An algorithm for decomposing a polynomial system into normal ascending sets

We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets.If the polynomial system is zero dimensional,the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.     

Enclosure of the Zero Set of Multivariate Exponential Interval Polynomials

The zero set of one general multivariate exponential polynomial with interval coefficients is enclosed by unions and intersections of closed half-spaces. Tighter enclosures are derived in the bivariate case. Common zeros of polynomial systems can be located by an appropriate intersection of these enclosure sets in an appropriate space. The resulting domains are directly brought into polynomial equation solvers.     

多项式组的特征分解

任一多项式理想的特征对是指由该理想的约化字典序Grobner基G和含于其中的极小三角列C构成的有序对(G,C).当C为正则列或正规列时,分别称特征对(G,C)为正则的或正规的.当G生成的理想与C的饱和理想相同时,称特征对(G,C)为强的.一组多项式的(强)正则或(强)正规特征分解是指将该多项式组分解为有限多个(强)正则或(强)正规特征对,使其满足特定的零点与理想关系.本文简要回顾各种三角分解及相应零点与理想分解的理论和方法,然后重点介绍(强)正则与(强)正规特征对和特征分解的性质,说明三角列、Ritt特征列和字典序Grobner基之间的内在关联,建立特征对的正则化定理以及正则、正规特征对的强化方法,进而给出两种基于字典序Grobner基计算、按伪整除关系分裂和构建、商除可除理想等策略的(强)正规与(强)正则特征分解算法.这两种算法计算所得的强正规与强正则特征对和特征分解都具有良好的性质,且能为输入多元多项式组的零点提供两种不同的表示.本文还给出示例和部分实验结果,用以说明特征分解方法及其实用性和有效性.     

多项式方程组的主项解耦消元法

本文提出多项式组符号求解的主项解耦 (主项只含主元 )消元法 :视多项式为变元不同幂积的线性组合 ,以主项解耦三角型多项式组 DTS为引导 ,用逐项伪除求余式 ,将多项式组 PS化为与其同解的 DTS.内容涉及 :消元算法、DTS的存在性与结构特性、零点集结构公式等 .亦对 Grobner基法、吴文俊消元法与本文方法之间的相互联系、区别以及特点进行了比较 .研究表明主项解耦消元法适用于一般多项式组且效率较高     

有限域上多项式集的简单分解

研究如何将任意有限域上的多项式集分解为有限多个简单列.为了解决这一问题,首先研究简单列和根理想之间的关系,然后基于已有的正则分解算法和有限域上理想的根的两种计算方法设计一个有限域上多项式集的简单分解算法.计算试验表明,文章给出的算法是有效的.     

Continuity properties for flat families of polynomials (I) Continuous parametrizations

Inspired by classical results in algebraic geometry, we study the continuity with respect to the coefficients, of the zero set of a system of complex homogeneous polynomials with a given pattern and when the Hilbert polynomial of the generated ideal is fixed. In this work we prove topological properties of some classifying spaces, e.g. the space of systems with given pattern, fixed Hilbert polynomial is locally compact, and we establish continuous parametrizations of Nullstellensatz formulae. In the general case we get local rational results but in the complex case we get global results using rational polynomials in the real and imaginary parts of the coefficients. In a second companion paper, we shall treat the continuity of zero sets for the Hausdorff distance, i.e., from a metric point of view.     

Regenerative cascade homotopies for solving polynomial systems

A key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set. In many problems involving a solution set of a polynomial system, the witness superset contains all the needed information. Sommese and Wampler gave the first numerical method to compute witness supersets, based on dimension-by-dimension slicing of the solution set by generic linear spaces, followed later by the cascade homotopy of Sommese and Verschelde. Recently, the authors of this article introduced a new method, regeneration, to compute solution sets of polynomial systems. Tests showed that combining regeneration with the dimension-by-dimension algorithm was significantly faster than naively combining it with the cascade homotopy. However, in this article, we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets. Computational tests give strong evidence that regenerative cascade is superior in practice to previous methods.     

Obstructions to determinantal representability

There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.     

A note on separation of algebraic sets and the ?ojasiewicz exponent for polynomial mappings

In the paper a global separation problem for affine algebraic sets is considered. As application an upper bound for the distance of the graph of polynomial mapping to its zero set in the form of a ?ojasiewicz inequality is given.     

