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In this paper we firstly review how to \textit{explicitly} solve a system of $3$ \textit{first-order linear recursions }and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of $3$ \textit{first-order nonlinear recursions} which also are \textit{explicitly solvable}. These systems might be of interest for practitioners in \textit{applied} sciences: they allow a complete display of their solutions, which may feature interesting behaviors, for instance be \textit{completely periodic} ("isochronous systems", if the independent variable $n=0,1,2,3...$is considered a \textit{ticking time}), or feature this property \textit{only asymptotically} (as\textit{\ }$n\rightarrow \infty $).

In this paper a class of simple, but nonlinear, systems of recursions involving $2$ dependent variables $x_{j}\left( n\right) $ is identified, such that the solutions of their initial-values problems -- with arbitrary initial data $x_{j}\left( 0\right) $ -- may be explicitly obtained.

Some simple nonlinear recursions which can be completely managed are identified and the behaviour of all their solutions is ascertained.

Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer $n_{\ell }$ is even, $n_{\ell +1}=n_{\ell }/2$. The so-called Collatz conjecture states that, starting from any positive integer $N$, the recursion characterized by the continued application of these rules ends up in the cycle $4,$ $2,1$. This conjecture is generally believed to be true (on the basis of extensive numerical checks), but it is as yet unproven. In this paper -- based on the assumption that the Collatz conjecture is indeed true -- we present a quite simple extension of it, which entails the possibility to divide all natural numbers into $3$ disjoint classes, to each of which we conjecture -- on the basis of (not very extensive) numerical checks -- $1/3$ of all natural numbers belong; or, somewhat equivalently, to $2$ disjoint classes, to which we conjecture that respectively $1/3$ and $2/3$ of all natural numbers belong.

In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs feature 2 right-hand sides which are the ratios of 2 homogeneous polynomials of first degree divided by the same homogeneous polynomial of second degree. The model features only 4 arbitrary parameters. We also report its isochronous variant featuring 4 nonlinearly-coupled first-order ODEs in 4 dependent variables, featuring 9 arbitrary parameters., Comment: 3 pages

A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we point out that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints., Comment: 10 pages

The generic monic polynomial of degree N features N a priori arbitrary coefficients $c_m$ and N zeros $z_n$. In this paper we limit consideration to $N = 8$ and $N = 9$. We show that if the $N$ -- a priori arbitrary -- coefficients $c_m$ of these polynomials are appropriately defined -- as it were, a posteriori -- in terms of 6 arbitrary parameters, then the $N$ roots of these polynomials can be explicitly computed in terms of radicals of these 6 parameters. We also report the constraints on the N coefficients $c_m$ implied by the fact that they are so defined in terms of 6 arbitrary parameters; as well as the explicit determination of these 6 parameters in terms of the N coefficients $c_m$., Comment: 5 pages

The initial-values problem of the following nonlinear autonomous recursion of order p , z (s + p) = c product of [z (s + l)]^a_l ; with p an arbitrary positive integer, z (s) the dependent variable (possibly a complex number), s the independent variable (a non negative integer), c an arbitrarily assigned, possibly complex, number, and the p exponents a_l arbitrarily assigned integers (positive, negative or vanishing, so that the right-hand side of the recursion be univalent)|is solvable by algebraic operations, involving the solution of a system of linear algebraic equations (generally explicitly solvable) and of a single polynomial equation of degree p (hence explicitly solvable for p = 1; 2; 3; 4 )., Comment: 7 pages

In this short paper we identify special systems of (an arbitrary number) N of first-order Difference Equations with nonlinear homogeneous polynomials of arbitrary degree M in their right-hand sides, which feature very simple explicit solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data, Comment: 3 Pages. arXiv admin note: substantial text overlap with arXiv:2106.06634

The asymmetric May-Leonard model is a prototypical system of 3 nonlinearly coupled first-order Ordinary Differential Equations with second-degree polynomial right-hand sides. In this short paper we identify a class of special solutions of this system which do not seem to have been previously advertised in spite of their rather elementary character., Comment: 3 pages

In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous--which feature periodic solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data., Comment: 4 pages

The generic monic polynomial of sixth degree features 6 a priori arbitrary coefficients. We show that if these 6 coefficients are appropriately defined in two different ways|in terms of 5 arbitrary parameters, then the 6 roots of the corresponding polynomial can be explicitly computed in terms of radicals of these parameters. We also report the 2 constraints on the 6 coefficients of the polynomial implied by the fact that they are so defined in terms of 5 arbitrary parameters; as well as the explicit determination of these 5 parameters in terms of the 6 coefficients of the sextic polynomial., Comment: 4 pages

