Distribution of nodes on algebraic curves in \({\mathbb C}^N\). (Q1415571)

Let \(A\subset\mathbb C^N\) be an irreducible curve and let \(K\subset A\) be a non-polar compact set. For \(d=0,1,\dots\), let \(m_d\) be the dimension of the space \(P_d| A\) of all complex polynomials of degree at most \(d\) restricted to \(A\). It is known that for sufficiently large \(d\) we have \(m_d=dD+c\), where \(D\) is the degree of \(A\) and \(c\) is an integer. Consider Lagrange interpolation polynomials \(L_df(z)= \sum_{j=1}^{m_d}f(A_{d,j})\ell^{(d)}_j(z)\) with nodes \(A_{d,j}\in K\). Let \(\varLambda_d:=\| \sum_{j=1}^{m_d}| \ell^{(d)}_j| \| _K\). Assume that the nodes are chosen in such a way that \(\limsup_{d\to+\infty}\varLambda_d^{1/d}\leq1\). Then \((1/m_d)\sum_{j=1}^{m_d}\delta_{A_{d,j}}\overset{\text{weak--}\ast}\)