Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-Geometric Dimension One
Publisher
Debrecen Publ. Math. Debrecen
Date
2015Funders
Ministerio de Ciencia e Innovación
Ministerio de Economía y Competitividad
Keywords
Algebraic ordinary differential equations
Rational solutions
Parametrizations of curves
Description / Notes
The final journal version of this paper appears in A. Lastra, J. R. Sendra, L. X. C. Ngô and F. Winkler
(2014). Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-
Geometric Dimension One. Publ. Math. Debrecen Publ. Math. Debrecen 2015 / 86 / 1-2 49–69. DOI:
10.5486/PMD.2015.6032
Project
info:eu-repo/grantAgreement/MICINN//MTM2011-25816-C02-01/ES/ALGORITMOS Y APLICACIONES EN GEOMETRIA DE CURVAS Y SUPERFICIES/
info:eu-repo/grantAgreement/MINECO//MTM2012-31439/ES/ANALISIS DE PERTURBACIONES SINGULARES: ESTUDIO ASINTOTICO, CAPAS LIMITE Y FENOMENOS MULTIESCALA/
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/acceptedVersion
Publisher's version
http://dx.doi.org/10.5486/PMD.2015.6032http://www.math.unideb.hu/publi/forthcoming/6032-Lastra-etal.pdfRights
© Publ. Math. Debrecen, 2014
Access rights
info:eu-repo/semantics/openAccess
Abstract
An algebro-geometric method for determining the rational solvability
of autonomous algebraic ordinary differential equations is extended from single equations
of order 1 to systems of equations of arbitrary order but dimension 1 in the algebrogeometric
sense. We provide necessary conditions, for the existence of rational solutions,
on the degree and on the structure at infinity of the associated algebraic curve. Furthermore,
from a rational parametrization of a planar projection of the corresponding
space curve one deduces, either by derivation or by lifting the planar parametrization,
the existence and actual computation of all rational solutions if they exist. Moreover, if
the differential polynomials are defined over the rational numbers, we can express the
rational solutions over the same field of coefficients.
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