Polynomial solutions of equivariant polynomial abel differential equations

© 2018 Walter de Gruyter GmbH, Berlin/Boston. Let a(x) be non-constant and let bj(x), for j = 0, 1, 2, 3, be real or complex polynomials in the variable x. Then the real or complex equivariant polynomial Abel differential equation a(x)y = b1(x)y + b3(x)y3, with b3(x) =/ 0, and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind a(x)yy = b0(x) + b2(x)y2, with b2(x) =/ 0, have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.