Point Separating Algebras of Polynomials
Donald J. Newman · American Mathematical Monthly · 1974
For a given collection of polynomials one would like to know whether all other polynomials are obtainable from them by operations. For this to hold it is clearly necessary that the collection separate points (including infinitely close points).Very often in approximation problems this separation condition is also sufficient, see e.g., [1], and it is natural to ask whether it is in our case. There is quite a different answer to this question depending on whether we construe polynomial to mean a in 1-variable or in several variables. In several variables polynomials have a complicated behavior at oo (unlike the situation for 1-variable) and one can give counter-examples based on this behavior. This will be shown below where we shall also obtain the affirmative result in the case of 1-variable. Our proof of this is fairly elementary, but we must mention that a much shorter proof is possible, based on local rings and a theorem of Nakayama. Before turning to the precise statements and proofs, however, allow us to make one curious observation. If our collection consisted of two polynomials, p(x) and q(x) say, then the point separation condition would amount to the fact that the system