  We prove explicit lower bounds for the smallest singular value and upper
bounds for the condition number of rectangular, multivariate Vandermonde
matrices with scattered nodes on the complex unit circle. Analogously to the
Shannon-Nyquist criterion, the nodes are assumed to be separated by a constant
divided by the used polynomial degree. If this constant grows linearly with the
spatial dimension, the condition number is uniformly bounded. If it grows only
logarithmically with the spatial dimension, the condition number grows slightly
stronger than exponentially with the spatial dimension. Both results are quasi
optimal and improve over all previously known results of such type.
