What is the min-max of the number of points at which two polynomials of given degree intersect?

Question

Let $m,n\in \Bbb N, m\geq n+1$ be given. Denote $\#(f)$ the number of roots of a functions $f:\Bbb R \to \Bbb R$, and ${\bf P}_k$ the space of polynomial of degree equal $k$. How much is the following min-max? $$ \phi(m,n) = \min_{p_m \in {\bf P}_m} \max_{p_n \in {\bf P}_n} \#(p_m-p_n) $$

Based on the result

• What is the maximum number of points at which an $n$-degree polynomial can intersect the power function $x^m, m>n$.

we can conclude that $$ n \leq \phi(m,n) \leq \begin{cases} n+1, & m-n \ \text{ odd}, \\ n+2, & m-n \ \text{ even}. \end{cases} $$

How much is $\phi(m,n)$ exactly?

Answer 1

Let $p_n\in \mathbf P_m$ maximise the number of real roots of $x^m-p_n$. Pick $a,b\in \Bbb R$ and $v>0$ such that $x^m-p_n$

• has all its real roots in $[a,b]$,
• assumes absolute value $>v$ between any two of these real roots,
• has absolute value $>v$ at $a$ and $b$.

Now let $p_m\in \mathbf P_m$ be arbitrary and let $c$ be its leading coefficient. For $t>0$, consider $c^{-1}t^{-m}p_m(tx)$. For $t$ big enough, this differs from $x^m$ by less than $v$ on $[a,b]$. Then $c^{-1}t^{-m}p_m(tx)-p_n(x)$ has at least as many real roots as $x^m-p_n(x)$, and so does $p_m(x)-ct^mp_n(t^{-1}x)$.

It follows that $$\min_{p_m\in \mathbf P_m}\max_{p_n\in\mathbf P_n}\#(p_m-p_n)= \max_{p_n\in\mathbf P_n}\#(x^m-p_n) =\begin{cases} n+1&n\not\equiv m\pmod2\\ n+2&n\equiv m\pmod2 \end{cases}$$ where the last step is shown in the referenced answer.