Classification of algebraic ODEs with respect to rational solvability

This is the author’s version of a work that was accepted for publication in Computational Algebraic and Analytic Geometry, AMS series Contemporary Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computational Algebraic and Analytic Geometry vol. 572 pp. 193-210, AMS series Contemporary Mathematics DOI 10.1090/conm/572/11361In this paper, we introduce a group of affine linear transformations and consider its action on the set of parametrizable algebraic ODEs. In this way the set of parametrizable ODEs is partitioned into classes with an invariant associated system, and hence of equal complexity in terms of rational solvability. We study some special parametrizable ODEs: some well-known and obviously parametrizable classses of ODEs, and some classes of ODEs with special geometric shapes, whose associated systems are characterized by classical ODEs such as separable or homogeneous ones

On Symbolic Solutions of Algebraic Partial Differential Equations

The final version of this paper appears in Grasegger G., Lastra A., Sendra J.R. and\ud Winkler F. (2014). On symbolic solutions of algebraic partial differential equations, Proc.\ud CASC 2014 SpringerVerlag LNCS 8660 pp. 111-120. DOI 10.1007/978-3-319-10515-4_9\ud and it is available at at Springer via http://DOI 10.1007/978-3-319-10515-4_9In this paper we present a general procedure for solving rst-order autonomous\ud algebraic partial di erential equations in two independent variables.\ud The method uses proper rational parametrizations of algebraic surfaces\ud and generalizes a similar procedure for rst-order autonomous ordinary\ud di erential equations. We will demonstrate in examples that, depending on\ud certain steps in the procedure, rational, radical or even non-algebraic solutions\ud can be found. Solutions computed by the procedure will depend on\ud two arbitrary independent constants

Birational transformations preserving rational solutions of algebraic ordinary differential equations

We characterize the set of all rational transformations with the property of pre- serving the existence of rational solutions of algebraic ordinary di erential equations (AODEs). This set is a group under composition and, by its action, partitions the set of AODEs into equivalence classes for which the existence of rational solutions is an invariant property. Moreover, we describe how the rational solutions, if any, of two different AODEs in the same class are related.Ministerio de Economía y CompetitividadVietnam Institute for Advanced Study in Mathematics (VIASM)Austrian Science Fund (FWF)Research Group ASYNAC

Transforming ODEs and PDEs from radical coefficients to rational coefficients

We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables so that solutions correspond one-to-one. Our method also applies to systems of linear ODEs. It is based on previous work on reparametrization of radical algebraic varieties.Agencia Estatal de InvestigaciónUniversidad de AlcaláJunta de Extremadur

Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-Geometric Dimension One

The final journal version of this paper appears in A. Lastra, J. R. Sendra, L. X. C. Ngô and F. Winkler\ud (2014). Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-\ud Geometric Dimension One. Publ. Math. Debrecen Publ. Math. Debrecen 2015 / 86 / 1-2 49–69. DOI:\ud 10.5486/PMD.2015.6032 and it is available at http://dx.doi.org/10.5486/PMD.2015.6032An algebro-geometric method for determining the rational solvability\ud of autonomous algebraic ordinary differential equations is extended from single equations\ud of order 1 to systems of equations of arbitrary order but dimension 1 in the algebrogeometric\ud sense. We provide necessary conditions, for the existence of rational solutions,\ud on the degree and on the structure at infinity of the associated algebraic curve. Furthermore,\ud from a rational parametrization of a planar projection of the corresponding\ud space curve one deduces, either by derivation or by lifting the planar parametrization,\ud the existence and actual computation of all rational solutions if they exist. Moreover, if\ud the differential polynomials are defined over the rational numbers, we can express the\ud rational solutions over the same field of coefficients.Vietnam Institute for Advanced Study in Mathematics (VIASM

A computer algebra approach to rational general solutions of algebraic ordinary differential equations

In this thesis, I approach to algebraic ODEs from Differential Algebra's point of view. I look for rational solutions of AODE, I present an algebro-geometric method to decide the existence of rational solutions of a first-order algebraic ODE and if they exist an algorithm to compute them. This method depends heavily on rational parametrizations, in particular for autonomous equations on the parametrization of algebraic curves, and for non-autonomous equations on the parametrization of algebraic surfaces. In the last case, I prove the correspondence between rational solutions of a parametrizable algebraic ODE and rational solutions of a first-order linear autonomous differential system of two equations in two variables. I provide an algorithm to compute rational solutions of such system based on its invariant algebraic curves. I also study a group of affine transformations which preserves the rational solvability, in order to reduce, when possible, an algebraic ODE to an easier one. Moreover I present the results of the implementation of all these algorithms in two computer algebra system: CoCoA and Singular

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