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We recover plane curves from their branch points under projection onto a line. Our focus lies on cubics and quartics. These have 6 and 12 branch points respectively. The plane Hurwitz numbers 40 and 120 count the orbits of solutions. We determine the numbers of real solutions, and we present exact algorithms for recovery. Our approach relies on 150 years of beautiful algebraic geometry, from Clebsch to Vakil and beyond. [ABSTRACT FROM AUTHOR]

This article presents the discovery of an explicit 16-stage Runge–Kutta method that numerically satisfies the Runge–Kutta order conditions (in Butcher form) to 10th order, conjecturally improving the best-known number of stages in an explicit 10th-order Runge–Kutta method from 17 to 16. Unlike the vast majority of published Runge–Kutta methods, the method presented in this paper was not constructed via symbolic analysis or the use of simplifying assumptions, but instead by directly applying numerical optimization and root-finding algorithms to the order conditions. A naïve implementation of these algorithms would be made computationally infeasible by the considerable size and complex structure of the Butcher equations. However, we present a collection of software optimizations that greatly accelerate the evaluation of the Butcher equations and their derivatives while mitigating the effects of destructive cancellation, pushing these techniques within the realm of feasibility. While we do not have a formal proof of order, we present the results of numerical experiments to demonstrate that our method satisfies the Butcher equations to an accuracy of over 3000 decimal digits. [ABSTRACT FROM AUTHOR]

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety X Σ . The algorithm lends its name from a construction, described by Cox, of X Σ as a GIT quotient X Σ = (ℂ k ∖ Z) / / G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ℂ k of X Σ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of X Σ . It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the G -orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of X Σ . In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound. [ABSTRACT FROM AUTHOR]

Modern research often requires the use of economic modelswith multiple agents that interact over time. In this paper we researchoverlapping generations models, hereinafter OLG. In these models, thephenomenon of the multiplicity of long-term equilibrium may arise. Thisfact proves to be important for the theoretical justification of some eco-nomic effects, such as the collapse of the market and others. However,there is little theoretical research on the possibility of multiple equilibriain these models. At the same time, the works that exist are devoted tomodels with only few periods. This is due to the fact that the complexityof algorithms that calculate all long-term equilibria grows too fast withrealistically selected lifespan values. However, solutions of some OLGmodels after the introduction of additional variables can become polyno-mial systems. Thus it is possible to represent many long-term equilibriaas an algebraic variety. In particular, the Gr¨obner basis method becamepopular. However, this approach can only be used effectively when thereare few variables. In this paper we consider the task of finding long-term equilibrium in overlapping generations models with many periods.We offer an algorithm for finding the system’s solutions and use it toinvestigate the presence of multiple solutions in realistically calibratedmodels with long-lived agents. We also examine these models for mul-tiple equilibria using the Monte Carlo method and replicate previouslyknown results using a new algorithm.

In this work, the existence of a polynomial system of degree four with an invariant triangle containing at least four small-amplitude limit cycles is proved. This result improves the result obtained in[2]. [ABSTRACT FROM AUTHOR]

We characterize the existence of first integrals and invariants (first integrals depending on the time) for the polynomial vector fields which are invariant under an involution. [ABSTRACT FROM AUTHOR]

Copyright of Automatisierungstechnik is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper discusses the stability analysis of fractional-order polynomial systems by using the sum of squares method. furthermore the feasibility of designing the problems demonstrates which can not be represented in LMIs. unlike the T-S fuzzy model which can only work with fixed matrices, this method deal with polynomial matrices. Therefore, displaying a nonlinear system model using polynomials is a more efficient way. The stabilization of fractional order systems based on the fuzzy T-S model is expressed according to Lyapunov theory of stability by linear matrix inequality (LMI) while stability analysis polynomial fuzzy is based on the sum of the square. The main advantage of the method is the stabilization of fractional order systems based on the fuzzy T-S model. the stabilization conditions are expressed according to Lyapunov theory of stability by linear matrix inequality (LMI) while stability analysis is based on the polynomial fuzzy model. the systems where LMI optimization methods do not work, stability analysis and controller design can be performed by SOSTOOLS. In this paper, the stability conditions of a fractional-order polynomial fuzzy system are investigated then obtained necessary and sufficient conditions for stability. Finally, shown an example of the accuracy and The correctness of the proposed method.

