Polynomial Factorization in Maple 2019

In the recent papers, a new efficient probabilistic semi-numerical absolute (i.e., complex) factorization algorithm for multivariate polynomials with integer coefficients is given. It is based on a simple property of the monomials arising after a generic linear change of coordinates for bivariate polynomials and on a deep result of complex algebraic geometry. Here, we consider the a priori simpler problem of factorization over the field of reals. We briefly review our algorithm for complex factorization and adapt it to solving the problem on the field of reals. This allows us to spare a significant part of the computations and to improve the range of tractability. The method provides factors with approximative coefficients and eventually exact factors in a suitable real algebraic extension of \(\mathbb{Q}\). Bibliography: 15 titles.

A new approach to the symbolic factorization of multivariate polynomials

In actual engineering, processing of big data sometimes requires building of mass physical models, while processing of physical model requires relevant math model, thus producing mass multivariate polynomials, the effective reduction of which is a difficult problem at present. A novel algorithm is proposed to achieve the approximation factorization of complex coefficient multivariate polynomial in light of characteristics of multivariate polynomials. At first, the multivariate polynomial is reduced to be the binary polynomial, then the approximation factorization of binary polynomial can produce irreducible duality factor, at last, the irreducible duality factor is restored to the irreducible multiple factor. As a unit root is cyclic, selecting the unit root as the reduced factor can ensure the coefficient does not expand in a reduction process. Chinese remainder theorem is adopted in the corresponding reduction process, which brought down the calculation complexity. The algorithm is based on approximation factorization of binary polynomial and calculation of approximation Greatest Common Divisor, GCD. The algorithm can solve the reduction of multivariate polynomials in massive math models, which can obtain effectively null point of multivariate polynomials, providing a new approach for further analysis and explanation of physical models. The experiment result shows that the irreducible factors from this method get close to the real factors with high efficiency.

Given a system of polynomial equations with parameters, we present a new algorithm for computing its Dixon resultant R. Our algorithm interpolates the monic square-free factors of R one at a time from monic univariate polynomial images of R using sparse rational function interpolation. In this work, we use a modified version of the sparse multivariate rational function interpolation algorithm of Cuyt and Lee.We have implemented our new Dixon resultant algorithm in Maple with some subroutines coded in C for efficiency. We present timing results comparing our new Dixon resultant algorithm with Zippel’s algorithm for interpolating R and a Maple implementation of the Gentleman & Johnson minor expansion algorithm for computing R.KeywordsDixon resultantParametric polynomial systemsResultantSparse rational function interpolationKronecker substitution

The expansion base algorithm, which was devised by Abhyankar, Kuo and McCallum is very efficient for analytic factorization of bivariate polynomials. The author had extended it to more than two variables but it was only for polynomials with non-vanishing leading coefficient at the expansion point. We improve it to be able to apply to polynomials including the case of vanishing leading coefficient, that is, singular leading coefficient, which comes to a specific problem only for more than two variables.

This paper describes the implementation of recursive algorithms in unique factorization domains, namely multivariate polynomial greatest common divisors (gcd) and factorization into irreducible parts in the Java computer algebra library (JAS). The implementation of gcds, resultants and factorization is part of the essential building blocks for any computation in algebraic geometry, in particular in automated deduction in geometry. There are various implementations of these algorithms in procedural programming languages. Our aim is an implementation in a modern object oriented programming language with generic data types, as it is provided by Java programming language. We exemplify that the type design and implementation of JAS is suitable for the implementation of several greatest common divisor algorithms and factorization of multivariate polynomials. Due to the design we can employ this package in very general settings not commonly seen in other computer algebra systems. As for example, in the coefficient arithmetic for advanced Grobner basis computations like in polynomial rings over rational function fields or (finite, commutative) regular rings. The new package provides factory methods for the selection of one of the several implementations for non experts. Further we introduce a parallel proxy for gcd implementations which runs different implementations concurrently.

While deriving rectangular transforms Agarwal and cooley have used polynomial factors with real integer coefficients which resulted in real convolution matrices. In this paper, it is shown that the use of polynomial factors with complex integer coefficients yields new algorithms with complex convolution matrices, which require less number of multiplications than rectangular transforms. The paper outlines the derivation of the new algorithms and presents the convolution matrices for N = 4,5,7,8 and 9. The results show that the new approach yields smaller theoretical minimum number of multiplications for N = 4 and 8, and the corresponding algorithms are optimum. In view of the simplest factors used in deriving the algorithms for N = 5,7 and 9, it is believed that the corresponding algorithms are the best among those which achieve the theoretical minimum number of multiplications. The matrices have been verified to satisfy the necessary and sufficient condition derived by Agarwal and Cooley.

