Sets of range uniqueness for multivariate polynomials and linear functions with rank k

Let F be a set of functions with a common domain X and a common range Y. A set S subset of X is called a set of range uniqueness (SRU) for F, if for all f, g is an element of F, f [S] = g[S] double right arrow f = g. Let P-n,P-k be the set of all real polynomials in n variables of degree at most k and let L-k(R-n, R-n) be the set of all linear functions f : R-n -> R-n with rank k. We show that there are SRU's for P-n,P-k of cardinality 2((n+k)(k)) - 1, but there are no such SRU's of size 2((n+k)(k)) - 2 or less. Moreover, we show that there are SRU's for L-k(R-n, R-n) of size {2n - 1 if k = 1, 2n - k + 1 ifk > 1, but there are no such SRU's of smaller size.