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# Factorization of polynomials

About: Factorization of polynomials is a research topic. Over the lifetime, 2270 publications have been published within this topic receiving 60693 citations.

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TL;DR: This paper presents a polynomial-time algorithm to solve the following problem: given a non-zeroPolynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q (X).

Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

3,513 citations

01 Jan 1982

TL;DR: In this paper, a polynomial-time algorithm was proposed to decompose a primitive polynomials into irreducible factors in Z(X) if the greatest common divisor of its coefficients is 1.

Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

3,248 citations

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TL;DR: In this paper, a self-contained derivation from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering, is presented, which asymptotically reduces the computational complexity of the transform by a factor two.

Abstract: This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z−1])=E(n;R[z, z−1])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.

2,357 citations

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2,032 citations

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06 Apr 2001

TL;DR: SingULAR as mentioned in this paper is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory.

Abstract: SINGULAR is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory. SINGULAR’s main computational objects are polynomials, ideals and modules over a large variety of rings, including important non-commutative rings. SINGULAR features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root–finding, visualization, and many more related functionalities.

1,394 citations