Let I = 〈 f 1 , ... , f m 〉 ⊂ Q [ x 1 , ... , x n ] be a zero dimensional radical ideal defined by polynomials given with exact rational coefficients. Assume that we are given approximations { z 1 , ... , z k } ⊂ C n for the common roots { ξ 1 , ... , ξ k } = V (I) ⊆ C n. In this paper we show how to construct and certify the rational entries of Hermite matrices for I from the approximate roots { z 1 , ... , z k }. When I is non-radical, we give methods to construct and certify Hermite matrices for I from the approximate roots. Furthermore, we use signatures of these Hermite matrices to give rational certificates of non-negativity of a given polynomial over a (possibly positive dimensional) real variety, as well as certificates that there is a real root within an ε distance from a given point z ∈ Q n. [ABSTRACT FROM AUTHOR]

This paper aims to study a specific kind of parallel robot: Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A focus is made on the kinematics of such mechanisms, especially taking into account uncertainties (e.g. on conception & fabrication parameters, measures) and their propagations. Such considerations are crucial if we want to control our robot correctly without any undesirable behavior in its workspace (e.g. effects of singularities). In this paper, we will consider two different approaches to study the kinematics and the singularities of the robot of interest: symbolic and semi-numerical. By doing so, we can compute a singularity-free zone in the work- and joint spaces, considering given uncertainties on the parameters. In this zone, we can use any control law to inertially stabilize the upper platform of the robot. [ABSTRACT FROM AUTHOR]

The delay-dependent state, Fault Estimation (FE) and Fault Tolerant Control problems for polynomial fuzzy systems with time delay are investigated in this paper. Firstly, a polynomial adaptive fuzzy observer is used to estimate system states and actuator faults. A polynomial Lyapunov–Krasovskii functional including double integral term is introduced in order to obtain delay-dependent sufficient conditions in terms of Sum Of Squares (SOS) which can be solved via SOSTOOLS and a Semi-Definite Program solver. Then, by exploiting the FE information, a polynomial fuzzy fault tolerant controller is designed to guarantee the stability of the system and to compensate the impact of actuator faults. Finally, several examples are given to illustrate the use of the present delay-dependent result. [ABSTRACT FROM AUTHOR]

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar kinematic properties (e.g. topology of the kinematic-singularity-free space). From these algorithms one also obtain solutions to connectivity queries of a specific parameter which is in turn related to kinematic-singularity free path-planning of a specific manipulator belonging to the family of robots with these properties; i.e. we obtain paths joining two given singularity free configurations lying in the same connected component of the singularity-free space. We prove in the paper how one reduces the problems related to connectivity queries of parameterized semi-algebraic sets to closed and bounded semi-algebraic sets. We then design an algorithm using computer-algebra methods for "solving" positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. The complexity of this algorithm is singly exponential in the dimension of the ambient space. The algorithm scales enough to analyze automatically the family of UR-series robots. Finally we provide manual analysis of the family of UR-series robots, proving that the number of connected components of the complementary of kinematic singularity set of a generic UR-robot is eight. [ABSTRACT FROM AUTHOR]

We consider the problem of computing the topology and describing the geometry of a parametric curve in R n. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most d and maximum bitsize of coefficients τ , then the worst case bit complexity of PTOPO is O ˜ B (n d 6 + n d 5 τ + d 4 (n 2 + n τ) + d 3 (n 2 τ + n 3) + n 3 d 2 τ). This bound matches the current record bound O ˜ B (d 6 + d 5 τ) for the problem of computing the topology of a plane algebraic curve given in implicit form. For plane and space curves, if N = max ⁡ { d , τ } , the complexity of PTOPO becomes O ˜ B (N 6) , which improves the state-of-the-art result, due to Alcázar and Díaz-Toca [CAGD'10], by a factor of N 10. In the same time complexity, we obtain a graph whose straight-line embedding is isotopic to the curve. However, visualizing the curve on top of the abstract graph construction, increases the bound to O ˜ B (N 7). For curves of general dimension, we can also distinguish between ordinary and non-ordinary real singularities and determine their multiplicities in the same expected complexity of PTOPO by employing the algorithm of Blasco and Pérez-Díaz [CAGD'19]. We have implemented PTOPO in maple for the case of plane and space curves. Our experiments illustrate its practical nature. [ABSTRACT FROM AUTHOR]

In this paper, a robust fault estimation for polynomial systems with time delay via a polynomial adaptive observer is presented and described. In fact, in contrast to previous research works on actuator faults, the learning rate utilized to govern the convergence speed of states and faults errors is not predetermined but is derived using the sum of squares optimization problem. Furthermore, the H ∞ optimization approach was created to reduce the impact of disturbances on state estimation error and fault estimation error. Finally, an illustrative practical example is given to validate the effectiveness of the suggested methodology. [ABSTRACT FROM AUTHOR]

