Consider the following subset of the $X Y$-plane.

$$ S=\left\{\left(|z-\mathrm{i} z|,|z|^2\right): \quad z \text { is a complex number }\right\} $$

Which one of the following statements is correct?

Let $a, b$ be nonzero real numbers. If $p(x)=x^n+c_{n-1} x^{n-1}+\cdots+c_0$, where $c_{n-1}, \ldots, c_0$ are integers and $n \geq 3$, then which of the following statements is correct?