Multivariate polynomials: A spanning question

Abstract

The main result of this paper is the following. Ifg is any given polynomial of two variables, then

$$span\left\{ {\left( {g\left( {. - a,. - b} \right)} \right)^k :\left( {a,b} \right) \in R^2 ,k \in {\rm Z}_ + } \right\}$$

contains all polynomials if and only if

$$span\left\{ {g\left( {. - a,. - b} \right):\left( {a,b} \right) \in R^2 } \right\}$$

separates points. This result is not valid inR ^d ford≥4.