Zero-dimensional Polynomial Research Articles

Squarefree normal representation of zeros of zero-dimensional polynomial systems

For any zero-dimensional polynomial ideal I and any nonzero polynomial F, this paper shows that the union of the multi-set of zeros of the ideal sum I+〈F〉 and that of the ideal quotient I:〈F〉 is equal to the multi-set of zeros of I, where zeros are counted with multiplicities. Based on this zero relation and the computation of Gröbner bases, a complete multiplicity-preserved algorithm is proposed to decompose any zero-dimensional polynomial set into finitely many squarefree normal triangular sets, resulting in a squarefree normal representation for the zeros of the polynomial set. In the representation the multiplicities of the zeros of the triangular sets can be read out directly. Examples and experiments are presented to illustrate the algorithm and its performance.

An Improvement of the Rational Representation for High-Dimensional Systems

Based on the rational univariate representation of zero-dimensional polynomial systems, Tan and Zhang proposed the rational representation theory for solving a high-dimensional polynomial system, which uses so-called rational representation sets to describe all the zeros of a high-dimensional polynomial system. This paper is devoted to giving an improvement for the rational representation. The idea of this improvement comes from a minimal Dickson basis used for computing a comprehensive Grobner system of a parametric polynomial system to reduce the number of branches. The authors replace the normal Grobner basis G satisfying certain conditions in the original algorithm (Tan-Zhang’s algorithm) with a minimal Dickson basis Gm of a Grobner basis for the ideal, where Gm is smaller in size than G. Based on this, the authors give an improved algorithm. Moreover, the proposed algorithm has been implemented on the computer algebra system Maple. Experimental data and its performance comparison with the original algorithm show that it generates fewer branches and the improvement is rewarding.

Segre class computation and practical applications

Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

The point decomposition problem over hyperelliptic curves

Computing discrete logarithms is generically a difficult problem. For divisor class groups of curves defined over extension fields, a variant of the Index-Calculus called Decomposition attack is used, and it can be faster than generic approaches. In this situation, collecting the relations is done by solving multiple instances of the Point m-Decomposition Problem (PDP$_m$). An instance of this problem can be modelled as a zero-dimensional polynomial system. Solving is done with Grobner bases algorithms, where the number of solutions of the system is a good indicator for the time complexity of the solving process. For systems arising from a PDP$_m$ context, this number grows exponentially fast with the extension degree. To achieve an efficient harvesting, this number must be reduced as much as as possible. Extending the elliptic case, we introduce a notion of Summation Ideals to describe PDP m instances over higher genus curves, and compare to Nagao's general approach to PDP$_m$ solving. In even characteristic we obtain reductions of the number of solutions for both approaches, depending on the curve's equation. In the best cases, for a hyperelliptic curve of genus $g$, we can divide the number of solutions by $2^{(n−1)(g+1)}$. For instance, for a type II genus 2 curve defined over $\mathbb{F}_{2^{93}}$ whose divisor class group has cardinality a near-prime 184 bits integer, the number of solutions is reduced from 4096 to 64. This is enough to build the matrix of relations in around 7 days with 8000 cores using a dedicated implementation.

On the last fall degree of zero-dimensional Weil descent systems

In this article we will discuss a mostly theoretical framework for solving zero-dimensional polynomial systems. Complexity bounds are obtained for solving such systems using a new parameter, called the last fall degree, which does not depend on the choice of a monomial order. The method is similar to certain MutantXL algorithms, but our abstract formulation has advantages. For example, we can prove that the cryptographic systems multi-HFE and HFE are insecure.More generally, let k be a finite field of cardinality qn and let k′ be the subfield of cardinality q. Let F⊂k[X0,…,Xm−1] be a finite subset generating a zero-dimensional ideal. We give an upper bound of the last fall degree of the Weil descent system of F from k to k′, which depends on q, m, the last fall degree of F, the degree of F and the number of solutions of F, but not on n. This shows that such Weil descent systems can be solved efficiently if n grows and the other parameters are fixed. In particular, one can apply these results to show a weakness in the cryptographic protocols HFE and multi-HFE.

