The number of polynomial solutions of polynomial Riccati equations
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Zhang, Xiang (Shanghai Jiao Tong University. Department of Mathematics)

Date:	2016
Abstract:	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.
Grants:	Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 European Commission 316338
Note:	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.
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Explicit solutions ; Number of polynomial solutions ; Polynomial differential equations ; Riccati differential equations ; Trigonometric polynomial differential equations
Published in:	Journal of differential equations, Vol. 261 (2016) , p. 5071-5093, ISSN 1090-2732

DOI: 10.1016/j.jde.2016.07.019

Postprint 21 p, 377.6 KB

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