If is compact, , |z| < 1, and then every element of B is expressible in the form In particular, if , then .
If , , then .
If is compact, , , and where int denotes the interior of S, then there is a neighborhood N of x such that
If for some , , |x| =1 and then
If |x| =1, , then .
Proving is trickier, because it will not do to take B as a disc of radius if is small compared to . We will instead take B as a parallelogram that becomes flatter and flatter as . The following two lemmas will be used in verifying the condition of Lemma 3.1.
Let . Let . Then for contains .
Let T, be as in Lemma 3.4. Let B be the square with vertices . Then for ,
.
We now combine all the results of this section.
There is an open neighborhood of contained in .
If for sufficiently small then z is a multiple zero of some 0,1 power series.