The number of polynomial solutions of polynomial Riccati equations.

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 η . We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 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 η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2 η (resp. 3) when η ≥ 2 (resp. η = 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. [ABSTRACT FROM AUTHOR]