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Pàgina inicial > Articles > Articles publicats > The number of polynomial solutions of polynomial Riccati equations |
Data: | 2016 |
Resum: | Consider real or complex polynomial Riccati differential equations a(x) y=b_0(x) b_1(x)y b_2(x)y^2 with all the involved functions being polynomials of degree at most . We prove that the maximum number of polynomial solutions is 1 (resp. 2) when 1 (resp. =0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2 (resp. 3) when 2 (resp. =1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain. |
Ajuts: | Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 European Commission 316338 |
Nota: | Agraïments: The third author is partially supported by the NNSF of China 11271252 and the Innovation program of Shanghai Municipal Education Commission of China 15ZZ02. |
Drets: | Tots els drets reservats. |
Llengua: | Anglès |
Document: | Article ; recerca ; Versió acceptada per publicar |
Matèria: | Explicit solutions ; Number of polynomial solutions ; Polynomial differential equations ; Riccati differential equations ; Trigonometric polynomial differential equations |
Publicat a: | Journal of differential equations, Vol. 261 (2016) , p. 5071-5093, ISSN 1090-2732 |
Postprint 21 p, 377.6 KB |