Polynomial Systems Research Articles

Classification of 3-Degree-of-Freedom 3-UPU Translational Parallel Mechanisms Based on Constraint Singularity Loci Using Gröbner Cover

Abstract A 3-degree-of-freedom (DOF) 3-UPU translational parallel mechanism (TPM) is one of the typical TPMs. Despite comprehensive studies on 3-UPU TPMs in which the joint axes on the base and the moving platform are coplanar, only a few 3-UPU TPMs with skewed base and moving platform have been proposed, and the impact of link parameters on constraint singularity loci of such TPMs has not been systematically investigated. The advances in computing comprehensive Gröbner system (CGS) or Gröbner cover of parametric polynomial systems provide an efficient tool for solving this problem. This paper presents a systematic classification of 3-UPU TPMs with skewed base and moving platform based on constraint singularity loci. First, the constraint singularity equation of a 3-UPU TPM is derived. Using Gröbner cover, the 3-UPU TPMs are classified into 12 types. Finally, a novel 3-UPU TPM is proposed. Reconfiguration analysis shows that unlike most existing 3-UPU TPMs which can transit from a 3-DOF translational mode to two or more 3-DOF operation modes, the proposed 3-UPU TPM can only transit from a 3-DOF translational mode to one general 3-DOF operation mode. The singularity locus divides the workspace of this 3-UPU TPM into two constraint singularity-free regions. As a by-product, a 3-UPU parallel mechanism that the moving platform can undergo 3-DOF translation and 1-DOF infinitesimal rotation is revealed. This work provides a solid foundation for the design of 3-UPU TPMs and a starting point for the classification of 3-UPU parallel mechanisms.

Betti Structures of Hypergeometric Equations

Abstract We study Betti structures in the solution complexes of confluent hypergeometric equations. We use the framework of enhanced ind-sheaves and the irregular Riemann–Hilbert correspondence of D’Agnolo–Kashiwara. The main result is a group theoretic criterion that ensures that enhanced solutions of such systems are defined over certain subfields of $\mathds{ C}$. The proof uses a description of the hypergeometric systems as exponentially twisted Gauß–Manin systems of certain Laurent polynomials.

A new interpolation algorithm for computing dixon resultants

Given a system of polynomial equations with parameters, we present a new interpolation algorithm for computing its Dixon resultant R. Our algorithm interpolates the monic square-free factors of R one at a time from monic univariate polynomial images of R using sparse rational function interpolation. We have implemented our new Dixon resultant algorithm in Maple with some subroutines coded in C for efficiency. Experimental results show that our algorithm significantly outperforms Zippel's sparse interpolation algorithm.

Computing zeta functions of large polynomial systems over finite fields

We improve the algorithms of Lauder-Wan [11] and Harvey [8] to compute the zeta function of a system of m polynomial equations in n variables, over the q element finite field Fq, for large m. The dependence on m in the original algorithms was exponential in m. Our main result is a reduction of the dependence on m from exponential to polynomial. As an application, we speed up a doubly exponential algorithm from a recent software verification paper [3] (on universal equivalence of programs over finite fields) to singly exponential time. One key new ingredient is an effective, finite field version of the classical Kronecker theorem which (set-theoretically) reduces the number of defining equations for a polynomial system over Fq when q is suitably large.

Towards signature-based gröbner basis algorithms for computing the nondegenerate locus of a polynomial system

Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[ x 1 ,..., x n ] over K and a finite sequence of polynomials f 1 ,..., f c in R with c ≤ n. Let V ⊂ K n be the algebraic set defined by the simultaneous vanishing of the f i 's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set V c which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f 1 ,..., f c .

State-Feedback Design for Nonlinear Saturating Systems

This article presents strategies for state-feedback control law design of nonlinear control laws with saturating inputs. The input constraints are handled by considering a generalized local sector inequality allowing the study of nonsymmetric saturation bounds. A numerical formulation is presented for polynomial systems and is based on the solution of Lyapunov inequalities with sum-of-squares programming.

Computer algebra methods for polynomial system solving at the service of image-based visual servoing

Visual servoing refers to different methods used for robot motion control based on data from computer vision, in general the projection of a number of characteristics from the scene on a camera mounted on the robot. When the control is carried out in the space of parameters extracted from the image, we refer to Image-Based Visual Servoing (IBVS).

Stability and Stabilization of 2D Linear Discrete Systems with Fractional Orders Based on the Discrimination System of Polynomials

This paper considers the stability and stabilization of two-dimensional (2D) fractional-order systems described by state-space model based on the discrimination system of polynomials. Necessary and sufficient conditions of stability and stabilization are established. We change the criterion for checking the stability of linear discrete-time 2D fractional-order systems into an easy checking criterion whether some polynomials are positive. We use the discrimination system of polynomials to check the new conditions. For the stabilization problem, we get a stable gain matrix region. The unstable system with the gain parameters of the stable gain matrix region is stable. We give the method of stability analysis and stabilization for the general 2D fractional-order system. An example shows the validity of the proposed stability and stabilization methods.

