VNƯ Journal of Science, Mathematics - Physics 24 (2008) 6-10

Local polynomial convexity of union of two graphs with CR
isolated singularities
K ieu Phuong Chi*
Department o f Mathematics, Vinh University, Nghe An, Vietnam
Received 26 October 2007; received in revised form 4 December 2007

A b s t r a c t, We give sufficient conditions so that the union of two graphs with CR isolated
singularities in
is locally polynomially convex at a singularly point. Using this result and
some ideas in previous work, we obtain a new result about local approximation continuous
function.

1. In tro d u c tio n
We recall that for a given compact K in

by K we denote the polynomial convcx hull o f

K i.e.,
K = { z E:

\ \p{z)\ < \\p \\k for every p o ly n o m ial p in c ^ } .

We say that K is polynom ially convex i f K — K . A compact K is called locally polynomially convex
at a G
if there exists the closed ball B{ a) centered at a such that D { a ) n K is polynomially convex.
A smooth real manifold S' c
is said to be to ta lly real at a e 5 if the tangent plane Ts { a) o f
5 at a contains no com plex line. A point a € S' is not totally real that will be called a C R singularity.
By the result o f Wermer, if K is contained in totally real smooth subm anifolds o f

Let / be a continuous function on D . We denote that [z^, p ; D] is the function algebra which
consisting o f uniform limit on D o f all polynomials in 2 ^ and /^ . Using polynomial convexity theory,
it can be shown that [z^, p ] D] = C { D ) for some choices a
function / , w hich behaves like z near
the origin (see [9-11] ...)■ By the known result about approximation o f O ’Farrell, Preskenis and Walsh
[12] :i,f X is p o h jnom ially convex subset o f the real m a n ifo ld M , K is a com pact subset o f X
such th a t X \ K I S totally real. Then, i f f is co ntinuous fu n c tio n on X a n d f can be u n ifo rm
approxim ated by polynom ials on K then f can be u n ifo rm approxim ated by p o lyn o m ia ls on X ,
and the techniques developed in [13], we give a class function / which behaves like ^ such that
\f-D]:=C{D).

2. T he m a in results
We alw ays take the graphs X i and X

2

o f the form (*). For each r > 0 we put

= X i n {(z, ti)) : 1^1 < r} ,

1

=

1

,2.

N ow vvc come to the main results o f this paper.
T h eo rem 2.1. Let r n ,n he positive integers with rn > n. Let ip be a

+ o{\zr).
,^^
k=—oo

€ R , we obtain
9/

Im p ( X 2 ) =
^

^
k=-ooI

+ o d z D ).
__


Kieu Phuorìg Chi / VNU Journal o f Science, Mathematics - Physics 24 (2008) 6-Ì0

C h o o se

a

=

27% . It fo llo w s that

i m p { X 2 ) > |^|2—

> 0
y we may get nontrivial hull o f X [ U X 2 .
P roposition 2.2. Lei n , p be positive integers and

= {(2, 2") ■.z e D ) - X 2 =

-.zeD).

Then X i u X 2 is not locally polynom ially convex at 0.
Proof. For each t > Q , let Wi = {{z, w) : z ^ w — t}. Consider the sets
Pt : = Wt n

= { { z , r ) : \z\ = t ầ } -

Qt : = VKi n X2 = {(2, z” + zPz^+P) : 1^1 = s},
where s is unique positive solution o f the equation

+ s 2p+ 2n _ ị

gy

maximum modulus

^ |a / c |.
k^i
Then the functions



it follows that h( a) = - h { - a ) . As m is even, we have
aka a

2^

-/(a )-/(-a )
= ---------- ----------- .

k= —oo

D ividing both sides by

'a ' we obtain

kỹíl

a‘

By the inequality (3) and the fact that f { z ) = o ( |z p ) , we arrive at a contradition if we choose the
disk D sufficiently small.
N ext vve consider for a small closed disk D the set X w hich is the inverse o f the compact
X = { ( 2 ^, g'^{z) ; 2 E D } under the map ( 2 , w ) ^ (z^, w"^). We have X — X i U X 2 u A'3 U X 4 where
X , = {{z,T + h{z)):zeD};
X

2

= { ( - z , - 2 " - h{z)) : z € Ơ } = { { z , r - h { - z ) ) : z G D} X, = { { - z X + h{z))):zeDy,


Now we consider the polynomial q { z , w) = z ^ w . Then q maps X i u X 2 to an angular sector
situated near the positive real axis, while p maps X 3 U X 4 to such sector situated near the negative real
axis. The sectors only meet at the origin. A pplying K allin’s lemm a w e get X ^
u X2 u X3 u X4
is polynom ially convex with D small enough. Furthermore, notice that X \ {0} is totally real (locally
contained in a totally real manifold), by an approximation theorem o f O T a ư e ll, Preskenis and Walsh
(mentioned in introduction), we get that every continuous function on X can be uniformly approximated
by polynomials. By the Lemma 2.4, we see that the same is true for X , which is equivalent to the
fact that our algebra equals C{ D) .
A cknow ledgem ents. The author is greatly indebted to Dr. Nguyen Quang Dieu for suggesting the
problem and for m any stim ulating conversations.

R eferences
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