Abstract. In that paper, we investigate the locally polynomial convexity of graphs of smooth functions in several variables. We also give a sufficient condition for real analytic function g defined near 0 in C which behaves like z n near the origin so that the algebra generated by z m and g is dense in the space of continuous functions on D for all disks D close enough to the origin in C
LOCAL POLYNOMIAL CONVEXITY OF GRAPHS OF FUNCTIONS IN SEVERAL VARIABLES KIEU PHUONG CHI Abstract. In that paper, we investigate the locally polynomial convexity of graphs of smooth functions in several variables. We also give a sufficient condition for real analytic function g defined near 0 in C which behaves like z n near the origin so that the algebra generated by z m and g is dense in the space of continuous functions on D for all disks D close enough to the origin in C. 1. Introduction ˆ we denote the polynomial convex We recall that for a given compact K in Cn , by K hull of K i.e., ˆ = {z ∈ Cn : |p(z)| ≤ p K K for every polynomial p in Cn }. ˆ = K. A compact K is called locally We say that K is polynomially convex if K polynomially convex at a ∈ K if there exists the closed ball B(a) centered at a such that B(a) ∩ K is polynomially convex. The interest for studying polynomial convexity stems from the celebrated Oka-Weil approximation theorem (see [1], page 36) which states that holomorphic functions near a compact polynomially convex subset of Cn can be uniformly approximated by polynomials in Cn . A compact K ⊂ C is polynomially convex if is C \ K connected. In higher dimensions, there is no such topological characterization of polynomially convex sets, and it is usual difficult to determine whether a given compact subset is polynomially convex. By a well-known result of Wermer ([19]; see also [1], Theorem. 17.1), every totally real manifold is locally polynomially convex. Recall that a C 1 smooth real manifold M is called totally real at p ∈ M if the real tangent space Tp M contains no complex line. In this paper, we are concerned with local polynomial convexity at the origin of the graph Γf of a C 2 smooth function f near 0 ∈ Cn such that f (0) = 0. By the theorem of Wermer just cited, we ∂f know that if (0) = 0 for all i = 1, 2, ..., n then Γf is locally polynomially convex at ∂z i ∂f the origin of Cn+1 . Thus it remains to consider the case where (0) = 0 for some i. ∂z i Our study is motivated by a similar problem in one complex variable. More precisely, let f be a C 2 smooth function near 0 ∈ C such that f (0) = 0. Under certain condition of f , one can show that Γf is locally polynomial convex at the origin of C2 . The work 2010 Mathematics Subject Classification. 46J10, 46J15, 47H10. Key words and phrases. polynomially convex, plurisubharmonic, totally real . 1 associated with these direction of research is too numerous to list here; instead, the reader is referred to [2, 3, 20] and the references given therein. In the section 3, we will refine the technical from [6] to attack the problem in several variables. For the readers convenience, we repeat a reasoning due to [6]. First, we construct nonnegative smooth functions vanishing exactly on Γf . These functions are, in general, plurisubharmonic only on open sets whose boundaries contain the origin. Secondly,under some technical assumptions, we may add small strictly plurisubharmonic functions to obtain plurisubharmonic functions on certain open sets containing the (local) polynomially convex hull of Γf . Finally, by invoking the nontrivial fact of about equivalence of plurisubharmonic hull and polynomial hulls, we can conclude that Γf is locally polynomially convex at the origin. In this vein, we obtain some known results in one variable. We also give some examples to show that our results are effective. In section 4, we shall present some results about locally uniform approximation of continuous function. Let D be a small closed disk in the complex plane, centered at the origin and g be a C 2 function on D which behaves like z n near the origin. By [z m , g; D] we denote the function algebra consisting of uniform limits on D of all polynomials in z m and g. Our goal finding conditions on g to ensure that [z m , g; D] = C(D), where C(D)is the set of continuous complex valued functions on D. Of course a necessary condition is that the two functions z m and g must separate points of D. However, this condition is far from sufficient. Indeed, it can be shown that [z m , g; D] = C(D) for some choices of g (see [12, 16],...), while for other choices of g we have [z m , g; D] = C(D) ([9, 10],...) . Our goal is giving generalized class of functions g defined near the origin in C such that [z m , g; D] = C(D). As in the previous work, we rely heavily on the theory of polynomial convexity. It may be useful to recall the