Tateaki Sasaki

Approximate factorization of multivariate polynomials and absolute irreducibility testing

A concept of approximate factorization of multivariate polynomial is introduced and an algorithm for approximate factorization is presented. The algorithm handles polynomials with complex coefficients represented approximately, hence it can be used to test the absolute irreducibility of multivariate polynomials. The algorithm works as follows: given a monic square-free polynomialF(x,y,…,z), it calculates the roots ofF(x,yo, ...,z0) numerically, whereyo, … z0 are suitably chosen numbers, then it constructs power series F1, …, Fn such thatF(x,y, …, z) ∈F1 (x,y., ...,z)...Fn(x,y, ...z) (mod Se+2), where n=degx (F),S=(y-y0, ...,z-z0), ande=max{degy(F), …, degx (F)}; finally it finds the approximate divisors ofF as products of elements of {F1, …,Fn}.

Approximate multivariate polynomial factorization based on zero-sum relations

Conventional algorithms for approximate factorization of multivariate polynomial suffer from a dilemma: a polynomial-time algorithm which is based on zero-sum relations among power-series roots is practically very time-consuming and unstable, while practically stable algorithms are of combinatorial nature. In this paper, we present two ideas: one is a numeric matrix manipulation method to find zero-sum relations efficiently and the other is a method to utilize power-series roots expanded at different points. We analyze the methods theoretically and investigate their practicality by applying to several examples. We also discuss numerical stability of the matrix method.

Solving Multivariate Algebraic Equation by Hensel Construction

Given a multivariate polynomial F(x, y, ...,z), this paper deals with calculating the roots ofF w.r.t.x in terms of formal power series or fractional-power series iny, ...,z. If the problem is regular, i.e. the expansion point is not a singular point of a root, then the calculation is easy, and the irregular case is considered in this paper. We extend the generalized Hensel construction slightly so that it can be applied to the irregular case. This extension allows us to calculate the roots of bivariate polynomial F(x, y) in terms of Puiseux series iny. For multivariate polynomial F(x, y, ...,z), we consider expanding the roots into fractional-power series w.r.t. the total-degree ofy, ...,z, and the roots are expressed in terms of the roots of much simpler polynomials.

Analysis of approximate factorization algorithm I

In [2], a concept of approximate factorization of multivariate polynomial was introduced and two algorithms of approximate factorization were proposed. One algorithm determines the irreducible factors by handling the combinations of roots of the form λ1ϕ1i+...+λnϕni, where ϕ1,...,ϕn are the roots of a given polynomial, λ1,...,λn are numbers, andi=1, 2,…, and it seems to be practical and important. However, [2] gave only an introductory description of the algorithm and the mathematical as well as computational analysis of the algorithm was postponed. This paper proves completeness of the algorithm by assuming that the numerical coefficients are calculated with an enough accuracy.

A unified method for multivariate polynomial factorizations

Recently, Sasaki et al. presented an approximate factorization algorithm of multivariate polynomials. The algorithm calculates irreducible factors by investigating linear combinations of the same power of approximate roots. In this paper, we show that various kinds of multivariate polynomial factorizations can be performed by this method. We present algorithms for factorization of multivariate polynomials over power-series rings, over the integers, over algebraic number fields including algebraically closed fields, and over algebraic function fields. Furthermore, we discuss applicability of this method to univariate polynomial factorization.

Polynomial remainder sequence and approximate GCD

Let <i>P</i><inf>1</inf> and <i>P</i><inf>2</inf> be polynomials, univariate or multivariate, and let (<i>P</i><inf>1</inf>, <i>P</i><inf>2</inf>, <i>P</i><inf>3</inf>,…, <i>P<inf>i</inf></i>,…) be a polynomial remainder sequence. Let <i>A<inf>i</inf></i> and <i>B<inf>i</inf></i> (<i>i</i> = 3, 4,…) be polynomials such that <i>A<inf>i</inf></i><i>P</i><inf>1</inf> + <i>B<inf>i</inf></i><i>P</i><inf>2</inf> = <i>P<inf>i</inf>,</i> deg(<i>A<inf>i</inf></i>) < deg(<i>P</i><inf>2</inf>) - deg(<i>P<inf>i</inf></i>), deg(<i>B<inf>i</inf></i>) < deg(<i>P</i><inf>1</inf>) - deg(<i>P<inf>i</inf></i>), where the degree is for the main variable. We derive relations such as <i>C<inf>i</inf>P</i><inf>1</inf> = -<i>B<inf>i</i>+1</inf><i>P<inf>i</inf></i> + <i>B<inf>i</inf>P<inf>i</i>+1</inf> and <i>C<inf>i</inf>P</i><inf>2</inf> = <i>A<inf>i</i>+1</inf><i>P<inf>i</inf></i> - <i>A<inf>i</inf>P<inf>i</i>+1</inf>, where <i>C<inf>i</inf></i> is independent of the main variable. Using these relations, we discuss approximate common divisors calculated by polynomial remainder sequence.

Computing floating-point gröbner bases stably

Computing floating-point gröbner bases stably.

An analysis of cancellation error in multivariate Hensel construction with floating-point number arithmetic

Algebraic computation with oating-point numbers often causes large numerical errors due to cancellation of almost the same numbers. In this paper, we take up multivariate Hensel construction F (x; u) G(x; u)H(x; u) (mod (u s)), where (u) denotes (u1; : : : ; ur) and (s) = (s1; : : : ; sr) is a set of numbers, and analyze the cancellation errors contained in the coe cients of G and H, in the case that the expansion point (i.e. (u) = (s)) is near a singular point. Let be the distance between the expansion point and the nearest singular point, hence 1. We investigate the -dependencies of norms kGk, kHk and the terms appearing in the computation, and show that the initial errors are magni ed by about O( ), where is a nonnegative number. We nd = 0 in some cases but 1 in other cases; the value of depends on the type of singularity and choice of initial factors G andH. The theory developed is con rmed by several numerical examples.

Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation

Computation of Grobner bases of polynomial systems with coefficients of floating-point numbers has been a serious problem in computer algebra for many years; the computation often becomes very unstable and people did not know how to remove the instability. Recently, the present authors clarified the origin of instability and presented a method to remove the instability. Unfortunately, the method is very time-consuming and not practical. In this paper, we first investigate the instability much more deeply than in the previous paper, then we give a theoretical analysis of the term cancellation which causes loss of accuracy in various cases. On the basis of this analysis, we propose a practical method for computing Grobner bases with coefficients of floating-point numbers. The method utilizes multiple precision floating-point numbers, and it removes the drawbacks of the previous method almost completely. Furthermore, we present a practical method of estimating the ill-conditionedness of the input system.

Three new algorithms for multivariate polynomial GCD

Three new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is to calculate a Grobner basis with a certain term ordering. The second is to calculate the subresultant by treating the coefficients w.r.t. the main variable as truncated power series. The third is to calculate a PRS (polynomial remainder sequence) by treating the coefficients as truncated power series. The first algorithm is not important practically, but the second and third ones are efficient and seem to be useful practically. The third algorithm has been implemented naively and compared with the trial-division PRS algorithm and the EZGCD algorithm. Although it is too early to derive a definite conclusion, the PRS method with power series coefficients is very efficient for calculating low degree GCD of high degree non-sparse polynomials.