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

Abstract

Let $F$ be a set of functions with a common domain X and a common range Y. A set $S\subseteq X$ is called a set of range uniqueness (SRU) for $F$ , if for all $f,g\in F$ , $f\left[S\right]=g\left[S\right]\Rightarrow f=g.$ Let $P_{n,k}$ 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 $f:R^n\to R^n$ with rank k. We show that there are SRU's for $P_{n,k}$ of cardinality $2\left(\frac{n+k}{k}\right)-1$ , but there are no such SRU's of size $2\left(\frac{n+k}{k}\right)-2$ or less. Moreover, we show that there are SRU's for $L_k(R^n,R^n)$ of size $\left(\begin{array}{cc}2n-1& ifk=1,\\ 2n-k+1& ifk>1,\end{array}\right)$ but there are no such SRU's of smaller size.

Keywords: 11C20; 26C05; Sets of range uniqueness; Vandermonde; magic sets; polynomials; unique range.