Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We are given two distinct real numbers $c$ and $d$ and we set: $$J = \begin{cases} [c,d] & \text{if } c < d \\ [d,c] & \text{if } d < c. \end{cases}$$ Let $A \in \mathbb{C}_n[X]$ and $B \in \mathbb{C}_m[X]$ be two non-zero polynomials. Show that there exist polynomials $C \in \mathbb{C}_n[X]$ and $D \in \mathbb{C}_m[X]$ satisfying the following properties: $$\begin{gathered}
\|A\|_I = \|C\|_J, \quad \|B\|_I = \|D\|_J, \quad \|AB\|_I = \|CD\|_J \\
A(a) = C(c), \quad B(b) = D(d).
\end{gathered}$$
Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We are given two distinct real numbers $c$ and $d$ and we set:
$$J = \begin{cases} [c,d] & \text{if } c < d \\ [d,c] & \text{if } d < c. \end{cases}$$
Let $A \in \mathbb{C}_n[X]$ and $B \in \mathbb{C}_m[X]$ be two non-zero polynomials. Show that there exist polynomials $C \in \mathbb{C}_n[X]$ and $D \in \mathbb{C}_m[X]$ satisfying the following properties:
$$\begin{gathered}
\|A\|_I = \|C\|_J, \quad \|B\|_I = \|D\|_J, \quad \|AB\|_I = \|CD\|_J \\
A(a) = C(c), \quad B(b) = D(d).
\end{gathered}$$