A CRITERION FOR POLYNOMIAL DECOMPOSITION - Lazov, Petar R. (Matematichki Bilten, 1995)
Abstract:Let \(B=B(x)\) be a complex polynomial for which\[\deg B(x)=m\cdot n,\qquad m\geq 1,\quad n\geq 2,\quad m,n\in N.\]In this work we state a criterion for the following proposition to hold:\[\left\{\begin{aligned}&\text{there exist complex polynomials } y=y(x),\quad \deg y(x)=m\\&\text{and } u=u(x),\quad \deg u(x)=n,\quad \text{such that}\\&B(x)=u(y(x)).\end{aligned}\right.\]In addition, as an auxiliary result we obtain a theorem that completely solves the problem of the polynomial solutions of the algebraic equation\[B(x)=c_0+c_1\cdot y+\cdots+c_{n-1}\cdot y^{n-1}+c_n\cdot y^n,\]giving also an algorithm for finding them.