Mather has given both
algebraic and geometric characterizations of finitely determined germs. We conjecture
analogous characterizations of infinitely determined germs and prove parts of this
conjecture. Recall that two mapgerms f and g are (right-left) equivalent if there are
germs of diffeomorphisms l and r such that f = l ∘g ∘r. A mapgerm f at x is finitely
determined if there is a k such that every germ having the same k-jet as f at x is
equivalent to f; f is infinitely determined if every germ having the same Taylor series
at x as f is equivalent to f.
Let En denote the space of germs at 0 in Rn of C∞ real valued functions and let
mn denote the unique maximal ideal in En. Let Enp denote the set of p-tuples
of elements of En; mnk may denote k-tuples or may be the k-th power of
mn—which should be clear from context. If f is analytic, let fc denote its
complexification.
