4 problems

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in\mathbb{C}[x]$ have no repeated zeros, and let $\mathcal{S}=(f(x),f(x),\ldots,f(x))$. Let…

Let $\mathcal{F}=\{\mathcal{S}_j\}_{j=1}^{\infty}$ be a collection of tuples in $\mathbb{R}[x]$, and let $\mathcal{S}\in\mathcal{F}$ satisfy $\operatorname{deg}(\mathcal{S})>1$. Ma…

Let $\{\mathcal{S}_j\}_{j=1}^{\infty}$ be the collection of all tuples in $\mathbb{R}[x]$, and let $\mathcal{Z}[(\gamma^{-1}\circ\beta\circ\gamma\circ\nabla)^n(\mathcal{S}_j)]$ be…

Let $\mathcal{S}$ be a tuple of polynomials in $\mathbb{R}[x]$ whose integer coefficients all have the same parity, and let $\mathcal{R}(\mathcal{S})$ denote the rank of its expans…