Hensel lifting and bivariate polynomial factorisation over finite fields

This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials.

     

连续代数扩域上多项式因式分解的Trager算法

多项式的因式分解是符号计算中最基本的算法,二十世纪六十年代开始出现的关于多项式因式分解的工作被认为是符号计算领域的起源．目前多项式的因式分解已经成熟,并已在Maple等符号计算软件中实现,但代数扩域上的因式分解算法还有待进一步改进．代数扩域上的基本算法是Trager算法．Weinberger等提出了基于Hensel提升的算法．这些算法是在单个扩域上做因式分解．而在吴零点分解定理中,多个代数扩域上的因式分解是非常基本的一步,主要用于不可约升列的计算．为了解决这一问题,吴文俊,胡森、王东明分别提出了基于方程求解的多个扩域上的因式分解算法．王东明、林东岱提出了另外一个算法Trager算法相似,将问题化为有理数域上的分解．他们应用了吴的三角化算法,因此算法的终止性依赖于吴方法的计算．支丽红则将提升技巧用于多个扩域上的因式分解算法．本文将Trager的算法直接推广为连续扩域上的因式分解,只涉及结式计算与有理数域上的因式分解,给出了多个代数扩域上的因式分解一个直接的算法．     

Sharp precision in Hensel lifting for bivariate polynomial factorization

Popularized by Zassenhaus in the seventies, several algorithms for factoring polynomials use a so-called lifting and recombination scheme. Concerning bivariate polynomials, we present a new algorithm for the recombination stage that requires a lifting up to precision twice the total degree of the polynomial to be factored. Its cost is dominated by the computation of reduced echelon solution bases of linear systems. We show that our bound on precision is asymptotically optimal.

     

Lifting and recombination techniques for absolute factorization

In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.     

Algorithms for Commutative Algebras Over the Rational Numbers

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel–Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.     

A Fast and Stable Updating Algorithm for Bivariate Nonparametric Curve Estimation

Abstract

An updating algorithm for bivariate local linear regression is proposed. Thereby, we assume a rectangular design and a polynomial kernel constrained to rectangular support as weight function. Results of univariate regression estimators are extended to the bivariate setting. The updates are performed in a way that most of the well-known numerical instabilities of a naive update implementation can be avoided. Some simulation results illustrate the properties of several algorithms with respect to computing time and numerical stability.     

A lifting and recombination algorithm for rational factorization of sparse polynomials

We propose a new lifting and recombination scheme for rational bivariate polynomial factorization that takes advantage of the Newton polytope geometry. We obtain a deterministic algorithm that can be seen as a sparse version of an algorithm of Lecerf, with a polynomial complexity in the volume of the Newton polytope. We adopt a geometrical point of view, the main tool being derived from some algebraic osculation criterion in toric varieties.     

On the Key Equation Over a Commutative Ring

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by Algorithm MR of Norton.WhenR is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal ofR . We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring.     

Gcd of multivariate polynomials via Newton polytopes

We study geometric criteria to determine coprimality between multivariate polynomials. Our main contribution is the development of a polynomial-time algorithm (on the number of monomials) that detects coprimality of multivariate polynomials using Newton polytopes. We also show how to construct the gcd of two bivariate polynomials using their Newton polygons.     

多变量多项式矩阵素分解的结果和问题

多变量矩阵素分解问题是多维系统和信号处理学科中的基本问题,从提出以来,已经经过众多学者的研究.近年来随着符号代数计算的快速发展,情况有了根本的改变.概述近几年来该领域的研究进展情况和一些相关未解决的问题,从中可以看出该领域的研究密切依赖于机械化数学的研究进展.作为多变多项式矩阵分解理论的应用,给出两个变量多项式矩阵分解存在性定理的新证明.从中可以看出,2个变量多项式矩阵和3个以上变量矩阵的本质区别.     

Piecewise Rational Approximations of Real Algebraic Curves

1.IntroductionAnaJgebraicplanecurveCofdegreedinn2isimplicitlydefinedbyasinglepolynomialequationf(x,y)=Oofdegreedwithcoefficientsinn.Arationalalgebraiccurveofdegreedinn2canadditionaJlybedefinedbyrationalparametricequationswhicharegivenas(x=G1(u),y=G2(u)),whereG1andG2arerationalfunctionsinuofdegreed,i.e-,eachisaquotientofpolynomiaJ8inuofmtalmumdegreedwithcoefficientsinn.ffetionalcurvesaxeonlyasubsetofimplicitalgebraiccurvesofdegreed+1.Whi1eaJldegreetwocurves(conics)arerational,oIilyasubsetof…     

Finite segmentp-adic number systems with applications to exact computation

Proceedings - Mathematical Sciences - A fractional weighted number system, based on Hensel’sp-adic number system, is proposed for constructing a unique code (called Hensel’s code) for...     

