Algebraic notes on testing sets for lower and upper grids

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