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The interplay among the time-evolution of the coefficients y m t and the zeros x n t of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently, this tool has been extended to the case of nongeneric polynomials characterized by the presence, for all time, of a single double zero; subsequently, significant progress has been made to extend this finding to the case of polynomials featuring a single zero of arbitrary multiplicity. In this paper, we introduce an approach suitable to deal with the most general case, i.e., that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of which features, for all time, an arbitrary (time-independent) multiplicity. We then focus on the special case of a polynomial of degree 4 featuring only 2 different zeros, and by using a recently introduced additional twist of this approach, we thereby identify many new classes of solvable dynamical systems of the following type: x ̇ n = P n x 1 , x 2 , n = 1,2 , with P n x 1 , x 2 being two polynomials in the two variables x 1 t and x 2 t . [ABSTRACT FROM AUTHOR]

Recently a convenient technique to relate the time evolution of the N zeros of a time-dependent polynomial p N z ; t of degree N in the (complex) variable z to the time evolution of its N coefficients has been exploited to identify large classes of dynamical systems solvable by algebraic operations. These models also include N-body problems that evolve in the complex plane (or, equivalently, in the real Cartesian plane) according to systems of nonlinearly-coupled equations of motion of Newtonian type (“accelerations equal forces”). Many of these models feature remarkable properties: for instance, they are Hamiltonian and integrable and/or multiply periodic or even isochronous (featuring completely periodic solutions with a fixed period largely independent of the initial data), or asymptotically isochronous (featuring isochrony only up to corrections vanishing in the remote future). In this paper, an analogous technique is introduced that focuses instead on the time evolution of the N zeros of an appropriate class of time-dependent rational functions R N z ; t , thereby opening large vistas of new dynamical systems solvable by algebraic operations and featuring remarkable properties. A few examples are reported. [ABSTRACT FROM AUTHOR]

A technique is introduced which allows to generate--starting from any solvable discrete-time dynamical system involving N time-dependent variables--new, generally nonlinear, generations of discrete-time dynamical systems, also involving N time-dependent variables and being as well solvable by algebraic operations (essentially by finding the N zeros of explicitly known polynomials of degree N). The dynamical systems constructed using this technique may also feature large numbers of arbitrary constants, and they need not be autonomous. The solvable character of these models allows to identify special cases with remarkable time evolutions: for instance, isochronous or asymptotically isochronous discrete-time dynamical systems. The technique is illustrated by a few examples. [ABSTRACT FROM AUTHOR]

Recently a simple differential algorithm to compute all the zeros of a generic polynomial was introduced. In this paper an analogous, but finite-difference, algorithm is introduced and discussed. At the end of the paper a minor generalization of the differential algorithm is also mentioned. [ABSTRACT FROM AUTHOR]

We define the generalized basic hypergeometric polynomial of degree N in terms of the generalized basic hypergeometric function, by choosing one of its parameters to allow the termination of the series after a finite number of summands. In this paper, we obtain a set of nonlinear algebraic equations satisfied by the N zeros of the polynomial. Moreover, we obtain an N × N matrix M defined in terms of the zeros of the polynomial, which, in turn, depend on the parameters of the polynomial. The eigenvalues of this remarkable matrix M are given by neat expressions that depend only on some of the parameters of the polynomial; that is, the matrix M is isospectral. Moreover, in case the parameters that appear in the expressions for the eigenvalues of M are rational, the matrix M has rational eigenvalues, a Diophantine property. [ABSTRACT FROM AUTHOR]

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers zn, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two operators acting on a function f (z) as follows: [f (z + a) - f (z)]/a respectively [ f (q z) - f (z)]/[(q - 1)z]. These representations are exact--in a sense explained in the paper--when the function f (z) is a polynomial in z of degree less than N. This formalism allows to transform difference equations valid in the space of polynomials of degree less than N into corresponding matrix-vector equations. As an application of this technique, several remarkable square matrices of order N are identified, which feature explicitly N arbitrary numbers zn, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character. [ABSTRACT FROM AUTHOR]

A class of N-body problems is identified, characterized by second-order discrete-time evolution equations determining the motion in the complex z-plane of an arbitrary number N of points zn ≡ zn (l), where l = 0,±1,±2,... is the discrete-time independent variable. Both these equations of motion, and the solution of their initialvalue problem, only involve algebraic operations: finding the zeros of explicitly known polynomials of degree N in z, finding the eigenvectors and eigenvalues of explicitly known N × N matrices. These models feature an arbitrarily large number of arbitrary parameters ("coupling constants"). [ABSTRACT FROM AUTHOR]

A simple algorithm to compute all the zeros of a generic polynomial is proposed. [ABSTRACT FROM AUTHOR]

A new class of dynamical systems are presented, together with their solutions. Some of these models are isochronous, namely, their generic solutions are all completely periodic with the same period; others are characterized by friction, all solutions vanishing in the remote future; and still others are “asymptotically isochronous,” approaching an isochronous behavior in the remote future. [ABSTRACT FROM AUTHOR]

We take advantage of the simple approach, recently discussed, which associates to (solvable) matrix equations (solvable) dynamical systems interpretable as (interesting) many-body problems, possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles. Starting from a solvable matrix evolution equation, we obtain the corresponding many-body model and note that in one case the auxiliary variables can be altogether eliminated, obtaining thereby an (also Hamiltonian) extension of the “goldfish” model. The solvability of this novel model, and of its isochronous variant, is exhibited. A related, as well solvable, model, is also introduced, as well as its isochronous variant. Finally, the small oscillations of the isochronous models around their equilibrium configurations are investigated, and from their isochronicity certain diophantine relations are evinced. [ABSTRACT FROM AUTHOR]

A simple approach is discussed which associates to (solvable) matrix equations (solvable) dynamical systems, generally interpretable as (interesting) many-body problems, possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles. We then focus on cases in which the auxiliary variables can be altogether eliminated, reobtaining thereby (via this unified approach) well-known solvable many-body problems, and moreover a (solvable) extension of the “goldfish” model.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

New upper and lower limits are given for the number of S-wave bound states yielded by an attractive (monotonic) potential in the context of the Schrödinger or Klein-Gordon equation. [ABSTRACT FROM AUTHOR]

Some properties of a solvable N-body problem featuring several free parameters ('coupling constants') are investigated. Restrictions on its parameters are reported which guarantee that all its solutions are completely periodic with a fixed period independent of the initial data (isochrony). The restrictions on its parameters which guarantee the existence of equilibria are also identified. In this connection a remarkable second-order ODE-generally not of hypergeometric type, hence not reducible to those characterizing the classical polynomials-is studied: if its parameters satisfy a Diophantine condition, its general solution is a polynomial of degree N, the N zeros of which identify the equilibria of the N-body system. [ABSTRACT FROM AUTHOR]