The name is absent

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 {(x_n, g(x_n))} where

g(t) = exp(-l∕t²), l ≠ 0,

and x_n → 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 a_i0 = 0 for all i ≥ 0 in the expansion

h(x,y) = Σ a_ijx^ty^j.

>J

For i = 0 this is clear by continuity of h at 0, since 0 = lim„_._œh(x_n,y_n)
= ∕ι(0,0) = a₀₀. If p > 0 and it is true for all i < p, then for each n,

° = ⅛^fc(^χ∣>.yn) = ⅛ ^{ς β}v^xW + ∑ a_ljxⁱ_n~^py^j_n.

^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
a_po. Therefore as n increases the right side tends to a_po, so that a_po = 0.

This proves that any function real-analytic in a connected neighborhood
of 0 and which vanishes on {(x_n, y_ll)} 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) = y^pk(^χ>y)

for some real-analytic function к and all (x,y) near 0. Thus for all n,
0 = y_nk(x_n,y_n), and since y_n ≠ 0, к also vanishes on {(x_n,y_n)}. 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.

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