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

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

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