we substitute the numbers found into the system (\[SRS\]). Finally, extracting any $n$ rows from (\[SRS\]) leads to a linear system of equations with unknowns $\mu_k$, which has a unique solution with non-vanishing components. We now give known formulas for calculation of the numbers $\mu_k$ (see, for example, [@DanDod2013]). Let $\sigma_m$ and $\sigma_m^{(k)}$ denote elementary symmetric polynomials of the form $$\sigma_m=\sigma_m(\lambda_1,\ldots,\lambda_n)= \sum_{1 {\leqslant}j_1<\ldots<j_m {\leqslant}n}{\lambda_{j_1}\ldots \lambda_{j_m}}, \quad
m=\overline{1,n},$$ $$\sigma_0=1,\quad
\sigma_m^{(k)}=\sigma_m(\lambda_1,\ldots,\lambda_{k-1},0,
\lambda_{k+1},\ldots,\lambda_n), \quad k=\overline{1,n}.$$

\[400\] The numbers $\mu_k$ are the scalar products $\mu_k=({\mathcal{L}}_k\cdot {\mathcal{S}})$, where ${\mathcal{S}}=
(s_0,\ldots, s_{n-1})$ and $$\label{444} {\mathcal{L}}_k=\frac{{g}_n}{{G}'_n(\lambda_k)}
\left((-1)^{n-1}\sigma_{n-1}^{(k)},  \ldots ,
(-1)^{n-m}\sigma_{n-m}^{(k)}, \ldots , -\sigma_{1}^{(k)},\;
1\right).$$

If $V=V(\lambda_1,\lambda_2,\ldots,\lambda_n)$ is a Vandermonde matrix of the first $n$ equations in the system (\[SRS\]), then the elements of the $k$th row ${\mathcal{L}}_k$ of the matrix $V^{-1}$ have the form (\[444\]); see [@DanDod2013] for more details.

We now formulate a known criterion of regularity in terms of roots of the polynomial $G_n$. Originally this criterion was obtained in an algebraic form by Sylvester [@Sylvester] (see also [@Kung1 Ch. 5]); later on Lyubich [@Lyubich; @Lyubich2] stated it in the analytical terms, which we use in the present paper.

\[Criterion\_Syl\] The system $(\ref{SRS})$ is regular if and only if the generating polynomial $G_n$ is of 