I wonder how to construct a polyomial $P(x)$ of even degree $N$ with
$n = \frac{N}{2}$ equal minima $x_i$, i.e. $P(x_i) = \text{m}$ for all $x_i$
$n -1$ equal maxima $X_i$, i.e. $P(X_i) = \text{M}$ for all $X_i$
with $x_1 < X_1 < x_2 < \dots < x_{n-1} < X_{n-1} < x_n$ and $\text{m} < \text{M}$.
For $N=6$ there is such a polynomial: $P(x) = x^2(x−1)^2(x−2)^2$
I suppose there is essentially one such polynomial for each even degree $N$ (upto a scaling factor and translations).
My questions are:
Is this true?
How to prove this constructively (by giving the polynomial for each $N$)?