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Plücker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications. [ABSTRACT FROM AUTHOR]

Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schäfli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. We propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant.We define a related polynomial called the multivariate iterated discriminant, generalizing the classical Schäfli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre--Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant. [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilization problem of a class of polynomial systems of odd degree in dimension three. The constructed stabilizing feedback is homogeneous and guarantee the homogeneity of the closed loop system.mynotered In the end of the paper, we show the efficiency of such a study in the local stabilization of nonlinear systems affine in control. [ABSTRACT FROM AUTHOR]

We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We characterize the complexity of our algorithm in terms of an algebraic property of this polynomial system—the multigraded regularity. We prove effective bounds for many tensor formats and ranks, which are of independent interest for overconstrained polynomial system solving. Moreover, we conjecture a general formula for the multigraded regularity, yielding a (parameterized) polynomial time complexity for the tensor rank decomposition problem in the considered setting. Our numerical experiments show that our algorithm can outperform state-of-the-art numerical algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption. [ABSTRACT FROM AUTHOR]

We present a number of new piecewise-polynomial kernels for image interpolation. The kernels are constructed by optimizing a measure of interpolation quality based on the magnitude of anisotropic artifacts. The kernel design process is performed symbolically using the Mathematica computer algebra system. An experimental evaluation involving 14 image quality assessment methods demonstrates that our results compare favorably with the existing linear interpolators. [ABSTRACT FROM AUTHOR]

Copyright of Automatisierungstechnik is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

We consider planar polynomial systems of ordinary differential equations of the form $\dot x = x + P_n(x,y)$, $\dot y = y + Q_n(x,y)$, where $P_n(x,y),\ Q_n(x,y)$ are homogeneous polynomials of degree $n$. We study the algebraic and non-algebraic invariant curves of these systems with emphasis on limit cycles.

In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu (C R Acad Sci Ser I Math 328(6):495–499, 1999). We use Jacobi's technique from (Math Z 237(2):259–273, 2001) to provide an alternative proof with an effective degree bound on the sums of squares weights in such certificates. As a consequence, it allows one to define a hierarchy of semidefinite relaxations for a general polynomial optimization problem. Convergence of this hierarchy to a neighborhood of the optimal value as well as strong duality and analysis are guaranteed. In a second contribution, we introduce a new numerical method for solving systems of polynomial inequalities and equalities with possibly uncountably many solutions. As a bonus, one can apply this method to obtain approximate global optimizers in polynomial optimization. [ABSTRACT FROM AUTHOR]

This paper investigates both fault estimation and fault-tolerant control problems for polynomial systems with state delay in presence of faults. Before studying the control problem, a polynomial observer is proposed to jointly estimate state and fault vectors. Based on the obtained informations, a polynomial controller is designed to compensate fault effects and to stabilize the closed-loop systems even in presence of state delay. The existence of both polynomial observer and controller is proved via satisfying sufficient conditions based on a sum of square (SOS) approach. Polynomial observer and controller gains are computed using SOSTOOLS. Finally, a numerical example is given to demonstrate that the SOS approach provides a significant improvement in reaction to fault occurrence for time delay systems. Furthermore, a tunnel diode circuit and a mass–spring–damper system are used to illustrate the applicability of the design method. [ABSTRACT FROM AUTHOR]

This work addresses the lateral stabilisation problem of four-wheels ground vehicles. The objective is to estimate the largest state-space region such that the closed-loop vehicle lateral stability can be guaranteed. Sum-of-squares (SOS) programming technique is applied to find these maximum invariant sets while accounting for steering and yaw moment input saturations. The algorithm allows the region of attraction (RoA) to be approximated by a level set of a Lyapunov function (LF) and the computation of polynomial state feedback control laws. The method is applied for both straight-line motion and cornering manoeuver. Finally, a Monte-Carlo analysis is presented to show that the proposed SOS-based methodology can be used as a valid analysis and design tool considering a real vehicle application. [ABSTRACT FROM AUTHOR]