After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which is explicitly solvable. It is identified by 2 explicit algebraic constraints on the 6 a priori arbitrary parameters that characterize this system. Simple extensions of this model to cases with nonhomogeneous quadratic right-hand sides are also identified, including isochronous cases., Comment: 18 pages

The explicit solution of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODE s with second-degree polynomial right-hand sides, hence featuring 12 a prior arbitrary (time-independent) coefficients. The solution is explicitly provided if the 12 coefficients are expressed by explicitly provided formulas in terms of 10 a prior arbitrary parameters; the inverse problem to express these 10 parameters in terms of the 12 coefficients is also explicitly solved, but it is found to imply as it were, a posterior that the 12 coefficients must then satisfy 4 algebraic constraints, which are explicitly exhibited.Special sub cases are also identified the general solutions of which are completely periodic with a period independent of the initial data, or are characterized by additional restrictions on the coefficients which identify particularly interesting models., Comment: 17 Pages, Submitted to Mathematical Physics, Analysis and Geometry

The solution $x_n\left(t\right)$, $n=1,2,$ of the \textit{initial-values} problem is reported of the \textit{autonomous} system of $2$ coupled first-order ODEs with \textit{homogeneous cubic polynomial} right-hand sides, \begin{eqnarray} \dot{x}_n = c_{n1} \left(x_1\right)^3 + c_{n2}\left( x_1\right)^2 x_2 + c_{n3} x_1 \left(x_2\right)^2+c_{n4} \left(x_2\right)^3\ ,\quad n=1,2\ , \nonumber \end{eqnarray} when the $8$ (time-independent) coefficients $c_{n\ell}$ are appropriately defined in terms of $7$ \textit{arbitrary} parameters, which then also identify the solution of this model. The inversion of these relations is also investigated, namely how to obtain, in terms of the $8$ coefficients $c_{n\ell},$ the $7$ parameters characterizing the solution of this model; and $2$ \textit{constraints} are \textit{explicitly} identified which, if satisfied by the $8$ parameters $c_{n\ell },$ guarantee the \textit{solvability by algebraic operations} of this dynamical system. Also identified is a related, \textit{appropriately modified}, class of (generally \textit{complex}) systems, reading \begin{eqnarray} \dot{{\tilde{x}}_{n}} = \mathbf{i}\omega \tilde{x}_{n} + c_{n1}\left(\tilde{x}_{1}\right) ^{3}+c_{n2}\left( \tilde{x}_{1}\right) ^2 \tilde{x}_2 + c_{n3}\tilde{x}_1 \left( \tilde{x}_2\right)^2 + c_{n4}\left(\tilde{x}_2 \right)^3\ ,\quad n=1,2\ , \nonumber \end{eqnarray} with $\mathbf{i}\omega $ an \textit{arbitrary imaginary} parameter, which feature the remarkable property to be \textit{isochronous}, namely their \textit{generic} solutions are -- as functions of \textit{real time} -- \textit{completely periodic} with a period which is, for each of these models, a \textit{fixed} \textit{integer multiple} of the basic period $\tilde{T}=2\pi /\left\vert \omega \right\vert$., Comment: 25 pages, To be published in Journal of Mathematical Physics (2021)

The evolution equations mentioned in the title of this paper read as follows: x~n = P(n)(x1; x2) , n = 1, 2 , where l is the "discrete-time" independent variable taking integer values (l =0, 1, 2, ...), xn = xn (l) are the 2 dependent variables, x~n = xn (l + 1), and the 2 functions P(n)(x1, x2), n = 1, 2, are 2 polynomials in the 2 dependent variables x1 (l) and x2 (l). The results reported in this paper have been obtained by an appropriate modification of a recently introduced technique to obtain analogous results in continuous-time t in which case xn = xn (t) and the above recursion relations are replaced by first-order ODEs. Their potential interest is due to the relevance of this kind of evolution equations in various applicative contexts., Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1904.00496

In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows: x_n = P(n) (x1, x2) , n = 1, 2 , with P(n) (x1, x2) specific polynomials of relatively low degree in the 2 dependent variables x1 = x1 (t) and x2 = x2 (t) . These findings are obtained via a new twist of a recent technique to identify dynamical systems solvable by algebraic operations, themselves explicitly identified as corresponding to the time evolutions of the zeros of polynomials the coefficients of which evolve according to algebraically solvable (systems of) evolution equations., Comment: 11 pages, Sabmitted 27 August 2018, to be published as a chapter in a collective book to celebrate the 65th birthdate of Emma Previato (in press)

