DSpace at VNU: Local polynomial convexity of union of two graphs with CR isolated singularities

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

then K is locally
polynom ially convex at all point a ^ K (see [1], chapter 17). N ote that union o f two polynomially
convex sets w hich can be not polynomially convex set. Let z? be a small closed disk in the complex
plane, centered at the origin and

Ml ^ { { z, z) : 2: e - D} ; M2 = { ( z , z ^ ( p ( z ) ) : z e D } ,
where if is ã
function in neighborhood o f 0, (f{z) = o ( | 2:|). Then M l, M 2 are totally real(locally
contained in a totally real manifold), so that M l, M 2 are locally polynom ially convex at 0. The local
polynomially convex hull o f M l u M 2 is essentially studied by Nguyen Q uang Dieu (see [2,3]).
Let

= {{z,^) : z e D } , X 2 = {{z,^ + ^z)) :zeD},
where n > 1 is interger and (yơ is a
function in neighborhood o f 0, (fi{z) = o ( |z |” ). If n > 1 then
X \ and X 2 is not totally real at 0, so we can not deduce that X ị and X 2 are locally polynomially at 0
by the W erm er’s work. However, using the results about local approxim ation o f De Paepe (see [4;) or
the work o f Bharali (see [5]), we can conclude that X i and X 2 are locally polynom ially convex at 0.
In this paper, we will investigate the local polynomially hull o f X i L I X 2 at 0. The ideas o f proof takes
* E-mail:


Kieu Phuong Chi / VNU Journal o f Science, Mathematics - Physics 24 (2008) 6-10

from [2] and [3]. An appropriate tool in this context is K allin’s lemma (see [6,7]); Suppose X i and
Ẩ 2 are po lyn o m ia lly convex subsets o f C ” , suppose there is po lyn o m ia l p m a p p in g X i and X 2
in to two p o lynom ialhj convex subsets Y \ and Y 2 o f the com plex plane such th a t 0 is a boundary
p o in t o f both Y i and Y 2 and w ith Yi n >2 = {0}. / / p “ -(0) n (X i u X 2 ) is polynornially convex,
then X i u X 2 IS p olynom ially convex. Several instances o f such a situation, m otivated by questions
o f local approxim ation, were studied by O ’Farell, De Paepe and Nguyen Quang D ieu (see [8-10],...).

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

fu n ctio n which is defined

near 0 o f the fo rm
ip{z) =
where f { z ) is a

0

z = 0,

fu n ctio n and f { z ) = o(l-sl"*). Suppose that there exists I < f
ai

such that

(1)

ak
k^l

and

is integer. Then X ị U X

2

is locally polynom ially convex at 0.

Proof. C onsider the polynomial p (z , w ) =

■^^m- 21 +n _|_
2 i+n belongs to real axis and

—_m —2í-Ị-n

p{X 2) = az

— _m —2/H-n ,

From p { X i ) = 0 2 " " '^ '+ " +

m —2/+n

w ith a choose later. Thus p { X i ) =

+

+CO
+ a (z "+ ^
a k Z ^z^-^ + f { z ) ) ^ + ^ =
k=—oo
^

+

01

+ 1 ) ^ - 2; y
71

+ 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

(2 )

k^i
for any z 7^ 0 in a small neighborhood o f 0, by (1). It implies that p { X 2 ) n R ^ {0}. On the other
hand, from the inquality (2) we see that
p - ' ( 0 ) n X 2^ = { 0 }.
It is elm entary to check that

p “ ^(0) n X I = {{pex.p{i9), p"^exp{-niO)) : 0 <

< r},

with a constant 9. Obviously,
p - '( 0) n x [
is p o ly n o m ia lly c o n v e x for r sm all enough. T h u s p ~ ^ ( 0 ) n ( X [ U X 2 ) is p o ly n o m ia lly c o n v e x fo r r sm all

enough. By K allin ’s lemma (mentioned in introduction) we conclude that X I u X
convex for r sm all enough. The proof is completed.

2

is polynom ially

R e m a rk . 1) In the Theorem 1 we can replace X i by
X{ = { { z X - ^ i z ) ) : z E D } .
Then, as p in T heorem 1 we obtain the estimate
ĩm p ix ị)

<

0,

for any 2: / 0 in sm all neighborhood o f 0. On the other hand p~ ^(0) n (X 'ỵ u x ^ ) = {0} for r small
enough. By K allin ’s lemma we may conclude that X Ị u X 2 is locally polynom ially convex.
2)
This result includes the more restricted case n = 1 that is studied by N guyen Q uang Die
(see [2]).
The follow ing Proposition shows that if we replace i > 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

principle we see that the hull of X Ị u
will contain an open subset o f Wt bounded by tw o closed
curves Pị and Qt for any Í > 0 small enough and hence X i u X 2 is not locally polynom ially convex
at 0 .


