The number of polynomial solutions of polynomial Riccati equations

TL;DR

This paper establishes sharp bounds on the maximum number of polynomial and trigonometric polynomial solutions for polynomial Riccati differential equations, revealing new insights into their solution structure.

Contribution

It provides the first precise bounds for the number of polynomial solutions in both polynomial and trigonometric cases, including sharpness and the complexities involved.

Findings

Maximum polynomial solutions for polynomial Riccati equations is η+1 for η≥1.

Maximum trigonometric polynomial solutions is 2η for η≥2.

Bounds are proven to be sharp and involve complex algebraic considerations.

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.