Extended Hensel Construction Research Articles

On the Extended Hensel Construction and its application to the computation of real limit points

The Extended Hensel Construction (EHC) is a procedure which, for an input bivariate polynomial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm, and, for the multivariate case, can be seen as an effective variant of Jung-Abhyankar Theorem. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. This work is motivated and illustrated by two applications. The first one is the computation of real branches of space curves. The second one is the computation of limits of real multivariate rational function.For the latter, we present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point p which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and Vélez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components.

Improvement of EZ-GCD algorithm based on extended hensel construction

In [2], we proposed the EZ-GCD algorithm based on extended Hensel construction, in order to compute the GCD of multivariate polynomials. The extended Hensel construction is a specify factorization into algebraic function, and it is efficient for sparse multivariate polynomials. However, it is slower than Maple's GCD routine. In this paper, we improve our EZ-GCD algorithm efficiency.

A New Algorithm for Computing the Extended Hensel Construction of Multivariate Polynomials

This paper presents a new algorithm for computing the extended Hensel construction (EHC) of multivariate polynomials in main variable x and sub-variables u1, u2, · · ·, um over a number field $$\mathbb{K}$$ . This algorithm first constructs a set by using the resultant of two initial coprime factors w.r.t. x, and then obtains the Hensel factors by comparing the coefficients of xi on both sides of an equation. Since the Hensel factors are polynomials of the main variable with coefficients in fraction field $$\mathbb{K}$$ (u1, u2, · · ·, um), the computation cost of handling rational functions can be high. Therefore, the authors use a method which multiplies resultant and removes the denominators of the rational functions. Unlike previously-developed algorithms that use interpolation functions or Grobner basis, the algorithm relies little on polynomial division, and avoids multiplying by different factors when removing the denominators of Hensel factors. All algorithms are implemented using Magma, a computational algebra system and experiments indicate that our algorithm is more efficient.

Enhancing the extended hensel construction by using Gröbner basis

Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factors, the extended Hensel construction (EHC: [8]) uses multivariate initial Hensel factors determined by the Newton polygon (see below) of the given multivariate polynomial F (x, u ) ∈ K[ x , u ], where ( u ) = ( u 1 ,..., u ℓ ), with ℓ ≥ 2, and K is a number field. The F ( x, u ) may be such that its leading coefficient may vanish at ( u ) = ( 0 ) = (0,...,0), and even may be F ( x , 0 ) = 0. The EHC was used so far for computing series expansion of multivariate algebraic function determined by F ( x, u ) = 0, at critical points [8, 5] and for factorization [4, 1] and GCD computation [7] of F ( x , u ), without shifting the origin of u . It allows us to construct efficient algorithms for sparse multivariate polynomials [1, 7]. The EHC is another and promising approach than Zippel's sparse Hensel lifting [9, 10].

Sparse bivariate polynomial factorization

Motivated by Sasaki’s work on the extended Hensel construction for solving multivariate algebraic equations, we present a generalized Hensel lifting, which takes advantage of sparsity, for factoring bivariate polynomial over the rational number field. Another feature of the factorization algorithm presented in this article is a new recombination method, which can solve the extraneous factor problem before lifting based on numerical linear algebra. Both theoretical analysis and experimental data show that the algorithm is efficient, especially for sparse bivariate polynomials.

An approach to singularity from the extended Hensel construction

Let F ( x,u ), with ( u ) = ( u 1 ,..., u ℓ ), be an irreducible multivariate polynomial over C, having singularity at the origin, and let F New ( x,u ) be the so-called Newton polynomial for F ( x,u ). The extended Hensel construction (EHC in short) of F ( x,u ) allows us to compute the Puiseux-series roots if ℓ = 1 and, for ℓ ≥ 2, the roots which are fractional-power series w.r.t. the (weighted) total-degree of u 1 ,..., u ℓ . This paper investigates the behavior of algebraic function χ( u ) around the origin, where χ( u ) is a root of F ( x,u ), w.r.t. the variable x , and clarifies a close relationship between the singularity of χ( u ) and the corresponding Hensel factor. Let the irreducible factorization of F New in C[ x,u ] be F New ( x,u ) m 1 ... H r ( x,u ) m . It is shown that: 1) the procedure of EHC distributes the factors of the leading coefficient of F ( x,u ) to mutually prime factors of F New in a unique way, and the "scaled-root" x ( u ) becomes infinity only at the zero-points of the leading coefficient of the corresponding factor of F New ( x,u ); 2) the behavior of χ( u ) around the origin changes singularly at zero-points of resultant( H i , H j ) (∀ i ≠ j ), resultant( H i ,∂ H i /∂ x ) ( i = 1,..., r ), and fcirc; ℓ (ℓ = 1,...,ρ), where f ℓ is the sum of terms plotted at the ℓ-th vertex of bottom sides of the "Newton polygon". We explain these points not only theoretically but also by examples.

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.

Hensel construction of F(x, u 1 , ..., x l ) l ≥ 2 at a singular point and its applications

In 1993, Sasaki and Kako proposed a Hensel-like construction of F ( x, u 1 ,¡­, u l ), l ¡Ý 2, at a singular point where the conventional generalized Hensel construction breaks down. In this paper, we first extend Sasaki-Kako's method so as to apply to polynomials with vanishing leading coefficients. Then, we investigate a special case that the initial factors are polynomials. We prove that the multivariate polynomial can be decomposed at any singular point into factors which are polynomials in a main variable with coefficients being (infinite) series of rational functions such that ¡Æ k =0 ¡Þ [ N k ( u 1 ¡­, u l )/ D k ( u 1 ,¡­, u l )]. Here, N k and D k are homogeneous polynomials in u 1 - s 1 ,¡­, u l - s l and tdeg( N k ) - tdeg( D k ) = k, where ( s 1 ,¡­, s l ) is the expansion point and tdeg denotes the total-degree. The extended Hensel construction can be used to factorization of multivariate polynomials having a singular point at the origin. After performing the extended Hensel construction at the origin, we search for the smallest subsets of Hensel factors such that the product of the members of each subset contains no rational function. Then, we obtain the factorization in K { u 1 ,¡­, u l }[ x ], with K a number field. Next, we search for the smallest subsets such that the product of the members of each subset contains no infinite series. Then, we obtain the factorization in K [ x, u 1 ,¡­, u l ], without employing the so-called nonzero substitution.

Analytic continuation and Riemann surface determination of algebraic functions by computer

Lety=f(x) be an algebraic function which is defined as a root of bivariate polynomialP(x, y). GivenP(x, y), we present a practical computation method for analytic continuation and Riemann surface determination off(x). First, we determine the branch points off(x) by calculating the roots of a univariate polynomial numerically. Next, we expandf(x) into the Puiseux series at each branch point by the extended Hensel construction method proposed by Sasaki and Kako recently. Then, using well-known Smith’s theorem, we determine which series are connected the each other for each pair of branch points, by numerically estimating the values of the series at some middle point between the branch points. This allows us to determine the Riemann surface off(x) completely. Analytic continuation is performed similarly. We employ floating-point number arithmetic to perform these calculations.