Kicu Phiiong Chi / VNV Journal o f Science, Mathematics - Physics 24 (2008) 6-Ỉ0

T heorem 2.3. Let m be a posiiive even integer and let n be a odd integer such that m > n. Let g
be a
function which is defined near 0 o f ihe fo rm
9Ì^) =

0

z =

0

,

where f is a c ’ fu n c tio n and f { z ) = o ( | 2:|"^). Suppose that there exists I such that

is positive

integer and
a ; l > ^ |a / c |.
k^i
Then the functions

(3)

and g'^{z) separate points near 0. Morever: [z^,g'^-,D] = C { D ) fo r D sm all

enough.
We need the next lemma (see [7,8]) for the proof o f Theorem 2.1.
L em m a 2.4. Let X be a com pact subset o f c ^ , and let TĨ \
be defined by tt{z , w ) =
{ z ' ^, w^ ) . Lei TT~^{X) = X u u ... u Xk i u ...
with X m n compact, and X k i = {{p'^z, t'-w ) :
{ z , w) e Xrr^n} f o r I < k < m , 1 < I < n, where p = exp

and T = exp

If

P ( ^ - 1 (X )) = c '(7 r-i (X )), then P { X ) = C { X ) .
P ro o f o f Theorem 2.3. First we show that the functions
and g'^iz) separate points near 0. Clearly
points a and b w ith a Ỷ - h are separated by z^. Now assume that g'^{z) takes the same value at a
and —a for some a ^ 0. Set
h( z ) =

z =

0

0

,

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,

X

4

= {{z, - z " - h{z) ) : z e D } = { ( - z , z" - h { - z ) ) : z € -D};


Kieu Phuong Chi / VNU Journal o f Science, Mathematics - Physics 24 (2008) 6-Ỉ0

10

By Rem ark ]), X ị U X

2

is polynomially convex for D small enough . We have X s u X

o f X i U J ^ 2 u n d er the b ih o lo m o rp h ic m ap

{z ,w )\-^

{-z ,w ).

4

is the image

S 0 X 3 U X 4 is also p o ly n o m ia lly c o n v e x

w ith D sufficiently small.

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
[1] 11. Alexander, J. Wermer, Several Complex Variables and Banach Algebras, Grad. Texts in Math., springer-Verlag, New
York, 35 (1998).
[2] Nguyen Quang Dieu, Local polynomial convexity of tangcntials union o f totaliy real graphs in c ^ , ỉndag. Math., 10
(1999) 349.
[3] Nguyen Quang Dicu, Local hulls of union of totally real graphs lying in real hypcrsurfaces, Michigan Math. Journal,
47 (2) (2000) 335.
[4] P.J. de Paepe, Approximation on a disk I, Math. Zeit., 212 (1993) 145.
[5] G. Bharali, Surfaces with degenerate CR sigularities that arc locally polynomially convcx, Michigan Math. Journal., 53
(2005) 429.
[6] E. Kallin, Fat polynomially convex sets, Function Algebras, (Proc. Inter. Symp. on Function Algebras, Tulane Univ,
1965), Scott Foresman, Chicago, (1966) 149.
[7] RJ. de Paepe, Eva Kallin’s lemma on polynomial convexity, B ull o f London Math. Soc., 33 (2001) 1.
[8] Kieu Phuong Chi, Function algebras on a disk, VNU Journal o f Sciences, Mathematics - Physics No3 (2002) 1.
[9] Nguyen Quang Dieu, P.J. de Paepe, Function algebras on disks, Complex Variables 47 (2002) 447.
[10] Nguyen Quang Dieu, Kieu Phuong Chi, Function algebras on disks II, Indag. Math., 17 (2006) 557.
[11] P.J. de Paepe, Algebras of continuous functions on disks, Proc. o f the R. Irish. Acad., 96A (1996) 85.
[12] A.G. O T aưell, K.J. Preskenis, Uniform approximation by polynoimials in two functions, Math Ann., 284 (1989) 529.

[13] A.G. O ’Farrcll, PJ. de Pacpe, Approximation on a disk n, Math. Zeit., 212 (1993) 153.
[14] Kieu Phuong Chi, Polynomial approxiamtion on polydisks, VNƯ Journal o f Sciences, Mathematics - Physics No3 (2005)
11.

[15] A.G. OTarrell, K.J. Preskenis, D. Walsh, Holomoq^hic approximation in Lipschitz nomis, Contemp. Math., 32 (1984)
187.