The Buchberger-Möller algorithm is a famous symbolic method for finding all polynomials that vanish on a point cloud. It has even been extended to noisy samples. However, the resulting variety does not necessarily represent the topological or geometric structure of the data well. By making use of the Vandermonde matrix, it is possible to find polynomials of a prescribed degree vanishing on the samples. As this matrix is severely ill-conditioned, modifications are necessary. By making use of statistical and algebro-geometric techniques, an algorithm for learning a vanishing ideal that represents the data points‘ geometric properties well is presented. It is investigated that this method -- among various other desirable properties -- is more robust against perturbations in the data than the original algorithm.