System Of Multivariate Polynomial Equations Research Articles

Rational Univariate Reduction via toric resultants

We describe algorithms for solving a given system of multivariate polynomial equations via the Rational Univariate Reduction (RUR). We compute the RUR from the toric resultant of the input system. Our algorithms derandomize several of the choices made in similar prior algorithms. We also propose a new derandomized algorithm for solving an overdetermined system. Finally, we analyze the computational complexity of the algorithm, and discuss its implementation and performance.

On the probability distribution of data at points in real complete intersection varieties

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multi-variate polynomial equations and convergence radius of Newton's operator for under-determined system of multi-variate polynomial equations.

Public-key cryptography using paraunitary matrices

In this paper, we propose an algebraic approach for designing multivariate cryptosystems. Our method is based on formulating a general system of multivariate polynomial equations by paraunitary matrices. These matrices are a special family of invertible polynomial matrices that can be completely parameterized and efficiently generated by primitive building blocks. Using the general formulation that involves paraunitary matrices, we design a one-way function that operates over the fields of characteristic two. In order to include a trapdoor, we make some approximations to the paraunitary matrix. The result is a trapdoor one-way function that is efficient to evaluate but hard to invert unless secret information about the trapdoor is known. Using this function, we propose a paraunitary asymmetric cryptosystem (PAC). We present an instance of the PAC and show how it can be efficiently implemented. This instance operates on the finite field GF(256). The message block consists of 16 to 32 symbols from GF(256), i.e., the block size n is an integer between 16 and 32. The ciphertext block takes its elements from the same field and has at least ten extra symbols. We show that the encryption and decryption can be efficiently performed with complexities O(n3) and O(n2), respectively, where n is the size of the message block. Comparing complexities of the PAC to those in the hidden-field equation (HFE) family, we show that the PAC is faster in public-key generation and decryption. We study the computational security of the PAC. In addition, we show that the attacks developed for the HFE are not applicable on the PAC

On the probability distribution of singular varieties of given corank

We exhibit sharp upper bounds for the probability distribution of the distance from a system of multivariate polynomial equations to the strata of all systems having a critical zero of given corank. We also prove sharp upper bounds for the probability distribution of the condition number of singular systems of multivariate polynomial equations. We finally state a new and sharp technique of the Geometry of Numbers. Using this technique we show that rational systems of multivariate polynomial equations are equidistributed with respect to singular systems having a critical zero of given corank.

Determining the number of real roots of polynomials through neural networks

The ability of feedforward neural networks to identify the number of real roots of univariate polynomials is investigated. Furthermore, their ability to determine whether a system of multivariate polynomial equations has real solutions is examined on a problem of determining the structure of a molecule. The obtained experimental results indicate that neural networks are capable of performing this task with high accuracy even when the training set is very small compared to the test set.

Calculation of the slowness vector from the ray vector in anisotropic media

The wave quantities needed in constructing wave fields propagating in anisotropic elastic media are usually calculated as a function of the slowness vector, or of its direction called the wave normal. In some applications, however, it is desirable to calculate the wave quantities as a function of the ray direction. In this paper, a method of calculating the slowness vector for a specified ray direction is proposed. The method is applicable to general anisotropy of arbitrary strength with arbitrary complex wave surface. The slowness vector is determined by numerically solving a system of multivariate polynomial equations of the sixth order. By solving the equations, we obtain a complete set of slowness vectors corresponding to all wave types and to all branches of the wave surface including the slowness vectors along the acoustic axes. The wave surface can be folded to any degree. The system of equations is further specified for rays shot in the symmetry plane of an orthorhombic medium and for a transversely isotropic medium. The system is decoupled into two polynomial equations of the fourth order for the P –SV waves, and into equations for the SH wave, which yield an explicit closed-form solution. The presented approach is particularly advantageous in constructing ray fields, ray-theoretical Green functions, wavefronts and wave fields in strong anisotropy.

Deformation techniques to solve generalised Pham systems

In Heintz et al. (Electron. J. SADIO 1(1) (1998) 37), Castro et al. (Found., Comput. Math. (2003) to appear) and Pardo (Proceedings EACA’2000, 2000, pp. 25–51), the authors have shown that universal solving procedures require exponential running time. Roughly speaking, a universal solving procedure takes as input a system of multivariate polynomial equations and outputs complete symbolic information on the solution variety. Here, we introduce a non-universal solving procedure adapted to Generalised Pham Systems. The aim is to compute partial information of the variety defined by the input system. The Algorithm is based on an homotopic deformation and on a non-Archimedean lifting procedure from a non-singular zero of the homotopy curve. The complexity of the procedure is also stated and it depends on some intrinsic quantity called the deformation degree of the given input system.

