TY - JOUR

T1 - The number of polynomial solutions of polynomial Riccati equations

AU - Gasull, Armengol

AU - Torregrosa, Joan

AU - Zhang, Xiang

PY - 2016/11/5

Y1 - 2016/11/5

N2 - © 2016 Elsevier Inc. 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.

AB - © 2016 Elsevier Inc. 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.

U2 - https://doi.org/10.1016/j.jde.2016.07.019

DO - https://doi.org/10.1016/j.jde.2016.07.019

M3 - Article

VL - 261

SP - 5071

EP - 5093

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 9

ER -