Let $(x_1,\dots,x_n) \in \mathbb{C}^n$ be the set of points $S$, where the polynomial $P(x_1,\dots,x_n,t)=0$ has (at least) two roots $t$ with same magnitude.

Is it true (in general), that the discriminant $D(x_1,\dots,x_n) = Discr_t(P)$ satisfy $D(x_1,\dots,x_n)=0$ on the boundary of $S$?

What happens if we restrict $S$ to be the set where the two roots of $P$ with largest magnitude must have the same magnitude?

Let $P(t)$ be a polynomial in $t$ of degree $n$, with some contiguous coefficients (not the first or last) being $x_1,\dots,x_k$ and the rest of the coefficients are fixed.

(E.g. $p(t)=1+2t+x_1t^2+x_2t^3+(5i+7)t^6$ is ok).

Consider the set $S$ defined as $(x_1,\dots,x_k) \in \mathbb{C}^k$ with the property that the roots $t_1,\dots,t_{n}$ of $P(t)=0$ may be ordered increasingly w.r.t modulus, and such that $$|t_j|=|t_{j+1}| = \dots = |t_{k+j}|,$$ for some fixed $j.$

This set is real $k$-dimensional.

Is it true, (in general), that the discriminant $D(x_1,\dots,x_n) = Discr_t(P)$ satisfy $D(x_1,\dots,x_n)=0$ on the $k-1$-dimensional boundary of $S$?