Explicit polynomial preserving trace liftings on a triangle

We give an explicit formula for a right inverse of the trace operator from the Sobolev space H¹(T) on a triangle T to the trace space H^1/2(?T) on the boundary. The lifting preserves polynomials in the sense that if the boundary data are piecewise polynomial of degree N, then the lifting is a polynomial of total degree at most N and the lifting is shown to be uniformly stable independently of the polynomial order. Moreover, the same operator is shown to provide a uniformly stable lifting from L₂(?T) to H^1/2(T). Finally, the lifting is used to construct a uniformly bounded right inverse for the normal trace operator from the space H (div; T) to H^–1/2(?T) which also preserves polynomials. Applications to the analysis of high order numerical methods for partial differential equations are indicated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Accurate evaluation algorithm for bivariate polynomial in Bernstein–Bézier form

In this paper it is presented a compensated de Casteljau algorithm to accurately evaluate a bivariate polynomial in Bernstein–Bézier form. The principle is to apply error-free transformations to improve the traditional de Casteljau algorithm. A forward error and a running error analysis are performed. Finally, some numerical experiments illustrate the accuracy of the proposed algorithm in ill-conditioned problems.     

Amortized multi-point evaluation of multivariate polynomials

The evaluation of a polynomial at several points is called the problem of multi-point evaluation. Sometimes, the set of evaluation points is fixed and several polynomials need to be evaluated at this set of points. Several efficient algorithms for this kind of “amortized” multi-point evaluation have been developed recently for the special cases of bivariate polynomials or when the set of evaluation points is generic. In this paper, we extend these results to the evaluation of polynomials in an arbitrary number of variables at an arbitrary set of points. We prove a softly linear complexity bound when the number of variables is fixed. Our method relies on a novel quasi-reduction algorithm for multivariate polynomials, that operates simultaneously with respect to several orderings on the monomials.     

Quantitative estimations of bivariate summation‐integral–type operators

In this paper, we study the approximation properties of bivariate summation‐integral–type operators with two parameters . The present work deals within the polynomial weight space. The rate of convergence is obtained while the function belonging to the set of all continuous and bounded function defined on ([0],∞)(×[0],∞) and function belonging to the polynomial weight space with two parameters, also convergence properties, are studied. To know the asymptotic behavior of the proposed bivariate operators, we prove the Voronovskaya type theorem and show the graphical representation for the convergence of the bivariate operators, which is illustrated by graphics using Mathematica. Also with the help of Mathematica, we discuss the comparison by means of the convergence of the proposed bivariate summation‐integral–type operators and Szász‐Mirakjan‐Kantorovich operators for function of two variables with two parameters to the function. In the same direction, we compute the absolute numerical error for the bivariate operators by using Mathematica and is illustrated by tables and also the comparison takes place of the proposed bivariate operators with the bivariate Szász‐Mirakjan operators in the sense of absolute error, which is represented by table. At last, we study the simultaneous approximation for the first‐order partial derivative of the function.     

Spherically complete models of Hensel minimal valued fields

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is 0-h^mix-minimality (which, in equi-characteristic 0, amounts to 0-h-minimality).     

The use of finite-segmentp-adic arithmetic for exact computation

A finite number system for doing exact computer arithmetic, due to Krishnamurthy, Rao, and Subramanian, is described. For each rational numbera/b, with |a| and |b| suitably bounded, the firstr digits of the (infinite)p-adic expansion ofa/b are used as a coded representation fora/b (the Hensel code). Arithmetic operations on the Hensel codes produce Hensel codes for the exact results of the arithmetic operations.     

Abelsche Galoiserweiterungen von R[X]

Our main result states that for a commutative ring R and a finite abelian group G the following conditions are equivalent: (a) Gal(R,G)=Gal (R[X],G), i.e. every commutative Galois extension of R[X]with Galois group G is extended from R. (b) The order of G is a non-zero-divisor in R/Nil(R). The proof uses lifting properties of Galois extensions over Hensel pairs and a Milnor-type patching theorem.     

Constrained multi-degree reduction of triangular Bézier surfaces

This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.