A polynomial
$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
where $a_{n-1}, a_0$ are known. How to determine whether polynomial $f$ has integer roots? Can we give a bound of $\max\{a_i \mid i=1,\dots,n-2\}$ and $\min\{a_i \mid i=1,\dots,n-2\}$ decide by $n,a_{n-1}$, and $a_0$ such that the polynomial roots are integers.
Best Answer
I'm not sure if I understand just what you're asking, but perhaps it is the following:
If that's what you're asking, then the answer is no. Consider for example the polynomial $$ g(x) = x (x^2 - c^2)^{k}, $$ where $c \in \mathbb{Z}$. Then this has $a_0 = a_{n-1} = 0$. But even for fixed $n$, the other coefficients are unbounded if you let $c$ be large.
This part added in response to Timothy Chow's comment:
Timothy Chow made the suggestion that allowing $a_{0} = a_{n-1} = 0$ feels like cheating, and on reflection, I agree!
If we allow $a_0 = 0$, then we can't get any bound on the other coefficients
Namely, just consider any monic degree $n-2$ polynomial of the form $p(x) = x^{n-2} + b_{n-4} x^{n-4} + b_{n-5} x^{n-5} + \cdots + b_0$. Then look at the polynomial $x (x+a) p(x) = x^n + a x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + 0$. This will have integer roots, and the coefficient $a_{n-1}$ is free to be any integer you like. Yet the other coefficients can be wild. For concreteness, consider this $$ h(x) = x (x+a)(x^2 - c^2)^k, $$ which, for fixed $a$ and $k$, will have unbounded coefficients if you pick $c$ large enough [e.g., the sum of these coefficients is $h(1)$ which is unbounded].
If we disallow $a_0 = 0$, then we do get a bound
However! If we disallow the case $a_0 = 0$, then yes you can get a bound on the remaining coefficients. Namely, if $f(x)$ factors as $f(x) = \prod_{i} (x +r_i)$, then we know $a_0 = \prod_{i} r_i$. If none of the roots is $0$ (i.e., if $a_0 \neq 0$), then each root is at least $1$ in absolute value, and thus $|a_0|$ is an upper bound on every product of the form $\prod_{i \in S} |r_i|$. Therefore, we have
$$ |a_{n-k}| = \left| \sum_{|S| = k} \prod_{i \in S} r_i \right| \leq \sum_{|S| = k} \prod_{i \in S} |r_i| \leq \sum_{|S| = k} |a_0| = {n \choose k} |a_0|. $$ This bound is tight of course by the polynomials $(x+1)^n$ and $(x-1)^n$. (This reminds me of Newton's inequalities, which probably can provide an alternate proof.)