20 research outputs found

### Solving first order autonomous algebraic ordinary differential equations by places

Given a first order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial values, classified in terms of the number of distinct formal power series solutions extending them. In addition, if a particular initial value is given, we present a second algorithm that computes all the formal power series solutions, up to a suitable degree, corresponding to it. Furthermore, when the ground field is the field of the complex numbers, we prove that the computed formal power series solutions are all convergent in suitable neighborhoods.Agencia Estatal de InvestigaciónAustrian Science Fund (FWF

### On real and observable realizations of input-output equations

Given a single algebraic input-output equation, we present a method for
finding different representations of the associated system in the form of
rational realizations; these are dynamical systems with rational right-hand
sides. It has been shown that in the case where the input-output equation is of
order one, rational realizations can be computed, if they exist. In this work,
we focus first on the existence and actual computation of the so-called
observable rational realizations, and secondly on rational realizations with
real coefficients. The study of observable realizations allows to find every
rational realization of a given first order input-output equation, and the
necessary field extensions in this process. We show that for first order
input-output equations the existence of a rational realization is equivalent to
the existence of an observable rational realization. Moreover, we give a
criterion to decide the existence of real rational realizations. The
computation of observable and real realizations of first order input-output
equations is fully algorithmic. We also present partial results for the case of
higher order input-output equations

### Rational Solutions of Parametric First-Order Algebraic Differential Equations

In this paper we give a procedure for finding rational solutions of a given
first-order ODE with functional and constant coefficients which occur in a
rational way. We derive an associated system with the same solvability, and
sufficient and necessary conditions for the existence of rational solutions are
given. In the case where all parametric coefficients are constant, we give an
algorithm to compute the rational solutions. In the case where one functional
coefficient appears, we algorithmically find rational general solutions which
rationally depend on the appearing transcendental constant. In the other cases,
the presented procedure is not completely algorithmic

### Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0.Ministerio de Ciencia, Innovación y UniversidadesMinisterio de Economía, Industria y CompetitividadAustrian Science Fun

### Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0.Agencia Estatal de InvestigaciónMinisterio de Economía y CompetitividadAustrian Science Fun

### Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables

In this paper we study systems of autonomous algebraic ODEs
in several differential indeterminates. We develop a notion of
algebraic dimension of such systems by considering them as
algebraic systems. Afterwards we apply differential elimination
and analyze the behavior of the dimension in the resulting
Thomas decomposition. For such systems of algebraic dimension
one, we show that all formal Puiseux series solutions can be
approximated up to an arbitrary order by convergent solutions. We
show that the existence of Puiseux series and algebraic solutions
can be decided algorithmically. Moreover, we present a symbolic
algorithm to compute all algebraic solutions. The output can
either be represented by triangular systems or by their minimal
polynomials.Agencia Estatal de InvestigaciónAustrian Science Fun