What is the maximum number of real roots in the interval (0,1) of a monic polynomial with degree 2022 and integer coefficients: $f(x) = x^{2022} + a_{2021}x^{2021} … + a_{0}$?
Vieta's formulas, $\sum_{i_{1}<i_{2}…<i_{k}}^{}\prod_{j=1}^{k} r_{i_{j}} = (-1)^{k}\frac{a_{n-k}}{a_{0}}$, k = 1,2…n where the i's are form the set {1…n}.
Show the upper bound is 2021 since, $ r_{1}r_{2}…r_{2022} = a_{0}$
must have one root greater than one, otherwise $a_{0} \notin \mathbb{Z},$ since the left would be less than one.
A Quadratics maximum is 1 by the above, $( x- (\sqrt{2}-1))(x -(-\sqrt{2}-1))$ shows there can be one root in the interval.
I assume the lower bound is 1011 by multiplying quadratics with distant roots in the interval to attain an example.
The rational root theorem shows the roots must be irrational. I can't find any progress in differentiating, integrating or treating roots as a module over the Integer Ring.
Best Answer
For any $N$, we claim there exists some monic $f \in \mathbb Z[X]$ of degree $N$ with $N-1$ roots in $(0,1)$.
Indeed, consider the polynomial $g(x) = \prod_{i=1}^{N-1} (x-\frac{i}{N})$, and define $y_j = \frac{2j-1}{2N}$ for $j=1,2,\dots, N$. One can check via the intermediate value theorem on each interval $(y_j, y_{j+1})$ that
$$ f(x) = x^N + M\cdot g(x) $$
works for all sufficiently large integers $M$ for which $M\cdot g \in \mathbb Z[X]$. Explicitly, we could take $M$ to be any multiple of $N^{N-1} $ greater than $(\min_{1 \leq j \leq N} |g(y_j)|)^{-1}$.
As you note, this is the best possible bound.