For a given finite dimensional subspace \( \mathcal P \) of \( k[x_1, \ldots, x_n] \), where \( k \) is a field, a subset  \( \mathcal N\subseteq k^n \) is a \( \mathcal P \)-testing set if any member of \( \mathcal P \) that vanishes at all points of \( \mathcal N \), vanishes all over; and we say \( \mathcal N \) is optimal if it has the smallest cardinality among all \( \mathcal P \)-testing sets. This is related to Lagrangian interpolation of data on a set \( \mathcal N \) of nodes using functions from \( \mathcal P \). We consider a generic version of this interpolation problem, when \( \mathcal P \) has a monomial basis \( \mathcal B \) that we identify with a grid (i.e. a finite subset of \( \mathbb N_0^n \) ), each node is an \( n \)-tuple of independent variables and the set of nodes is identified with a grid \( \mathcal C\subseteq \mathbb N_0^n \) . A corollary to our main result offers an explicit formula for the determinant of the linear system corresponding to the generic interpolation problem in case \( \mathcal B=\mathcal C \) is a \( \sigma \)-lower (or \( \sigma \)-upper) grid, where we say \( \mathcal B \) is a \( \sigma \)-lower (resp., \( \sigma \)-upper ) grid if it is a union of intervals of \( \mathbb N_0^n \) having  as common origin (resp., endpoint). We give explicit (optimal) \( \mathcal P \)-testing sets for spaces having monomial bases determined by \( \sigma \)-lower (or \( \sigma \)-upper) grids. The corollaries at the end, for the finite field case, have potential use in Number Theory and Coding Theory.