Explicit H∞ controllers for 1st to 3rd order single-input single-output systems with parameters

The purpose of this paper is to explicitly characterize H∞ controllers for single-input single-output (SISO) systems of order 1, 2 and 3 in terms of their coefficients considered as unknown parameters. In the SISO case, computing H∞ controllers requires to find the real positive definite solution of an algebraic Riccati equation. Due to the system parameters, no purely numerical method can be used to find such a solution, and thus parametric H∞ controllers. Using elimination techniques of zero-dimensional polynomial systems, we first give a parametrization of all the solutions of the algebraic Riccati equation associated with the H∞ control problem. Since the problem reduces to solving an univariate polynomial of degree less than or equal to 4, closed-form solutions are then obtained for the solutions of the algebraic Riccati equation by means of radicals. Using the concept of discriminant variety, we show that the maximal real root of this polynomial is always defined by the same closed-form expression, which yields the positive definite solution of the algebraic Riccati equation. Finally, we use the above results to explicitly compute the H∞ criteria γpt and H∞ controllers in terms of the system parameters and study them with respect to parameter variations.

Explicit H∞ controllers for 4th order single-input single-output systems with parameters and their application to the two mass-spring system with damping

The purpose of this paper is to explicitly characterize H∞ controllers for 4th order single-input single-output (SISO) systems in terms of their coefficients considered as unknown parameters. In the SISO case, computing H∞ controllers requires to find the real positive definite solution of an algebraic Riccati equation (ARE). Due to the system parameters, no purely numerical method can be used to find such a solution, and thus parametric H∞ controllers. Using elimination techniques for zero-dimensional polynomial systems, we first give a rational parametrization of all the solutions of the ARE. Then, as the problem reduces to solving polynomials of degree 4, closed-form solutions are obtained for all the solutions of this ARE by using expressions by radicals. Using the concept of discriminant variety, we then show that the maximal real root of one of these polynomials is encoded by two different closed-form expressions depending on the values of the system parameters, which yields to different positive definite solution of the ARE. The above results are then used to explicitly compute the H∞ criterion γopt and H∞ controllers in terms of the system parameters. Finally, we study in detail a particular system: the two-mass-spring system with damping. Due to the low number of parameters, we can plot the variations of γopt in function of the parameters, compute approximations of γopt at a working point, and derive the expression of a weight function of the parameters to set γopt to a desired value.

A generic position based method for real root isolation of zero-dimensional polynomial systems

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is O˜B(N10) for the bivariate case, where N=max⁡(d,τ), d resp., τ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.

Certified rational parametric approximation of real algebraic space curves with local generic position method

In this paper, an algorithm is given for determining the topology of an algebraic space curve and to compute a certified G1 rational parametric approximation of the algebraic space curve. The algorithm works by extending to dimension one the local generic position method for solving zero-dimensional polynomial equation systems. Here, certified means that the approximation curve and the original curve have the same topology and their Hausdorff distance is smaller than a given precision. The main advantage of the algorithm, inherited from the local generic position method, is that the topology computation and approximation for a space curve are directly reduced to the same tasks for two plane curves. In particular, an error bound of the approximation space curve is deduced explicitly from the error bounds of the approximation plane curves. The complexity of the algorithm is also analyzed. Its effectivity is shown on some non-trivial examples.

Localization and Perturbations of Roots to Systems of Polynomial Equations

We establish estimates for the sums of absolute values of solutions of a zero-dimensional polynomial system. By these estimates, inequalities for the counting function of the roots are derived. In addition, bounds for the roots of perturbed systems are suggested.

Root isolation of zero-dimensional polynomial systems with linear univariate representation

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the complex roots of the polynomial system are represented as linear combinations of the roots of several univariate polynomial equations. An algorithm is proposed to compute such a representation for a given zero-dimensional polynomial equation system based on Gröbner basis computation. The main advantage of this representation is that the precision of the roots of the system can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain roots of the polynomial system with a given precision. As a consequence, a root isolating algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.

Pivoting in Extended Rings for Computing Approximate Gröbner Bases

It is well known that in the computation of Grobner bases arbitrarily small perturbations in the coefficients of polynomials may lead to a completely different staircase, even if the solutions of the polynomial system change continuously. This phenomenon is called artificial discontinuity in Kondratyev’s Ph.D. thesis. We show how such phenomenon may be detected and even “repaired” by using a new variable to rename the leading term each time we detect a “problem”. We call such strategy the TSV (Term Substitutions with Variables) strategy. For a zero-dimensional polynomial ideal, any monomial basis (containing 1) of the quotient ring can be found with the TSV strategy. Hence we can use TSV strategy to relax term order while keeping the framework of Grobner basis method so that we can use existing efficient algorithms (for instance the F5 algorithm) to compute an approximate Grobner basis. Our main algorithms, named TSVn and TSVh, can be used to repair artificial \({\epsilon}\)-discontinuities. Experiments show that these algorithms are effective for some nontrivial problems.

