A CRITERION FOR POLYNOMIAL DECOMPOSITION

Abstract

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.