Theorem 3 in the paper of Bary-Soroker et al. (https://arxiv.org/pdf/2007.14567.pdf) states the following about probability of a polynomial over $\mathbb{Z}$ being irreducible:
Let $H \ge 3$ and $n \ge 3$ be integers, and let $\mu$ be a probability measure on $\mathbb{Z}$ such that:
- (support not too large) $supp (\mu) \subseteq [−H, H]$;
- (support not too sparse) $\|\mu\|_2^2 \le \min\{H^{−4/5}, n^{1/16}/H\}/(\log H)^2$.
There are absolute constants $c > 0$ and $H_0 \ge 3$ such that if $H > H_0$, then $\mathbb{P}(A(T) \text{ is irreducible } | a_0 \ne 0) \ge 1 - n^{-c}$
Here $A(T) \in \mathbb{Z}[T]$ is a random monic polynomial of degree $n$, such that all its coefficients (with exception of the leading coefficient) are distributed according to the probability measure $\mu$ and $\|\mu\|_2^2 = \sum_{a\in \mathbb{Z}} \mu(a)^2$
I have following questions:
- Why the theorem is stated about monic polynomials? Is the non-monic case trivial? Is it, on the contrary, too hard to prove?
- Is the base of the logarithm in $\min\{H^{−4/5}, n^{1/16}/H\}/(\log H)^2$ non important? Can I use any fixed base or just any convenient base for each $H$? Particularly, I want to know if the theorem applies for the binomial distribution for K trials with probability $p = 1/2$.