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,x~n = F^(n)(x1; x2) , n = 1, 2 , which are algebraically solvable. Here l is the "discrete-time" independent variable taking integer values (l = 0, 1, 2,...), xn = xn (l) are 2 dependent variables, and x~n = xn (l + 1) are the corresponding 2 updated variables. In a previous paper the 2 functions F^(n)(x1, x2), n = 1, 2 were defined as follows: F^(n)(x1; x2) = P2 (xn, xn+1), n = 1,2 mod[2]; with P2 (x1; x2) a specific second-degree homogeneous polynomials in the 2 (indistinguishable!) dependent variables x1 (`) and x2 (l). In the present paper we further clarify some aspects of that model and we present its extension to the case when F(n)(x1, x2) = Q^(n)_k(x1, x2), n = 1, 2 mod[2], with Q(n)_k(x1; x2) a specific homogeneous function of arbitrary (integer ) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x1 (l) and x2 (l)., Comment: 13 pages, submitted 13 March 2019

We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of 2 first-order ODEs with polynomial right-hand sides, x_ n= P(n)(x1, x2) , n = 1, 2 , satisfied by the 2 (possibly complex ) dependent variables xn = xn (t). Here P(n)(x1, x2) indicates some specific polynomial. These examples are analogous, but different, from those previously reported., Comment: 9 pages, submitted 29 January 2019

The interplay among the time-evolution of the coefficients and the zeros 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; and 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., Comment: 36 pages, Submitted 11 December, 2018

Recently new solvable systems of nonlinear evolution equations -- including ODEs, PDEs and systems with discrete time -- have been introduced. These findings are based on certain convenient formulas expressing the $k$-th time-derivative of a root of a time-dependent monic polynomial in terms of the $k$-th time-derivative of the coefficients of the same polynomial and of the roots of the same polynomial as well as their time-derivatives of order less than $k$. These findings were restricted to the case of generic polynomials without any multiple root. In this paper some of these findings -- those for $k=1$ and $k=2$ -- are extended to polynomials featuring one double root; and a few representative examples are reported of new solvable systems of nonlinear evolution equations.

It has been recently demonstrated that it is possible to construct isochronous cosmologies, extending to general relativity a result valid for non-relativistic Hamiltonian systems. In this paper we review these findings and we discuss the Newtonian limit of these isochronous spacetimes, showing that it reproduces the analogous findings in the context of non-relativistic dynamics., Comment: arXiv admin note: text overlap with arXiv:1406.7156

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em peculiar\/} polynomials. We further show that the problem of determining the peculiar polynomials of degree $N$ simplifies when any of the coefficients is either 0 or 1. We proceed to estimate the numbers of peculiar polynomials of degree $N$ having one coefficient zero, or one coefficient equal to one, or neither., Comment: 15 pages

A simple algorithm to compute all the zeros of a generic polynomial is proposed.

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.

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type ("acceleration equals forces") which determine the motion of points in the complex plane. These models are solvable, namely their configuration at any time can be obtained from the initial data by algebraic operations, amounting to the determination of the zeros of a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronous or asymptotically isochronous; others display scattering phenomena., Comment: arXiv admin note: text overlap with arXiv:1510.05017

The notion of generations of monic polynomials such that the coefficients of the polynomials of the next generation coincide with the zeros of the polynomials of the current generation is introduced, and its relevance to the identification of endless sequences of new solvable many-body problems of "goldfish type" is demonstrated.

We define the generalized hypergeometric polynomial of degree N in terms of the generalized hypergeometric function that depends on p parameters a_1, ..., a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly complex, numbers. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros z_n of this polynomial. We moreover manufacture an NxN matrix L in terms of the 1+p+q parameters N, a_j, b_k characterizing this polynomial, and of its N zeros z_n. We show that the matrix L features N eigenvalues that depend only on the q parameters b_k, implying that this matrix is isospectral for the variations of the p parameters a_j. These eigenvalues are integer (or rational) numbers if the q parameters b_k are themselves integer (or rational) numbers: a nontrivial diophantine property., Comment: arXiv admin note: text overlap with arXiv:1504.01748

A new solvable many-body problem of goldfish type is introduced and the behavior of its solutions is tersely discussed.

We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the ordering assignment of the N zeros of the Hermite polynomial of order N. We construct two NxN matrices M_1 and M_2 defined in terms of the N zeros of the polynomial p_N(z). We prove that the eigenvalues of M_1 and M_2 are the first N integers respectively the first N squared-integers, a remarkable isospectral and Diophantine property. The technique whereby these findings are demonstrated can be extended to other named polynomials.

