6 problems

Let $q>2$ be an integer, and let $r_n$ denote the probability that a random monic degree-$n$ $p$-adic polynomial splits into linear factors. Let $\bar W_q$ be a continuous function…

Higher-arity recursion conjecture. The same recursion should hold when $q>2$ is integral:

Let $P(t) \in \mathbb{Z}_p[t]$ be a non-constant monic square-free polynomial, written as … where each $P_j(t)\in\mathbb{Z}_p[t]$ is monic and its reduction $\bar{P}_j(t)$ modulo…

Asymptotic independence conjecture. We have

Strong phase-transition conjecture. Under these assumptions, the displayed intersection is nonempty.

Uniqueness conjecture. There exists a unique $p$-adic Gibbs measure corresponding to this model.