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Takashi SfflOYA

By eliminating lemons one gets the following:

1.11. Proposition. Let a be an almost regular curve.

(i) If ind(a) = °°, then rot(a) = «.

(ii) If ind(a) °°, then rot(a) = ind(a) - 1 if ~n{a) - ind(a) is odd, and

rot(a) = ind(a) ifn(cc) - ind(a) is even or infinite.

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Corollary. A semi-regular curve a is regular if and only if rot(a) = ind(a) =

n(cc).

D

1.12. Proposition. Let a be a semi-regular curve and K a closed disk such

that K n a = 0 and that, when a is not simple, K C Int(Bi). Let N := C1(M -

K) and let A ^ be the universal covering of N. Then any lifting a of a is a semi-

regular curve in the differentiable plane N - dN. Moreover if a is almost regular, the

lifting a is simple.

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Exercise. For a general semi-regular curve a characterize the set of integers i e

\n(a)] such that the inverse image of the double point pt e a i s a double point of a,

and give necessary and sufficient conditions on the sequence {&(/)} for & to be

simple (resp. regular, almost regular).