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Rational general solutions of first-order algebraic ODEs by L. X. Châu Ngô is a Computer Science article available to read on EtoBox.

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We consider the following main problem: "Given a trivariate polynomial F (x, y, z), decide the existence of a rational general solution of the algebraic ODE F (x, y, y ) = 0 of order 1; in the affirmative case, compute a rational general solution explicitly". First we neglect the differential aspect of the problem, and we consider the algebraic solution surface defined by F (x, y, z) = 0. Then, geometrically, the problem is equivalent to looking for rational solution curves on the algebraic surface F (x, y, z) = 0. We consider algebraic ODEs of order 1, whose solution surfaces admit a rational parametrization. The structure of the rational solution curves on the algebraic surface F (x, y, z) = 0 is encoded into a so-called associated system of autonomous ODEs via the parametrization map. This associated system is much simpler than the given differential equation because it is an autonomous system of order 1 and of degree 1 in the derivatives. We investigate several aspects of such an associated system. Its rational general solutions are in one-to-one correspondence with the rational general solutions of the given differential equation. Moreover, it is rather simple to solve by just

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Author

L. X. Châu Ngô

Publisher

ACM

Published

2012

Language

EN

Field

Computer Science (Physical Sciences)