178 RICE UNIVERSITY STUDIES
Thus a sequence in R2 with “infinite order of contact to the x-axis at 0”
but which is never in the x-axis is a determining set. Examples are the
sequences of the type {(xn, g(xn))} where
g(t) = exp(-l∕t2), l ≠ 0,
and xn → 0. In particular, Lemma 3.1 shows that the graph of g is a deter-
mining set for functions real-analytic near 0 in R2. Later, other graphs
constructed from g will be shown to be determining sets for real-analytic
functions of three and four variables.
Proof. It will be shown first that ai0 = 0 for all i ≥ 0 in the expansion
h(x,y) = Σ aijxtyj.
>J
For i = 0 this is clear by continuity of h at 0, since 0 = lim„_.œh(xn,yn)
= ∕ι(0,0) = a00. If p > 0 and it is true for all i < p, then for each n,
° = ⅛fc(χ∣>.yn) = ⅛ ς βvxW + ∑ aljxin~pyjn.
xn xn l<p.J≥O ,i≥p,j≥O
By the induction hypothesis, the first sum on the right is taken only over
positive j, so that this sum is divisible by y„. It therefore becomes y„lx„ times
a function analytic in a neighborhood of zero. By hypothesis, the first sum
thus tends to zero. The constant term of the second sum on the right is
apo. Therefore as n increases the right side tends to apo, so that apo = 0.
This proves that any function real-analytic in a connected neighborhood
of 0 and which vanishes on {(xn, yll)} is divisible by y.
Now h can be rewritten as an infinite sum of homogeneous monomials.
If h is not identically zero, there is in this expression a non zero monomial
of least (total) degree m, and consequently a largest integer p ≤ in such that
h(χ,y) = ypk(χ>y)
for some real-analytic function к and all (x,y) near 0. Thus for all n,
0 = ynk(xn,yn), and since yn ≠ 0, к also vanishes on {(xn,yn)}. Therefore к
is divisible by y, which contradicts the choice of p. This proves that h is zero.
This result is used to construct a determining set in R3.
Lemma 3.2. If h is a real-analytic function on a connected neigh-
borhood of 0 in R3 which vanishes on
{(i,g(i)sin(l∕t), g(t)cos(l∕t)): t > 0}
then h = 0.