On the density of the set of generators of a polynomial algebra

Abstract:

Let $K[X] = K[x_1,...,x_n], ~n \ge 2,$ be the polynomial algebra over a field $K$ of characteristic $0$. We call a polynomial $~p \in K[X]$ coordinate (or a generator) if $K[X] = K[p, p_2, ..., p_n]$ for some polynomials $~p_2, ..., p_n$. In this note, we give a simple proof of the following interesting fact: for any polynomial $~h~$ of the form $~(x_i + q),$ where $q$ is a polynomial without constant and linear terms, and for any integer $~m \ge 2$, there is a coordinate polynomial $~p~$ such that the polynomial $(p-h)$ has no monomials of degree $\leq m$. A similar result is valid for coordinate $k$-tuples of polynomials, for any $k < n$. This contrasts sharply with the situation in other algebraic systems. On the other hand, we establish (in the two-variable case) a result related to a different kind of density. Namely, we show that given a non-coordinate two-variable polynomial, any sufficiently small perturbation of its non-zero coefficients gives another non-coordinate polynomial.