Is the set of coefficients connected for monic polynomial with roots bounded by some interval?

Question

Suppose we have a monic polynomial with real coefficients \begin{align*} p(x) = x^n + \alpha_{n-1} x^{n-1} + \dots + \alpha_1 x + \alpha_0. \end{align*} Let $M > 0$ be some fixed real number and let \begin{align*} E = \{ (\alpha_{n-1}, \dots, \alpha_0)^T: \text{modulus of roots of } p(x) \text{ lie in } (-M, M)\}. \end{align*} Is $E$ a connected set considered as a subset of $\mathbb R^n$? Intuitively, it should be so since the roots continuously depend on the coefficients.

Answer 1

Answer: $E$ is connected.

Proof: Let $\Delta$ denote the set $\left\{z \in \mathbb{C} : |z| < M \text{ and } \Im(z) \geq 0 \right\}$ - which is connected.
Now, for $0 \leq k \leq \lfloor \frac{n}{2} \rfloor$, let $\Gamma_{k}$ denote the image of $\Delta^{k} \times (-M, M)^{n -2 k}$ by the continuous function $f_{k} : \mathbb{C}^{k} \times \mathbb{C}^{n -2 k} \rightarrow \mathbb{C}^{n}$ defined by $$f_{k}\left(\alpha_{1}, ..., \alpha_{k}, \beta_{1}, ..., \beta_{n -2 k}\right) = \left(\alpha_{1}, \overline{\alpha_{1}}, ..., \alpha_{k}, \overline{\alpha_{k}}, \beta_{1}, ..., \beta_{n -2 k}\right)$$ Then, for every integer $0 \leq k \leq \lfloor \frac{n}{2} \rfloor$, the set $\Gamma_{k}$ is connected and contains $(-M, M)^{n}$.
It follows that the set $$\Gamma = \bigcup_{k = 0}^{\lfloor \frac{n}{2} \rfloor} \Gamma_{k}$$ is connected.
Thus, the set $$E = \left\{ \left(a_{0}, ..., a_{n -1}\right) \in \mathbb{R}^{n} : \text{ all the roots of } x^{n} +\sum_{k = 0}^{n -1} a_{k} x^{k} \in \mathbb{R}[x] \text{ have modulus } < M \right\}$$ - which is the image of $\Gamma$ by the continuous function $g : \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ defined by $$g\left(x_{1}, ..., x_{n}\right) = \left( (-1)^{n -k} \sum_{1 \leq i_{1} < ... < i_{n -k} \leq n} x_{i_{1}} ... x_{i_{n -k}} \right)_{0 \leq k \leq n -1}$$ - is connected.

P.S: The image of $\left\{ z \in \mathbb{C} : |z| < M\right\}^{n}$ by the function $g$ is the set $$\left\{ \left(a_{0}, ..., a_{n -1}\right) \in \mathbb{C}^{n} : \text{ all the roots of } x^{n} +\sum_{k = 0}^{n -1} a_{k} x^{k} \in \mathbb{C}[x] \text{ have modulus } < M \right\}$$ which is different from $E$.