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 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 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 if the tangent plane Ts { a) o f 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 are totally real(locally contained in a totally real manifold), so that M l, M are locally polynom ially convex at The local polynomially convex hull o f M l u M is essentially studied by Nguyen Q uang Dieu (see [2,3]) Let = {{z,^) : z e D } , X = {{z,^ + ^z)) :zeD}, where n > is interger and (yơ is a function in neighborhood o f 0, (fi{z) = o ( |z |” ) If n > then X \ and X is not totally real at 0, so we can not deduce that X ị and X are locally polynomially at 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 are locally polynom ially convex at In this paper, we will investigate the local polynomially hull o f X i L I X at The ideas o f proof takes * E-mail: kpchidhv@yahoo.com 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 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 in to two p o lynom ialhj convex subsets Y \ and Y o f the com plex plane such th a t is a boundary p o in t o f both Y i and Y and w ith Yi n >2 = {0} / / p “ -(0) n (X i u X ) is polynornially convex, then X i u X 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 ^ 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) T he m a in results We alw ays take the graphs X i and X o f the form (*) For each r > we put = X i n {(z, ti)) : 1^1 < r} , = ,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 o f the fo rm ip{z) = where f { z ) is a z = 0, fu n ctio n and f { z ) = o(l-sl"*) Suppose that there exists I < f such that (1) ak k^l and is integer Then X ị U X is locally polynom ially convex at Proof C onsider the polynomial p (z , w ) = ■^^m- 21 +n _|_ i+n belongs to real axis and —_m —2í-Ị-n p{X 2) = az — _m —2/H-n , From p { X i ) = " " '^ '+ " + m —2/+n w ith a choose later Thus p { X i ) = + +CO + a (z "+ ^ a k Z ^z^-^ + f { z ) ) ^ + ^ = k=—oo ^ + 01 + ) ^ - 2; y 71 + o{\zr) ,^^ k=—oo € R , we obtain 9/ Im p ( X ) = ^ ^ 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 ) k^i for any z 7^ in a small neighborhood o f 0, by (1) It implies that p { X ) n R ^ {0} On the other hand, from the inquality (2) we see that p - ' ( ) n X 2^ = { } It is elm entary to check that p “ ^(0) n X I = {{pex.p{i9), p"^exp{-niO)) : < < r}, with a constant 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 ~ ^ ( ) n ( X [ U X ) 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 is polynom ially R e m a rk 1) In the Theorem we can replace X i by X{ = { { z X - ^ i z ) ) : z E D } Then, as p in T heorem we obtain the estimate ĩm p ix ị) < 0, for any 2: / in sm all neighborhood o f 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 is locally polynom ially convex 2) This result includes the more restricted case n = 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 P roposition 2.2 Lei n , p be positive integers and = {(2, 2") ■.z e D ) - X = -.zeD) Then X i u X is not locally polynom ially convex at 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 > small enough and hence X i u X is not locally polynom ially convex at 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 o f ihe fo rm 9Ì^) = z = , 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 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 , < I < n, where p = exp and T = exp If P ( ^ - (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 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 ^ Set h( z ) = z = 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 = { ( ^, g'^{z) ; E D } under the map ( , w ) ^ (z^, w"^) We have X — X i U X u A'3 U X where X , = {{z,T + h{z)):zeD}; X = { ( - z , - " - h{z)) : z € Ơ } = { { z , r - h { - z ) ) : z G D} X, = { { - z X + h{z))):zeDy, X = {{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 is polynomially convex for D small enough We have X s u X o f X i U J ^ u n d er the b ih o lo m o rp h ic m ap {z ,w )\-^ {-z ,w ) is the image S X U X 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 to an angular sector situated near the positive real axis, while p maps X U X 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 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) [8] Kieu Phuong Chi, Function algebras on a disk, VNU Journal o f Sciences, Mathematics - Physics No3 (2002) [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  ... 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... and f { z ) = o(l-sl"*) Suppose that there exists I < f such that (1) ak k^l and is integer Then X ị U X is locally polynom ially convex at Proof C onsider the polynomial p (z , w ) = ■^^m- 21