Birkhoff interpolation on some perturbed roots of unity (Q2746995)

The problem of \((0,m)\) interpolation on the roots of unity and in particular, on the zeros of \(z^{2n}+1\), is regular. The present paper deals with the regularity of some interpolation problems with respect to nodes which are not uniformly distributed. It is proved that the problem of \((0,m)\) and \((0, 1,\dots,r-2,r)\) interpolation on the zeros of \((z^n+1) (z-\zeta)\), \(\zeta\in \mathbb{C}\) is regular, if and only if, \(\zeta(\zeta^n+1) (\zeta^n+ \alpha_{m,n})=0\), where \(\alpha_{m,n}\) is explicitly given. The regularity is also studied on the zeros of \((z^{2n}+1) (z-\omega_2) (z-\omega_2)\), where \(\omega_1,\omega_2\) are two distinct zeros of \(z^{2n}-1\), and on the zeros of \((z^{2n}+1) (z^2-\zeta^2)\) and \((z^{2n}+1) (z^n-\zeta^n)\) respectively. It is proved that the values of \(\zeta\) for which the above problems are regular are not zeros of certain polynomials which are also explicitly obtained.