Computation of vanishing ideals of multiple points comes with surprising challenges even over a finite field. Gröbner bases, a special type of generating sets, are the standard tool for computation. Although efficient techniques, such as the BM-algorithm, exist, for most sets of points the vanishing ideal has several equally “nice” generating sets. These sets yield multiple interpolating polynomials which is not optimal in the context of model selection. It is thus desirable to be able to find a unique or at least a small number of generating sets. In this talk, we will explore properties of vanishing ideals, particularly over finite fields, and identify properties of the points and their associated ideal that result in a unique reduced Gro ̈bner basis for the ideal. Additionally, we will see how these questions arise naturally in the design of experiments and selection of algebraic models of systems in mathematical biology.