The files in this data set support the paper "Truth table invariant cylindrical algebraic decomposition" by Russel Bradford, James H. Davenport, Matthew England, Scott McCallum and David Wilson. They include the following: 1. A Maple worksheet and library file concerning the Maple results for the worked examples throughout Sections 1-7 of the paper. 2. A zipped directory of files concerning the worked examples from Sections 1-7 of the paper, this time when studied with Qepcad-B. 3. The example set that is the subject of the experiments in Section 8.2 of the paper, whose results were summarised in Table 2, and an example Maple worksheet that uses this set as input. 4. A Maple worksheet that shows how the numbers from Maple in Table 3 in the paper were obtained. 5. A zipped directory of files showing how the numbers from Qepcad-B in Table 3 in the paper were obtained.

The work presented here is concerned with the antenna array design in collocated multiple-input multiple-output (MIMO) radars. After knowing the system requirements, the antenna array design problem is formulated as a standard polynomial factorization. In addition, an algorithm based on Newton-Schubert-Kronecker (NSK) polynomial factorization is proposed. The algorithm contains three steps. First, linear factors are extracted by extended Vieta theorem. Then, undermined high-order factors are confirmed with Newton interpolation and certain high-order factors should be searched for within the undermined ones. Finally, the antenna array configurations are determined according to the result of polynomial factorization. Simulations confirm the wide use of the proposed algorithm in MIMO radar antenna array design.

For odd square-free $n > 1$ the cyclotomic polynomial ${\Phi _n}(x)$ satisfies the identity of Gauss, \[ 4{\Phi _n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2.\] A similar identity of Aurifeuille, Le Lasseur, and Lucas is \[ {\Phi _n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2\] or, in the case that n is even and square-free, \[ \pm {\Phi _{n/2}}( - {x^2}) = C_n^2 - nxD_n^2.\] Here, ${A_n}(x), \ldots ,{D_n}(x)$ are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require $O({n^2})$ arithmetic operations and work over the integers. We also give explicit formulae and generating functions for ${A_n}(x), \ldots ,{D_n}(x)$, and illustrate the application to integer factorization with some numerical examples.

For odd square-free n > 1 n > 1 the cyclotomic polynomial Φ n ( x ) {\Phi _n}(x) satisfies the identity of Gauss, \[ 4 Φ n ( x ) = A n 2 − ( − 1 ) ( n − 1 ) / 2 n B n 2 . 4{\Phi _n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2. \] A similar identity of Aurifeuille, Le Lasseur, and Lucas is \[ Φ n ( ( − 1 ) ( n − 1 ) / 2 x ) = C n 2 − n x D n 2 {\Phi _n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2 \] or, in the case that n is even and square-free, \[ ± Φ n / 2 ( − x 2 ) = C n 2 − n x D n 2 . \pm {\Phi _{n/2}}( - {x^2}) = C_n^2 - nxD_n^2. \] Here, A n ( x ) , … , D n ( x ) {A_n}(x), \ldots ,{D_n}(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O ( n 2 ) O({n^2}) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n ( x ) , … , D n ( x ) {A_n}(x), \ldots ,{D_n}(x) , and illustrate the application to integer factorization with some numerical examples.

A new package for factoring polynomials with integer coefficients is described which yields significant improvements over previous implementations in both time and space requirements. For multivariate problems, the package features an inexpensive method for early detection and correction of spurious factors. This essentially solves the multivariate extraneous factor problem and eliminates the need to factor more than one univariate image, except in rare cases. Also included is an improved technique for coefficient prediction which is successful more frequently than prior versions at short-circuiting the expensive multivariate Hensel lifting stage. In addition some new approaches are discussed for the univariate case as well as for the problem of finding good integer substitution values. The package has been implemented both in Scratchpad II and in an experimental version of muMATH.

This paper gives an algorithm for finding the irreducible factors of any multivariate polynomial with integer coefficients. The algorithm begins by making substitutions for all but one of the variable. This univariate polynomial is then factored by a known method, which uses an algorithm of Berlekamp for factoring univariate polynomials over finite fields. After this factorization is done, the multivariate factors are recovered from the univariate ones by a kind of Hensel algorithm. A number of ideas are given which greatly speed the computation in some special cases.

@ = ƒ1 = ··· = ƒk = 0 has a finite number of roots. We show that the number of possible distinct vectors of multiplicities of these roots is small when r, d and k are small. As technical tools we design algorithms which produce Grobner bases and vectors of multiplicities of the roots for a parametric zero-dimensional system. The complexities of these algorithms are singly exponential. We also describe an algorithm for parametric absolute factorization of multivariate polynomials. This algorithm has subexponential complexity in the case of a small (relative to the number of variables) degree of the polynomials.

Computing the multilinear factors of lacunary polynomials without heights