In a recent paper, it has been shown that up to ⌊K over 2⌋ ⌈L over 2⌉two-dimensional (2-D) exponentials are almost surely identifiable from a K × L mixture, assuming regular sampling at or above Nyquist in both dimensions. This holds for damped or undamped exponentials. In this paper, we show that up to ⌊K over 2⌋ ⌈L over 2⌉ undamped exponentials can be uniquely recovered almost surely. Multidimensional conjugate folding is used to achieve this improvement. The main result is then generalized to N > 2 dimensions. The gain is interesting from a theoretical standpoint, but also for small 2-D sensor arrays or higher dimensions and odd sample sizes. Also important is that the proof implies an algebraic retrieval algorithm, called the MDF algorithm, which outperforms some of the best known algebraic 2-D harmonic retrieval algorithms. We illustrate this by comparing to MEMP, JAFE, and also our own earlier multidimensional embedding (MDE) algorithm.