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Rational ODEs with Movable Algebraic Singularities
Author(s) -
Filipuk G.,
Halburd R. G.
Publication year - 2009
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2009.00445.x
Subject(s) - mathematics , gravitational singularity , singularity , class (philosophy) , algebraic number , differential algebraic geometry , ordinary differential equation , algebraic function , algebraic equation , mathematical analysis , series (stratigraphy) , singular point of an algebraic variety , differential equation , pure mathematics , function field of an algebraic variety , rational function , differential algebraic equation , nonlinear system , paleontology , physics , quantum mechanics , artificial intelligence , biology , computer science
A class of second‐order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite‐length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations P II – P VI are poles.