Effective Parametrization of Low Order Bézier Motion Primitives for Continuous-Curvature Path-Planning Applications

We propose a new parametrization of motion primitives based on Bézier curves that suits perfectly path-planning applications (and environment exploration) of wheeled mobile robots. The individual motion primitives can simply be calculated taking into account the requirements of path planning and the constraints of a vehicle, given in the form of the starting and ending orientations, velocities, turning rates, and curvatures. The proposed parametrization provides a natural geometric interpretation of the curve. The solution of the problem does not require optimization and is obtained by solving a system of simple polynomial equations. The resulting planar path composed of the primitives is guaranteed to be C2 continuous (the curvature is therefore continuous). The proposed primitives feature low order Bézier (third order polynomial) curves. This not only provides the final path with minimal required turns or unwanted oscillations that typically appear when using higher-order polynomial primitives due to Runge’s phenomenon but also makes the approach extremely computationally efficient. When used in path planning optimizers, the proposed primitives enable better convergence and conditionality of the optimization problem due to a low number of required parameters and a low order of the polynomials. The main contribution of the paper therefore lies in the analytic solution for the third-order Bézier motion primitive under given boundary conditions that guarantee continuous curvature of the composed spline path. The proposed approach is illustrated on some typical scenarios of path planning for wheeled mobile robots.

The Development of Suitable Inequalities and Their Application to Systems of Logical Equations

In this paper, two not-difficult inequalities are invented and proved in detail, which adequately describe the behavior of discrete logical functions xor(x1, x2,…, xn) and and(x1, x2,…, xn). Based on these proven inequalities, infinitely differentiable extensions of the logical functions xor(x1, x2,…, xn) and and(x1, x2,…, xn) were defined for the entire ℝn. These suitable extensions were applied to systems of logical equations. Specifically, the system of m logical equations in a constructive way without adding any equations (not field equations and no others) is transformed in ℝn first into an equivalent system of m smooth rational equations (SmSRE) so that the solution of SmSRE can be reduced to the problem minimization of the objective function, and any numerical optimization methods can be applied since the objective function will be infinitely differentiable. Again, we transformed SmSRE into an equivalent system of m polynomial equations (SmPE). This means that any symbolic methods for solving polynomial systems can be used to solve and analyze an equivalent SmPE. The equivalence of these systems has been proved in detail. Based on these proofs and results, in the next paper, we plan to study the practical applicability of numerical optimization methods for SmSRE and symbolic methods for SmPE.

A New Analytic Approximation of Luminosity Distance in Cosmology Using the Parker–Sochacki Method

The luminosity distance dL is possibly the most important distance scale in cosmology and therefore accurate and efficient methods for its computation is paramount in modern precision cosmology. Yet in most cosmological models the luminosity distance cannot be expressed by a simple analytic function in terms of the redshift z and the cosmological parameters, and is instead represented in terms of an integral. Although one can revert to numerical integration techniques utilizing quadrature algorithms to evaluate such an integral, the high accuracy required in modern cosmology makes this a computationally demanding process. In this paper, we use the Parker–Sochacki method (PSM) to generate a series approximate solution for the luminosity distance in spatially flat ΛCDM cosmology by solving a polynomial system of nonlinear differential equations. When compared with other techniques proposed recently, which are mainly based on the Padé approximant, the expression for the luminosity distance obtained via the PSM leads to a significant improvement in the accuracy in the redshift range 0≤z≤2.5. Moreover, we show that this technique can be easily applied to other more complicated cosmological models, and its multistage approach can be used to generate analytic approximations that are valid on a wider redshift range.

Solving degree, last fall degree, and related invariants

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Gröbner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo–Mumford regularity.

New single traveling wave solution in birefringent fibers or crossing sea waves on the high seas for the coupled Fokas–Lenells system

The main work of this paper is focused on construction the single traveling wave solution of the coupled Fokas–Lenells system, which is usually used to simulate the propagation of ultrashort optical pulses in birefringent fibers or crossing sea waves on the high seas. Firstly, the coupled Fokas–Lenells system is simplified into a nonlinear ordinary differential equation by traveling wave transformation and linear transformation. Then, using the well-known complete discriminant system of third-order polynomials, some single traveling wave solutions of the coupled Fokas–Lenells system are obtained including implicit solutions, rational function solutions and Jacobian function solutions.

The number of limit cycles for some polynomial systems with multiple parameters

In this paper, by introducing double small parameters, under certain conditions, we study the maximum number of limit cycles bifurcating from the periodic orbits of the systems x˙=−y, y˙=x with (x,y)∈R2 and x˙=−y, y˙=x, z˙=0 with (x,y,z)∈R3 under nth degree polynomial perturbations, respectively. Moreover, we improve some known results.

