition is between the short vector approach, a different method due to Kaltofen [1982] (see Kaltofen [1983], von zur GathenKaltofen [1983]) which is also polynomial-time in the worst-case, and older algorithms (e.g., Musser [1975], Wang [1978], Zippel [1981]) which may require exponential time in some cases. For the case of sparse polynomials-of great practical importance-a different approach is necessary (von zur Gathen [1983]).…”

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confidence: 99%

Hensel and Newton Methods in Valuation Rings

Gathen¹

1984

Mathematics of Computation

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Abstract. We give a computational description of Hensel's method for lifting approximate factorizations of polynomials. The general setting of valuation rings provides the framework for this and the other results of the paper. We describe a Newton method for solving algebraic and differential equations. Finally, we discuss a fast algorithm for factoring polynomials via computing short vectors in modules.1. Introduction. Hensel and Newton methods have received quite a lot of attention in algebraic computing. We present them in their natural framework, that of valuation rings. The Hensel method deals with factorization of polynomials, the Newton method with zeros of polynomials over the given valuation ring. Both methods take an approximate solution and produce a new approximation which is better with respect to the given valuation. Apart from the pioneering paper by Zassenhaus [1969], these methods have usually only been treated in the setting of either the integers or a polynomial ring, thus requiring separate proofs for each case. The unified treatment avoids this, and incidentally obtains the Newton method as a special case of the Hensel method, also giving the aesthetical advantage of avoiding rational functions for the important application of inverting power series.The Hensel method presented in Section 2 describes a lifting of an approximate factorization of a given polynomial over a valuation ring, where the factors are approximately relatively prime. It results in two choices of an iterative procedure, one with linear and one with quadratic convergence behavior. It allows us to describe the factorization of certain polynomials that are not squarefree over the residue class field, a case not covered by the usual formulation.In Section 3, we present a Newton method for solving differential equations for formal power series in several variables, in the general case of systems of nonlinear partial differential equations. This includes the case of a system of algebraic equations. One obtains a simple condition which provides an iterative procedure to compute a solution.In Section 4, we discuss an important recently discovered tool for factoring polynomials: computing short vectors in modules over (valuation) rings. This tool has been introduced by Lenstra-Lenstra-Lovasz [1982] for factoring univariate integer polynomials, used in Chistov-Grigoryev [1982], for multivariate polynomials over finite fields, and in Lenstra [1983a] for multivariate integer

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Multivariate Polynomial Factorization

Cited by 88 publications

References 5 publications

On the Efficiency of Algorithms for Polynomial Factoring

On the Efficiency of Algorithms for Polynomial Factoring

Sharp precision in Hensel lifting for bivariate polynomial factorization

Hensel and Newton Methods in Valuation Rings

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