The Hardness of Polynomial Equation Solving

Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids “unnecessary” branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P$ cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.

The Distribution of Condition Numbers of Rational Data of Bounded Bit Length

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n 4 log n) + log w , the condition number k(M) satisfies k(M) ≤ w n 5/2 with probability at least 1-2w -1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length.

Kronecker's and Newton's Approaches to Solving: A First Comparison

These pages are a first attempt to compare the efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker's solution (from the symbolic approach) with approximate zero theory (introduced by S. Smale as a foundation of numerical analysis). For this purpose we show upper and lower bounds of the bit length of approximate zeros. We also introduce efficient procedures that transform local Kronecker solutions into approximate zeros and conversely. As an application of our study we exhibit an efficient procedure to compute splitting field and Lagrange resolvent of univariate polynomial equations. We remark that this procedure is obtained by a convenient combination of both approaches (numeric and symbolic) to multivariate polynomial solving.

Closed-form displacement and coupler curve analysis of planar multi-loop mechanisms using Gröbner bases

The displacement analysis problem for planar and spatial mechanisms can be written as a system of algebraic equations in particular as a system of multi-variate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods which have been used to solve this problem. This paper explores an alternate approach, based on Gröbner bases, to solve the displacement analysis problem for planar mechanisms. It is shown that the reduced set of generators obtained using Buchberger's algorithm for Gröbner bases not only yields the input–output polynomial for the mechanism, but also provides comprehensive information on the number of closures and the relationships between various links of the mechanism. Numerical examples illustrating the applicability of Gröbner bases to displacement analysis of 10- and 12-link mechanisms, and determination of coupler curve equation for an 8-link mechanism are presented. It is seen that even though the Gröbner bases method is versatile enough to handle finitely solvable as well as over-constrained systems of equations, it can run into computational problems due to rapidly growing numerical coefficients and/or the set of generators. The examples presented show how these difficulties can be overcome by artificially decoupling complex mechanisms to help facilitate their closed-form analysis.

The Projective Noether Maple Package: Computing the Dimension of a Projective Variety

Recent theoretical advances in elimination theory use straight-line programs as a data structure to represent multivariate polynomials. We present here the Projective Noether Package which is a Maple implementation of one of these new algorithms, yielding as a byproduct a computation of the dimension of a projective variety. Comparative results on benchmarks for time and space of several families of multivariate polynomial equation systems are given and we point out both weaknesses and advantages of different approaches.

Multivariate Polynomials, Duality, and Structured Matrices

We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues, and relations between them are studied. Finally, we show some applications of this study to rootfinding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians, and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to the matrix eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations.

A Gro¨bner-Sylvester Hybrid Method for Closed-Form Displacement Analysis of Mechanisms

The displacement analysis problem for planar and spatial mechanisms can be written as a system of multivariate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods that have been used to solve this problem. This paper presents a new approach to displacement analysis using the reduced Gro¨bner basis form of a system of equations under degree lexicographic (dlex) term ordering of its monomials and Sylvester’s Dialytic elimination method. Using the Gro¨bner-Sylvester hybrid approach, a finitely solvable system of equations F is transformed into its reduced Gro¨bner basis G using dlex term ordering. Next, using the entire or a subset of the set of generators in G, the Sylvester’s matrix is assembled. The vanishing of the resultant, given as the determinant of Sylvester’s matrix, yields the necessary condition for polynomials in G (as well as F) to have a common factor. The proposed approach appears to provide a systematic and rational procedure to the problem discussed by Roth, dealing with the generation of (additional) equations for constructing the Sylvester’s matrix. Three examples illustrating the applicability of the proposed approach to displacement analysis of planar and spatial mechanisms are presented. The first and second examples address the forward displacement analysis of the general 6-6 Stewart mechanism and the 6-6 Stewart platform, whereas the third example deals with the determination of the I/O polynomial of an 8-link 1-DOF mechanism that does not contain any 4-link loop. [S1050-0472(00)01204-6]

Lower bounds for diophantine approximations

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero-dimensional polynomial equation system. This result represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight-line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton's algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

Multidimensional structured matrices and polynomial systems

We apply and extend some well-known and some recent techniques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing, that is, computations with structured matrices and solving a system of polynomial equations. In the first part of our paper, we extend the Toeplitz and Hankel structures of matrices and some of their known properties to some new classes of structured (quasi-Hankel and quasi-Toeplitz) matrices, naturally associated to systems of multivariate polynomial equations. In the second part of the paper, we prove some relations between these structured matrices, which extend the classical relations of the univariate case.