In the area of symbolic-numerical computation within computer algebra, an interesting question is how "close" a random input is to the "critical" ones. Examples are the singular matrices in linear algebra or the polynomials with multiple roots for Newton's root-finding method. Bounds, sometimes very precise, are known for the volumes over R or C of such neighborhoods of the varieties of "critical" inputs; see the references below. This paper deals with the discrete version of this question: over a finite field, how many points lie in a certain type of neighborhood around a given variety? A trivial upper bound on this number is given by the product (size of the variety) ⋅ (size of a neighborhood of a point). It turns out that this bound is usually asymptotically tight, in particular for the singular matrices, polynomials with multiple roots, and pairs of non-coprime polynomials. The interesting question then is: for which varieties is this bound not asymptotically tight? We show that these are precisely those that admit a shift, that is, where one absolutely irreducible component of maximal dimension is a shift (translation by a fixed nonzero point) of another such component. Furthermore, the shift-invariant absolutely irreducible varieties are characterized as being cylinders over some base variety. Computationally, determining whether a given variety is shift-invariant turns out to be intractable, namely NP-hard even in simple cases. [ABSTRACT FROM AUTHOR]

In this article, for the incomplete H-functions, we obtain a set of new generating functions. The bilateral along with linear generating relations are derived for the incomplete H-functions. Many of the generating functions readily accessible in the literature are often deemed as implementations of the main findings. All the derived findings are of a natural type and can produce a variety of new results in generating function theory. [ABSTRACT FROM AUTHOR]

We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

This paper study the type of integrability of differential systems with separable variables x ˙ = h (x) f (y) , y ˙ = g (y) , where h , f and g are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems. In the particular case h (x) = (a x + b) m we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from Giné et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) and Llibre and Valls (Discrete Contin. Dyn. Syst., Ser. B 20:2657–2661, 2015). Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables x ˙ = x f (y) , y ˙ = g (y) , with f and g polynomials. [ABSTRACT FROM AUTHOR]

Copyright of Baghdad Science Journal is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

For the polynomial system $\dot x = ix + x \bar x ( a x^2 + b x \bar x + c \bar x^2)$ the study of critical period bifurcations is performed. Using calculations with algorithms of computational commutative algebra it is shown that at most two critical periods can bifurcate from any nonlinear center of the system.

We investigate the critical period bifurcations of the system $$ \dot x = ix + x \bar x ( a x^3 + b x^2 \bar x + \bar x \bar x^2+d \bar x^3) $$ studied in [6]. We prove that at most three critical periods can bifurcate from any nonlinear center of the system.

We consider planar polynomial systems of ordinary differential equations of the form x=x+Pn(x,y), y=y+Qn(x, y), where Pn(x, y), Qn(x, y) are homogeneous polynomials of degree n. We study the algebraic and non-algebraic invariant curves of these systems with emphasis on limit cycles. [ABSTRACT FROM AUTHOR]

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R / I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R / I. This is exploited in this paper to introduce the use of two special (non-monomial) types of basis functions with nice properties. This allows us, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments. [ABSTRACT FROM AUTHOR]

This paper presents an efficient algorithm called SolveFuzzySystem, or SFS, for finding real solutions of polynomial systems whose coefficients are symmetrical L - R fuzzy numbers with bounded support for which the spread functions L and R are bijective. The real solutions of such a system are deduced from solutions of some polynomial systems with real coefficients. This algorithm is based on new results that are universal because they are independent from the spread functions. These theoretical results include the management of the fuzzy system's solutions signs. An implementation in the Fuzzy package of the free computer algebra software SageMath and a parallel version of the algorithm are described. [ABSTRACT FROM AUTHOR]

We propose a new algorithm for numerical path tracking in polynomial homotopy continuation. The algorithm is "robust" in the sense that it is designed to prevent path jumping, and in many cases it can be used in (only) double precision arithmetic. It is based on an adaptive stepsize predictor that uses Padé techniques to detect local difficulties for function approximation and danger for path jumping. We show the potential of the new path tracking algorithm through several numerical examples and compare it with existing implementations. [ABSTRACT FROM AUTHOR]

Copyright of Automatisierungstechnik is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Consider a zero-dimensional ideal I in K [ X 1 , ... , X n ]. Inspired by Faugère and Mou's Sparse FGLM algorithm, we use Krylov sequences based on multiplication matrices of I in order to compute a description of its zero set by means of univariate polynomials. Steel recently showed how to use Coppersmith's block-Wiedemann algorithm in this context; he describes an algorithm that can be easily parallelized, but only computes parts of the output in this manner. Using generating series expressions going back to work of Bostan, Salvy, and Schost, we show how to compute the entire output for a small overhead, without making any assumption on the ideal I other than it having dimension zero. We then propose a refinement of this idea that partially avoids the introduction of a generic linear form. We comment on experimental results obtained by an implementation based on the C++ libraries Eigen, LinBox and NTL. [ABSTRACT FROM AUTHOR]