Real solution isolation with multiplicity of zero-dimensional triangular systems

Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions. In this paper, we define in a natural way the multiplicity of solutions of zero-dimensional triangular polynomial systems and prove that our definition is equivalent to the classical definition of local (intersection) multiplicity. Then we present an effective and complete algorithm for isolating real solutions with multiplicities of zero-dimensional triangular polynomial systems using our definition. The algorithm is based on interval arithmetic and square-free factorization of polynomials with real algebraic coefficients. The computational results on some examples from the literature are presented.

Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case

This paper presents algorithms for decomposing any zero-dimensional polynomial set into simple sets over an arbitrary finite field, with an associated ideal or zero decomposition. As a key ingredient of these algorithms, we generalize the squarefree decomposition approach for univariate polynomials over a finite field to that over the field product determined by a simple set. As a subprocedure of the generalized squarefree decomposition approach, a method is proposed to extract the p th root of any element in the field product. Experiments with a preliminary implementation show the effectiveness of our algorithms.

On factoring parametric multivariate polynomials

This paper presents a new algorithm for the absolute factorization of parametric multivariate polynomials over the field of rational numbers. This algorithm decomposes the parameters space into a finite number of constructible sets. The absolutely irreducible factors of the input parametric polynomial are given uniformly in each constructible set. The algorithm is based on a parametric version of Hensel's lemma and an algorithm for quantifier elimination in the theory of algebraically closed field in order to reduce the problem of finding absolute irreducible factors to that of representing solutions of zero-dimensional parametric polynomial systems. The complexity of this algorithm is single exponential in the number n of the variables of the input polynomial, its degree d w.r.t. these variables and the number r of the parameters.

Approximate computation of zero-dimensional polynomial ideals

The Buchberger–Möller algorithm is a well-known efficient tool for computing the vanishing ideal of a finite set of points. If the coordinates of the points are (imprecise) measured data, the resulting Gröbner basis is numerically unstable. In this paper we introduce a numerically stable Approximate Vanishing Ideal (AVI) Algorithm which computes a set of polynomials that almost vanish at the given points and almost form a border basis. Moreover, we provide a modification of this algorithm which produces a Macaulay basis of an approximate vanishing ideal. We also generalize the Border Basis Algorithm ([Kehrein, A., Kreuzer, M., 2006. Computing border bases. J. Pure Appl. Algebra 205, 279–295]) to the approximate setting and study the approximate membership problem for zero-dimensional polynomial ideals. The algorithms are then applied to actual industrial problems.

Evaluation techniques for zero-dimensional primary decomposition

This paper presents a new algorithm that computes the local algebras of the roots of a zero-dimensional polynomial equation system, with a number of operations in the coefficient field that is polynomial in the number of variables, in the evaluation cost of the equations and in a Bézout number.

On inverse systems and squarefree decomposition of zero-dimensional polynomial ideals

Let I be a zero-dimensional ideal in a polynomial ring F [ s ] : = F [ s 1 , … , s n ] over an arbitrary field F . We show how to compute an F -basis of the inverse system I ⊥ of I . We describe the F [ s ] -module I ⊥ by generators and relations and characterise the minimal length of a system of F [ s ] -generators of I ⊥ . If the primary decomposition of I is known, such a system can be computed. Finally we generalise the well-known notion of squarefree decomposition of a univariate polynomial to the case of zero-dimensional ideals in F [ s ] and present an algorithm to compute this decomposition.

Characterizations of border bases

This paper presents characterizations of border bases of zero-dimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, confluence of the corresponding rewrite relation, reduction of S-polynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebases is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute.

Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality

Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A=k[X 1 ,…,X n ]/ℐ, where ℐ is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the A-module structure of the dual space $\widehat{A}$ . An important feature of our algorithms is that we do not require $\widehat{A}$ to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 nD 5/2 ) and O(n2 nD 5/2 ) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.