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

Let [Image: see text] be a set of functions with a common domain X and a common range Y. A set [Image: see text] is called a set of range uniqueness (SRU) for [Image: see text] , if for all [Image: see text] , [Image: see text] Let [Image: see text] be the set of all real polynomials in n variables of degree at most k and let [Image: see text] be the set of all linear functions [Image: see text] with rank k. We show that there are SRU's for [Image: see text] of cardinality [Image: see text] , but there are no such SRU's of size [Image: see text] or less. Moreover, we show that there are SRU's for [Image: see text] of size [Image: see text] but there are no such SRU's of smaller size.