3 problems

Let $h\in\mathbb{Z}[x]$ be a nonzero polynomial, and let $X=h(\mathbb{Z})$ or $X=h(\mathcal{P})$, where $\mathcal{P}$ denotes the set of primes. For each $m\in\mathbb{N}$, let…

Let $K$ be a field, let $n\geq 2$ be an integer, and let $f(x_1,\dots,x_m)$ be a multilinear polynomial. Say that $f$ satisfies $(S1)$ when it is a central polynomial, so its image…

Let $K$ be a field, let $n\geq 2$ and $m\geq 1$ be integers, and let $f(x_1,\dots,x_m)$ be a nonzero multilinear polynomial in $K\langle x_1,\dots,x_m\rangle$. Mesyan conjecture. I…