An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

An Elimination Method for Polynomial Systems

AbstractWe present an elimination method for polynomial systems, in the form of three main algorithms. For any given system [P,Q] of two sets of multivariate polynomials, one of the algorithms computes a sequence of triangular forms T1,…,Te and polynomial sets U1,…,Ue such that Zero(P/Q) = ∪ei=1 Zero(Ti/Ui), where Zero(P/Q) denotes the set of common zeros of the polynomials in P which are not zeros of any polynomial in Q, and similarly for Zero(Ti/Ui). The two other algorithms compute the same zero decomposition but with nicer properties such as Zero(Ti/Ui) ≠ &0slash; for each i. One of them, for which the computed triangular systems [Ti, Ui] possess the projection property, provides a quantifier elimination procedure for algebraically closed fields. For the other, the computed triangular forms Ti are irreducible. The relationship between our method and some existing elimination methods is explained. Experimental data for a set of test examples by a draft implementation of the method are provided, and show that the efficiency of our method is comparable with that of some well-known methods. A few encouraging examples are given in detail for illustration

Resultant-based Elimination in Ore Algebra

We consider resultant-based methods for elimination of indeterminates of Ore polynomial systems in Ore algebra. We start with defining the concept of resultant for bivariate Ore polynomials then compute it by the Dieudonne determinant of the polynomial coefficients. Additionally, we apply noncommutative versions of evaluation and interpolation techniques to the computation process to improve the efficiency of the method. The implementation of the algorithms will be performed in Maple to evaluate the performance of the approaches.Comment: An updated (and shorter) version published in the SYNASC '21 proceedings (IEEE CS) with the title "Resultant-based Elimination for Skew Polynomials

A Methodology for Solving the Equations Arising in Nonlinear Parameter Identification Problems: Application to Induction Machines

This dissertation presents a method that can be used to identify the parameters of a class of systems whose regressor models are nonlinear in the parameters. The technique is based on classical elimination theory, and it guarantees that the solution for the parameters which minimize a least-squares criterion can be found in a finite number of steps. The proposed methodology begins with an input-output linear overparameterized model whose parameters are rationally related. After making appropriate substitutions that account for the overparameterization, the problem is transformed into a nonlinear least-squares problem that is not overparameterized. The extrema equations are computed, and a nonlinear transformation is carried out to convert them to polynomial equations in the unknown parameters. It is then show how these polynomial equations can be solved using elimination theory using resultants. The optimization problem reduces to a numerical computation of the roots of a polynomial in a single variable. This nonlinear least-squares method is applied to the identification of the parameters of an induction motor. A major difficulty with the induction motor is that the rotor’s state variables are not available measurements so that the system identification model cannot be made linear in the parameters without overparameterizing the model. Previous work in the literature has avoided this issue by making simplifying assumptions such as a “slowly varying speed”. Here, no such simplifying assumptions are made. This method is implemented online to continuously update the parameter values. Experimental results are presented to verify this method. The application of this nonlinear least-squares method can be extended to many research areas such as the parameter identification for Hammerstein models. In principle, as long as the regressor model is such that the system parameters are rationally related, the proposed method is applicable

Methods for Optimal Model Fitting and Sensor Calibration

The problem of fitting models to measured data has been studied extensively, not least in the field of computer vision. A central problem in this field is the difficulty in reliably find corresponding structures and points in different images, resulting in outlier data. This thesis presents theoretical results improving the understanding of the connection between model parameter estimation and possible outlier-inlier partitions of data point sets. Using these results a multitude of applications can be analyzed in respects to optimal outlier inlier partitions, optimal norm fitting, and not least in truncated norm sense. Practical polynomial time optimal solvers are derived for several applications, including but not limited to multi-view triangulation and image registration. In this thesis the problem of sensor network self calibration is investigated. Sensor networks play an increasingly important role with the increased availability of mobile, antenna equipped, devices. The application areas can be extended with knowledge of the different sensors relative or absolute positions. We study this problem in the context of bipartite sensor networks. We identify requirements of solvability for several configurations, and present a framework for how such problems can be approached. Further we utilize this framework to derive several solvers, which we show in both synthetic and real examples functions as desired. In both these types of model estimation, as well as in the classical random samples based approaches minimal cases of polynomial systems play a central role. A majority of the problems tackled in this thesis will have solvers based on recent techniques pertaining to action matrix solvers. New application specific polynomial equation sets are constructed and elimination templates designed for them. In addition a general improvement to the method is suggested for a large class of polynomial systems. The method is shown to improve the computational speed by significant reductions in the size of elimination templates as well as in the size of the action matrices. In addition the methodology on average improves the numerical stability of the solvers

Inverse kinematics and path planning of manipulator using real quantifier elimination based on Comprehensive Gr\"obner Systems

Methods for inverse kinematics computation and path planning of a three degree-of-freedom (DOF) manipulator using the algorithm for quantifier elimination based on Comprehensive Gr\"obner Systems (CGS), called CGS-QE method, are proposed. The first method for solving the inverse kinematics problem employs counting the real roots of a system of polynomial equations to verify the solution's existence. In the second method for trajectory planning of the manipulator, the use of CGS guarantees the existence of an inverse kinematics solution. Moreover, it makes the algorithm more efficient by preventing repeated computation of Gr\"obner basis. In the third method for path planning of the manipulator, for a path of the motion given as a function of a parameter, the CGS-QE method verifies the whole path's feasibility. Computational examples and an experiment are provided to illustrate the effectiveness of the proposed methods.Comment: 26 pages. arXiv admin note: text overlap with arXiv:2111.0038