Let $s_\lambda(x_1,\ldots,x_n)$ be a Schur polynomial in $\mathbb{C}[x_1,\ldots,x_n]$ with $\lambda=(\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n=0)$ and $\gcd(\lambda_1+n-1,\lambda_2+n-2,\dots,\lambda_n)=1$. Set of its roots on the unit $n$-circle is specified with $$ V=\left\{(\theta_1,\ldots,\theta_n)\in[0,2\pi]^n\big|s_{\lambda}\left(e^{j\ \theta_1},\ldots,e^{j\ \theta_n}\right)=0\right\}. $$ Considering $I=\{i\ |\lambda_i\geq i\}$, $$ V_i=\left\{(\theta_1,\ldots,\theta_n)\Bigg|\theta_k=\cfrac{2\pi k}{\lambda_i+n-i}\text{ for }k=1,\ldots,n\right\},\quad i\in I $$ are subsets of $V$, because the last and $i$th rows (by Wikipedia notation) of Generalized Vandermonde Determinant are identical to $[1,\ldots,1]$. Also it is obvious: $\dim \cup_{i\in I}V_i=1$.

Question:

1. Is it true that $\cup_{i\in I}V_i=V$?
2. If the first statement is not true, what is the dimension of $V$?
3. If specifying the $\dim V$ is not possible, does there exist an informative upper bound for it?