Suppose we are given strictly concave function $g$ on some closed interval $[a,b]$.
We are interesting in answering how many times a degree $n$ polynomial can touch the function $g$ without crossing it.
Formally, let $P_n(g)$ denote set of all polynomial of degree at most $n$ such that $p_n \in P_n$ satisfies $p_n(t) \le g(t)$ for all $t \in [a,b]$ and let
\begin{align}
S(p_n) = \{ x \in [a,b]: g(x)=p_n(x) \}
\end{align}
We are interested in
\begin{align}
K (n,g) = \sup_{p_n \in P_n(g)} | S(p_n)|
\end{align}
where $|\cdot|$ denotes cardinality.
To make the problem non-trivial. We assume that $g$ is analytic but not a polynomial.
Question: Can we find upper bounds on $K (n,g) $ as function of $n$?
Here is my thinking and rough calculation: We can first kiss at the end points. So this is two.
So we have $n-2$ more degrees of freedom.
Next, place maxima of the polynomial to touch $g$. We can have now at most $\frac{n-2}{2}$ maxima.
So, it looks like to function can kiss at most $\frac{n-2}{2}+2$ times.
Best Answer
There is no upper bound. Consider $p(t)=-t^2$ and $g(t) = 1 - \cos t - t^2$. Note that $g''(t) = - \cos t - 2 < 0$ and so $g$ is strictly concave. Yet $p(t) \le g(t)$ everywhere and $p(t) = g(t)$ at infinitely many points, namely every multiple of $2\pi$. (One can scale these functions horizontally so that there are arbitrarily many periods inside any prescribed compact interval.)