The exact solutions of the conformable time fractional version of the generalized Pochhammer–Chree equation

The time-fractional version of the generalized Pochhammer–Chree equation is analyzed. In this paper, the equation is converted into an ordinary differential equation by applying certain real transformation, then the discrimination of polynomials system is used to find exact solutions depending on the fractional order derivative. The obtained solutions are graphically illustrated for different values of the fractional order derivative keeping the other parameters fixed.

Exploring circulations yielding no vortex equilibria

A system of n planar point vortices with prescribed real circulations Γi≠0 may be arranged so that the interaction between vortices causes them all to move with a common velocity. Such solutions are called vortex equilibria, where the required configurations are referred to as stationary or translating depending on whether the velocity is zero or not. The necessary condition, ∑i<jΓiΓj=0, for stationary configurations is not sufficient, for general n. The necessary condition, ∑iΓi=0, for translating configurations is also not sufficient in general. For example, for n≥5, the set of circulations (24−n,⋯,24−n,2(2−n)n(4−n),1) yields no stationary configurations. For n≥4, the set of circulations (−1,⋯,−1,n−3,1) yields no translating configurations. In this paper, we generalize such results and find new classes of sets of circulations satisfying the respective necessary conditions, but yielding no vortex equilibria.Let α and m be two independent parameters of natural numbers. The sets of circulations we consider are of the form (x,…,x,y,1,…,1) with α 1's, no less than three x=−2m, and one y such that ∑i<jΓiΓj=0 or ∑iΓi=0 holds. Using the minimal polynomial systems defined by O'Neil, we obtain new systems involving only part of the variables in the minimal polynomial systems. Sufficient conditions for no vortex equilibria are derived from them. We explore α and m such that the n-vortex system has no vortex equilibria for general n with the help of a computer algebra system. As a byproduct of the derived systems, they also help to find vortex equilibria for some exceptional n's where no information is obtained from our sufficient conditions.

On integrability of the Nosé–Hoover oscillator and generalized Nosé–Hoover oscillator

The Nosé–Hoover system is a basic primitive model for the molecular dynamics simulations, which describes the equilibrium characterized by canonical distributions at a constant temperature. Its simplest form, called the Nosé–Hoover oscillator (NH), is a three-dimensional quadratic polynomial system that admits both regular invariant tori and chaotic trajectories. Recently, a similar system, called the generalized Nosé–Hoover oscillator (GNH), was constructed to show the coexistence of invariant tori and topological horseshoe for time-reversible systems in [Formula: see text]. This paper aims to study the integrability of both NH and GNH models. We show that (i) in the case of [Formula: see text], both NH and GNH models are integrable by quadratures and the general solutions are given; (ii) in the case of [Formula: see text] the non-existence of either global analytic first integrals or Darboux first integrals of two models are discussed and a complete characterization of Darboux polynomials and exponential factors are given; (iii) both NH and GNH models are not rationally integrable in an extended Liouville sense except for several parameter values by an extended Morales–Ramis theory; (iv) the reduced systems of NH and GNH models at the Poincaré compactification balls are integrable, which yields complete descriptions of dynamics at infinity for NH and GNH models. Interestingly, the topological structure of NH and GNH models at infinity strongly depends on the sign of [Formula: see text]. These results may help us better understand the complex and rich dynamics of nonlinear time-reversible systems.

Central Limit Theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems

We establish the Central Limit Theorem, as the degree goes to infinity, for the normalized volume of the zero set of a rectangular Kostlan–Shub–Smale random polynomial system. This paper is a continuation of Central Limit Theorem for the number of real roots of Kostlan–Shub–Smale random polynomial systems by the same authors in which the case of square systems was considered. Our main tools are Kac-Rice formula and an expansion of the volume of the level set into the Itô-Wiener Chaos.

Recovery from Power Sums

We study the problem of recovering a collection of n numbers from the evaluation of m power sums. This yields a system of polynomial equations, which can be underconstrained (m < n), square (m = n), or overconstrained (m > n). Fibers and images of power sum maps are explored in all three regimes, and in settings that range from complex and projective to real and positive. This involves surprising deviations from the Bézout bound, and the recovery of vectors from length measurements by p-norms.

Certified numerical real root isolation for bivariate nonlinear systems

In this paper, we present a new method for isolating real roots of a bivariate nonlinear system. It is a subdivision method based on analyzing the local geometrical properties of the given system. We propose the concept of opposite monotone system in a box and use it to determine the existence and the uniqueness of a simple real zero of the system in the box. We have implemented our method and the experiments show the effectivity and efficiency of our approach, especially for high degree sparse polynomial systems. This paper is an extended version of the ISSAC'19 conference paper (Cheng and Wen (2019)).