5 problems

Hurwitz stability conjecture. The polynomial $T_{n,r}(x)$ is Hurwitz stable for all $r\ge 2$ and $n\in\mathbb{N}$. The sequence is introduced after Hurwitz stability has been estab…

Real-root location conjecture. The inequality $x_0>\max(S_1)$ holds if and only if

Integral and classified real-root conjecture. If all roots of $p_S(x)$ are real, then all of them are integral. Moreover, $p_S(x)$ has only real roots if and only if

Bounded-roots conjecture. Every complex root of $p_S(z)$ lies in

Billey–Burdzy–Sagan's positivity conjecture. Each coefficient $(\Delta^j p_S)(m)$ is a positive integer for $1\le j\le m-1$.