Polynomial Solutions of Equivariant Polynomial Abel Differential Equations

Abstract

Let a⁢(x){a(x)} be non-constant and let bj⁢(x){b_{j}(x)}, for j=0,1,2,3{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{a(x)\dot{y}=b_{1}(x)y+b_{3}(x)y^{3}}, with b3⁢(x)≠0{b_{3}(x)\neq 0}, and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind a⁢(x)⁢y⁢y˙=b0⁢(x)+b2⁢(x)⁢y2{a(x)y\dot{y}=b_{0}(x)+b_{2}(x)y^{2}}, with b2⁢(x)≠0{b_{2}(x)\neq 0}, have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.

Keywords

• polynomial abel equations
• equivariant polynomial equation
• polynomial solutions
• 34a05
• 34c05
• 37c10