On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

On symmetric polynomials having triangular automorphisms

“…In [1] we considered the functional equation f (x, y) = f (y, x) = f (x, −y +u(x)) where u is a complex polynomial of degree at least two, and have shown that the only solutions are the constant solutions. The proof of this result required tools which are somewhat different from those we need in this paper.…”

“…How to extend this result? We note that Theorem 1 of [1] and the case (I) in the main Theorem hereafter provide two subgroups G 1 and G 2 of Aut C C(x, y) which are isomorphic (to the group ∆, see §2), with invariant fields of respective transcendence degree zero and one over C. The usual topology, used in infinite Galois theory, for which the subgroups of finite indices are the elements of a basis for the neighborhoods of zero, cannot explain this phenomenon. Does there exist some topology on G 1 , G 2 which gives light on this fact?…”

“…Let here and throughout the paper a, b ∈ C, let s and t be as in (1), and G be the subgroup of Aut C C(x, y) generated by s and t. Then, two cases occur:…”

“…In accordance with Theorem 5 of [1], since ψ(x, y) is invariant and of degree two, we must seek for an invariant polynomial of degree n(a) and we are led to define the polynomial:…”

“…Let us suppose, for a contradiction, that ϕ, ψ are algebraically dependent over C; then, by Castelnuovo's theorem [1,Theorem 3], there exists some rational function h ∈ C(x, y) such that C(ϕ, ψ) = C(h). Furthermore, the lemma of E. Noether [1, Lemma 3 (a)] shows that we can choose h as a polynomial since C(h) contains some non-constant polynomials say ϕ, ψ.…”