Vol. 102, No. 1, 1982

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Mapgerms infinitely determined with respect to right-left equivalence

Leslie Wilson

Vol. 102 (1982), No. 1, 235–245

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.

Mathematical Subject Classification
Primary: 58C27
Received: 1 October 1979
Revised: 9 January 1981
Published: 1 September 1982
Leslie Wilson