6 problems

Let $\mathbf{p}(x)\in\mathbb{Z}[x]$ be an irreducible monic polynomial of degree $3$. An integer-valued quadratic polynomial $q_{0}(x)$ is a quadratic polynomial taking integer val…

Monotonicity Conjecture. As we cross any primary ray of co-period $q$ into the face which contains $H_\mathfrak{p}$, the period $q$ orbit portrait is replaced by a strictly larger…

Let $\mathcal{S}_1$ denote the space of cubic polynomials with a fixed critical point, and let $f,g\in\mathcal{S}_1$ be postcritically finite polynomials. Two polynomials are matea…

Let $\mathcal{M}_3$ be the connectedness locus of affine conjugacy classes of cubic polynomials, let $\mathcal{PHD}_3$ be its principal hyperbolic domain, and let $\mathcal{CU}$ be…

Common-tail conjecture. The roots have common tails whenever the splitting field of $f$ has degree $3$ over $\mathbb{Q}$.

Let $\overline{\operatorname{PHD}}_3$ denote the closure of the principal hyperbolic domain for monic centered cubic polynomials, let $\mathrm{CU}$ denote the cubic main cubioid, a…