Daiju Inaba

Factorization of multivariate polynomials by extended Hensel construction

The extended Hensel construction is a Hensel construction at an unlucky evaluation point for the generalized Hensel construction, and it allows as to avoid shifting the origin in multivariate polynomial factorization. We have implemented a multivariate factorization algorithm which is based on the extended Hensel construction, by solving a leading coefficient problem which is peculiar to our method. We describe the algorithm and present some experimental results. Experiments show that the extended Hensel construction is quite useful for factoring multivariate polynomials which cause large expression swell by shifting the origin.

A numerical study of extended Hensel series

The extended Hensel construction is a Hensel construction at a singular point of the multivariate polynomial, and it allows us to expand the roots of a given multivariate poly-nomial into a kind of series which we call an extended Hensel series. This paper investigates the behavior of the extended Hensel series numerically, and clarifies the following four points. 1) The convergence domain of the extended Hensel series is very different from those of the Taylor series; both convergence and divergence domains coexist in the neighborhood of the expansion point. 2) The extended Hensel series truncated at 7 ~ 8 order coincides very well with the corresponding algebraic function in the convergence domain, while it behaves very wildly in the divergence domain. 3) In the case of non-monic polynomial, the factors of leading co-efficient are distributed among the extended Hensel series, and the singular behaviors of the roots at the zero-points of the leading coefficient are expressed nicely by the Hensel series. 4) Although many-valuedness of extended Hensel series is usually different from that of the corresponding exact roots, the Hensel series reproduce the behaviors of the exact roots by jumping from one branch to another occasionally.

Enhancing the Extended Hensel Construction by Using Gröbner Bases

Contrary to that the general Hensel construction (GHC) uses univariate initial Hensel factors, the extended Hensel construction (EHC) uses multivariate initial Hensel factors determined by the Newton polygon of the given multivariate polynomial. In the EHC so far, Moses-Yun’s (MY) interpolation functions (see the text) are used for Hensel lifting, but the MY functions often become huge when the degree w.r.t. the main variable is large. In this paper, we propose an algorithm which uses, instead of MY functions, Grobner bases of two initial factors which are homogeneous w.r.t. the main variable and the total-degree variable for sub-variables. The Hensel factors computed by the EHC are polynomials in the main variable with coefficients of mostly rational functions in sub-variables. We propose a method which converts the rational functions into polynomials by replacing the denominators by system variables. Each of the denominators is determined by the lowest order element of a Grobner basis. Preliminary experiments show that our new EHC method is much faster than the previous one.

Solving Parametric Sparse Linear Systems by Local Blocking

In solving parametric sparse linear systems, we want 1) to know relations on parametric coefficients which change the system largely, 2) to express the parametric solution in a concise form suitable for theoretical and numerical analysis, and 3) to find simplified systems which show characteristic features of the system. The block triangularization is a standard technique in solving the sparse linear systems. In this paper, we attack the above problems by introducing a concept of local blocks. The conventional block corresponds to a strongly connected maximal subgraph of the associated directed graph for the coefficient matrix, and our local blocks correspond to strongly connected non-maximal subgraphs. By determining local blocks in a nested way and solving subsystems from low to higher ones, we replace sub-expressions by solver parameters systematically, obtaining the solution in a concise form. Furthermore, we show an idea to form simple systems which show characteristic features of the whole system.

Series-Expansion of Multivariate Algebraic Functions at Singular Points: Nonmonic Case

In a series of papers, we have developed a method of expanding multivariate algebraic functions at their singular points. The method applies the Hensel construction to the defining polynomial of the algebraic function, so we named the resulting series “Hensel series”. In [1], we derived a concise representation of Hensel series for the monic defining polynomial, and clarified several characteristic properties of Hensel series theoretically. In this paper, we study the case of nonmonic defining polynomial. We show that, by determining the so-called Newton polynomial suitably, we can construct Hensel series which show reasonable behaviors at zero-points of the leading coefficients and we can derive a representation of Hensel series in the nonmonic case just similarly as in the monic case. Furthermore, we investigate the convergence/divergence behavior and many-valuedness of Hensel series in the nonmonic case.

Convergence domain of series expansions of multivariate algebraic functions

Power-series expansion is a fundamental method in handling analytic functions, not only in mathematics but also in numerical analysis and computer algebra, and knowing the convergence domain is desirable in developing various algorithms. Convergence domain of the power-series expansion of univariate analytic function is well known. For multivariate functions, however, there seems to be no general formula specifying the convergence domain explicitly. In this poster, given a generic multivariate polynomial F(x,u) def = F(x,u1, . . . ,u`) (`≥ 2), we consider a power-series root φ(∞)(u) satisfying F(φ(∞)(u),u) = 0, where the expansion is made at the origin, and we derive a formula which specifies the convergence domain of φ(∞)(u), assuming that F(x,u) is monic and square-free w.r.t. x and F(x,0, . . . ,0) is square-free. Our derivation is based on a new formulation of Hensel construction developed recently by the present authors; see [SI08] for details. Let F(x,u) be divided as F(x,u) = F0(x) + Fu(x,u), where F0(x) = F(x,0). Let n = degx(F) and the roots of F(x,0) be α1, . . . ,αn. Let F0(x) be factorized as F0(x) = G0(x)H0(x). Our new formulation is such that 1) the Hensel factors are expressed in the roots of F0(x) and 2) all the terms of Fu(x,u) are treated as a mass. The point 1) is realized by expressing polynomials Ai(x) and Bi(x) (i=0, . . . ,n−1) in α1, . . . ,αn, where Ai(x) and Bi(x) are determined uniquely to satisfy

Visualization of extended Hensel factors

Let F (x, u1, ..., u`), with ` ≥ 2, be an irreducible multivariate polynomial over a number field K of characteristic 0. A point (s1, ..., s`) ∈ K̄ is called a singular point for the Hensel construction, if F (x, s1, ..., s`) has multiple roots. The generalized Hensel construction breaks down for the multiple factors, and the extended Hensel construction is the Hensel construction at a singular point [Kuo89],[SK99],[SI00]. A very important feature of an extended Hensel factors is that it is a polynomial in x (and a basic algebraic function) with coefficients of homogeneous rational functions in u1, . . . , u`. In ISSAC2005 Poster Session, the present authors clarified the factors appearing in denominators of rational functions, and showed that the order of algebraic function increases at the zero-points of the denominators [SI06]. Since rational functions appear in coefficients of extended Hensel factors, one may think that the extended Hensel factors show very peculiar behaviors. In this poster, we investigate the behavior of extended Hensel factors by visualizing them; we compute the series roots by the extended Hensel construction (which we call Hensel-series roots) and compare them with exact roots. The visualization revealed the following interesting features. 1) Around the expansion point, the convergence domain of one extended Hensel factor is composed of sectors the edges of which are the expansion point, and the divergence domain is also composed of sectors which contain the zero-points of denominators of Hensel factors. 2) A convergence domain is, although a sector near the expansion point, complicated and reaches to rather far points from the expansion point. 3) In the convergence domain excluding its boundary area, the Hensel-series root reproduces the corresponding exact roots fairly well; the Hensel-series root truncated at order 7, for example, coincides with the exact one up to 6 decimal figures on an average. 4) In the divergence domain excluding its boundary area, the Hensel-series roots becomes very large; the Henselseries root truncated at order 7, for example, is 10 ∼ 10 times larger than the corresponding exact roots. 5) In an area where the exact root is complex while the Hensel-series root is real, for example, the Hensel-series root is very large as in 4).