The number of polynomial solutions of polynomial Riccati equations

Abstract

Consider real or complex polynomial Riccati differential equations $a(x)\dot{y}=b_0(x)+b_1(x)y+b_2(x)y^2$ with all the involved functions being polynomials of degree at most $\eta$. We prove that the maximum number of polynomial solutions is $\eta +1$ (resp. 2) when $\eta \ge 1$ (resp. $\eta =0$) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most $\eta \ge 1$ we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is $2\eta$ (resp. $3$) when $\eta \ge 2$ (resp. $\eta =1$) 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.

21 pages, 1 figure