The number of polynomial solutions of polynomial Riccati equations

Abstract

Consider real or complex polynomial Riccati differential equations a(x)y˙=b0(x)+b1(x)y+b2(x)y2 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 2η (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.