We define the generalized basic hypergeometric polynomial of degree $N \geq 1$ in terms of the generalized basic hypergeometric function, which depends on (arbitrary, generic, possibly complex) parameters $q \neq 1$, the $r \geq 0$ parameters $\alpha _{j}$ and the $s \geq 0$ parameters $\beta _{k}$. In this paper we obtain a set of $N$ nonlinear algebraic equations satisfied by the $N$ zeros $\zeta _{n}\equiv \zeta _{n}\left( \underline{\alpha },\underline{\beta };q;N\right) $ of this polynomial. We moreover identify an $\left( N\times N\right) $-matrix $\underline{M}\equiv \underline{M}\left( \underline{\alpha },\underline{\beta };\underline{\zeta };q;N\right) $ featuring the $N$ eigenvalues $\mu _{n}=-q^{\left( s-r\right) \left( N-n\right) }\left(q^{-n}-1\right) ~\prod\limits_{j=1}^{r}\left( \alpha _{j}~q^{N-n}-1\right)$, where $n=1,2,...,N.$ These $N$ eigenvalues depend only on the $r$ parameters $\alpha _{j}$ (besides $q$ and $N$), implying that the $\left( N\times N\right) $-matrix $\underline{M}$ is isospectral for variations of the $s$ parameters $\beta _{k}$; and they clearly are rational numbers if $q$ and the $r$ parameters $\alpha _{j}$ are themselves rational numbers: a nontrivial Diophantine property.

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers z_{n}, 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(qz)-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 z_{n}, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character., Comment: 29 pages

In this paper we provide properties -- which are, to the best of our knowledge, new -- of the zeros of the polynomials belonging to the q-Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted., Comment: This version differs from version v1 by one sentence at the end of section 4.1. arXiv admin note: text overlap with arXiv:1407.3379

In this paper we provide properties---which are, to the best of our knowledge, new---of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.

Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion ("accelerations equal forces") are integrable, i. e. they allow the explicit exhibition of $N$ constants of motion in terms of the dependent variables and their time-derivatives. Some of these models are moreover solvable by purely algebraic operations, by (explicitly performable) quadratures and, finally, by functional inversions. The techniques to manufacture these models are not new; some of these models are themselves new; others are reinterpretations of known models.

The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period $T$ yet mimic with good approximation, or even exactly, any given many-body problem (within a quite large class, encompassing most of nonrelativistic physics) over times $\tilde{T}$ which may also be arbitrarily large (but of course such that $\tilde{T}

The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period $T$ yet mimic with good approximation, or even exactly, any given many-body problem (within a large, physically relevant, class) over times $\tilde{T}$ which may also be arbitrarily large (but of course such that $\tilde{T}

The class of solvable many-body problems "of goldfish type" is extended by including (the additional presence of) three-body forces. The solvable $N$-body problems thereby identified are characterized by Newtonian equations of motion featuring 19 arbitrary "coupling constants". Restrictions on these constants are identified which cause these systems - or appropriate variants of them - to be isochronous or asymptotically isochronous, i.e. all their solutions to be periodic with a fixed period (independent of the initial data) or to have this property up to contributions vanishing exponentially as $t\rightarrow\infty$.

A new class of solvable $N$-body problems is identified. They describe $N$ unit-mass point particles whose time-evolution, generally taking place in the complex plane, is characterized by Newtonian equations of motion "of goldfish type" (acceleration equal force, with specific velocity-dependent one-body and two-body forces) featuring several arbitrary coupling constants. The corresponding initial-value problems are solved by finding the eigenvalues of a time-dependent $N\times N$ matrix $U(t)$ explicitly defined in terms of the initial positions and velocities of the $N$ particles. Some of these models are asymptotically isochronous, i.e. in the remote future they become completely periodic with a period $T$ independent of the initial data (up to exponentially vanishing corrections). Alternative formulations of these models, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited.

A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion ("acceleration equal force") featuring one-body and two-body velocity-dependent forces "of goldfish type" which determine the motion of an arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex) values $z_{n}(t)$ at time $t$ of the $N$ coordinates of these moving particles are given by the $N$ eigenvalues of a time-dependent $N\times N$ matrix $U(t)$ explicitly known in terms of the 2N initial data $z_{n}(0)$ and $\dot{z}_{n}(0)$. This model comes in two different variants, one featuring 3 arbitrary coupling constants, the other only 2; for special values of these parameters all solutions are completely periodic with the same period independent of the initial data ("isochrony"); for other special values of these parameters this property holds up to corrections vanishing exponentially as $t\rightarrow \infty$ ("asymptotic isochrony"). Other isochronous variants of these models are also reported. Alternative formulations, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited. Some mathematical findings implied by some of these results - such as Diophantine properties of the zeros of certain polynomials - are outlined, but their analysis is postponed to a separate paper.