The number of polynomial solutions of polynomial Riccati equations
Gasull i Embid, 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)

 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-PAgència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568European 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

DOI: 10.1016/j.jde.2016.07.019

 Postprint 21 p, 377.6 KB

Registre creat el 2017-01-23, darrera modificació el 2021-09-29

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