Connectedness structure of the solution sets of vector variational inequalities

By a scalarization method and properties of semi-algebraic sets, it is proved that both the Pareto solution set and the weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. Consequences of the results for vector optimization problems are discussed in details. The results of this paper solve in the affirmative some open questions for the case of general problems without requiring monotonicity of the operators involved.     

Symmetries of the Julia sets of Newton’s method for multiple root

The symmetries of Julia sets of Newton’s method is investigated in this paper. It is shown that the group of symmetries of Julia set of polynomial is a subgroup of that of the corresponding standard, multiple and relax Newton’s method when a nonlinear polynomial is in normal form and the Julia set has finite group of symmetries. A necessary and sufficient condition for Julia sets of standard, multiple and relax Newton’s method to be horizontal line is obtained.     

On fundamental solutions for multivariate singular interpolation

The objective of this paper is to give a constructive approach to the solution of the fundamental functions for cardinal interpolation from a shift-invariant space generated by the (multi-)integer translates of some compactly supported function whose polynomial symbol has a non-empty zero set. This problem was first introduced by Chui, Diamond, and Raphael, where explicit solutions were given for various zero sets. Later, de Boor, Höllig, and Riemenschneider gave an existence proof for zero sets which are more general. In this paper, we give an integral representation of the fundamental solutions that can be made explicit in some cases and we will also give a growth condition of such fundamental solutions. The four-directional box splines will be used as an illustrative example.     

Conical Kakeya and Nikodym sets in finite fields

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.)Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.     

Test sets of integer programs

This article is a survey about recent developments in the area of test sets of families of linear integer programs. Test sets are finite subsets of the integer lattice that allow to improve any given feasible non-optimal point of an integer program by one element in the set. There are various possible ways of defining test sets depending on the view that one takes: theGraver test set is naturally derived from a study of the integral vectors in cones; theScarf test set (neighbors of the origin) is strongly connected to the study of lattice point free convex bodies; the so-calledreduced Gröbner basis of an integer program is obtained from a study of generators of polynomial ideals. This explains why the study of test sets connects various branches of mathematics. We introduce in this paper these three kinds of test sets and discuss relations between them. We also illustrate on various examples such as the minimum cost flow problem, the knapsack problem and the matroid optimization problem how these test sets may be interpreted combinatorially. From the viewpoint of integer programming a major interest in test sets is their relation to the augmentation problem. This is discussed here in detail. In particular, we derive a complexity result of the augmentation problem, we discuss an algorithm for solving the augmentation problem by computing the Graver test set and show that, in the special case of an integer knapsack problem with 3 coefficients, the augmentation problem can be solved in polynomial time.Supported by a Gerhard-Hess-Forschungsförderpreis of the German Science Foundation (DFG).     

A New Approach to Extracting Roots in Garside Groups

We study the problem of extracting roots in Garside groups by reducing it to the calculation of certain ultra summit sets. Several properties concerning the roots and an effective algorithm are derived. In particular, in the case of braid groups, a conjecture on the bound of super summit set implies that, for fixed number of strands and ordinal number of root, the algorithm is polynomial in the word length.     

Sum of Cantor sets: Self-similarity and measure

In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension).

     

The order of Lebesgue constant of Lagrange interpolation on several intervals

We consider Lagrange interpolation on the set of finitely many intervals. This problem is closely related to the least deviating polynomial from zero on such sets. We will obtain lower and upper estimates for the corresponding Lebesgue constant. The case of two intervals of equal lengths is simpler, and an explicit construction for two non-symmetric intervals will be given only in a special case.     

基于二次多项式的积极集光滑化max函数及其在无约束minimax问题中的应用

本文基于分段二次多项式方程,构造了一种积极集策略的光滑化max函数.通过给出与光滑化max函数相关的分量函数指标集的直接计算方法,将分段二次多项式方程转化为一般二次多项式方程.利用二次多项式方程根的性质,给出了该光滑化max函数的稳定计算策略,证明了其具有一阶光滑性,其梯度函数具有局部Lipschitz连续性和强半光滑性.该光滑化max函数仅与函数值较大的分量函数相关,适用于含分量函数较多且复杂的max函数的问题.为了验证其效率,本文基于该函数构造了一种解含多个复杂分量函数的无约束minimax问题的光滑化算法,数值实验表明了该光滑化max函数的可行性及有效性.     

How to construct totally disconnected Markov sets?

We deduce a polynomial estimate on a compact planar set from a polynomial estimate on its circular projection, which enables us to prove Markov and Bernstein-Walsh type inequalities for certain sets. We construct