general scheme in proving [z m , g; D] = C(D) for appropriately chosen g. Roughly speaking we consider ˜ which is inverse of X := {(z m , g) : z ∈ D} under the proper the compact set X ˜ is a union of graphs (in C2 ) over D. If polynomial mapping (z, w) → (z m , w). Then X g behaves ”nicely” near the origin then we could show that each graph is polynomially convex. Notice that we can not apply a well known result of Wermer as in [1] or [19], since each graph may fail to be totally real at the origin. Thus in this case the ˜ is a union of polynomially convex compact sets. If we could prove that compact X ˜ X is polynomially convex, then by some known result about approximation on totally real compact sets (possibly with singularities) we could show that every continuous ˜ can be approximated uniformly by polynomials. Since X ˜ transforms function on X nicely to X, by a well known lemma (see [11, 16]) we may deduce the same conclusion ˜ For this, we shall on X. Hence, it remains to decide the polynomial convexity of X. use an appropriate tool which is the version of Kallin lemma. Acknowledgment. This work was done during a stay of the author at Vietnam Institute for Advance Study in Mathematics. He wishes to express his gratitude to the institute for the support. The author greatly thanks to Professor Nguyen Quang 2 Dieu for many stimulating conversations. The author is supported by a grant from the NAFOSTED program. 2. Some technical lemmas Consider the function g : [0, +∞) → R, is defined by −1 g(t) = e t , t>0 0 t = 0. For each x ≥ 0 set x χ(x) = g(t)dt. 0 n Let U be a open neighborhood of 0 in C , let the functions f = (f1 , f2 , ..., fm ) ∈ C 2 (U ) and let X = { z, f1 (z), ..., fm (z) : z ∈ U } be the common graph of the functions fi in Cn+m . Consider the function F (z) = k=1 |zn+k − fk ( z)|2 of class C 2 in the region U × Cm , where z = (z1 , ..., zn ) for all z = (z1 , .., zn , zn+1 , ..., zm ) ∈ Cn+m . It is easy to see that the zero set of F coincides with X. Let n+m LF (w) = ∂ 2F wν wµ ∂zν ∂z µ µ,ν=1 be the Levi form of F and n n Lf ( w) = ∂ 2 fm ∂ 2 f1 wµ wν , ..., wµ w ν , ∂zν ∂z ν ∂zν ∂z ν µ,ν=1 µ,ν=1 where w = (w1 , ..., wn ) for all w = (w1 , ..., wn+m ) ∈ Cn+m . For f = (f1 , f2 , ..., fm ) ∈ C 2 (U ), we mean that ∂f ( w) := ∂z n n ν=1 n ∂f1 ∂f2 ∂fm wν , wν , ..., wν . ∂z ν ∂z ν ∂z ν ν=1 ν=1 We have following the lemma. Lemma 2.1. The function ϕ(z) = χ ◦ F (z) is plurisubharmonic on Ω = z ∈ U × Cm : 2 ∂f (z) w + ∂z z − f ∂(z − f ) , (z) w |z − f | ∂z > 2Re z − f (z), Lf (z)( w) 2 , where z = (zn+1 , ..., zn+m ), for all z = (z1 , ..., zn+m ) ∈ Cn+m and w = (w1 , ..., wn+m ) ∈ Cn+m , w1 w2 ....wn = 0. Moreover, the following statements hold. i) If m = 1 then ϕ is plurisubharmonic on 3 2 ∂f (z) w > Re(zn+1 − f (z))Lf (z)( w) . ∂z ii) If m = n = 1 then ϕ is plurisubharmonic on Ω1 = z ∈ U × C : ∂f (z) Ω2 = (z, w) ∈ U × C : ∂z 2 ∂ 2f > Re(w − f (z)) (z) . ∂z∂z Proof. It is easy to check χ(0) = χ (0) = χ (0) = 0, χ (x) = x2 χ (x), ∀x ≥ 0 and so that χ is a convex and increasing. Since ϕ is convex, we reduce to show that F is plurisubharmonic. Consider the Levi form of F n+m LF (w) = ∂ 2F wν w µ . ∂z ∂z ν µ µ,ν=1 By the easy computations, we obtain n m LF (w) = k=1 ∂fk wν ∂z ν + ν=1 m − 2Re n+m m 2 ν=1 k=1 ∂(zn+k − fk ) wν ∂z ν 2 n (zn+k − fk ) k=1 ∂ 2f k wν w µ . ∂z ∂z ν µ µ,ν=1 So that in vector notation this takes the form 2 2 ∂f ∂(z − f ) (z) w + (z)w − 2Re z − f (z), Lf ( w) . ∂z ∂z For the plurisubhamonicity of F , it is sufficient that LF (w) be nonnegative on complex tangent vectors w, these vectors are defined by the condition (1) LF (w) = n+m k=1 ∂F wk = 0, ∂zk which for the function F takes the form ∂f ∂(z − f ) w = w, z − f . ∂z ∂z By applying Cauchy-Schwatz inequality, we obtain (2) z − f, ∂(z − f ) w |z − f | ≥ ∂z ∂(z − f ) w, z − f ∂z = z − f, and hence (3) ∂(z − f ) w ∂z 2 ≥ 2 z − f ∂(z − f ) , (z) w . |z − f | ∂z 4 ∂f w , ∂z Since (1) and (3), we have (4) LF (w) ≥ 2 ∂f (z) w + ∂z 2 z − f ∂(z − f ) , (z) w − 2Re z − f (z), Lf (z)( w) . |z − f | ∂z Hence, the form LF (w) is nonnegative in the open set z ∈ U × Cm : ∂f 2 w + ∂z z − f ∂(z − f ) , (z) w |z − f | ∂z 2 > 2Re z − f (z), Lf (z)( w) , i.e. the function F is plurisubharmonic on Ω. If m = 1 then the right hand of (4) reduces to 2 2 ∂f (z) w − Re(zn+1 − f (z))Lf (z)( w) ∂z so that F is plurisubharmonic on Ω1 = z ∈ U × C : ∂f (z) w ∂z 2 > Re(zn+1 − f (z))Lf (z)( w) . ∂ 2f (z)w for z ∈ U and If m = n = 1 then the fact is desired from Lf (z)( w) = ∂z∂z w ∈ C. We need some the following fact, it is a consequence of the solution of the Levi problem in Cn (see [4], Theorem 4.3.3). Lemma 2.2. ([6, 18]) Let K ⊂ Cn be compact. ˆ then there exists a continuous plurisubharmonic function u on Cn such 1) If K = K that u = 0 on K and u > 0 on Cn \ K. ˆ if and only if u(z) supK u for every plurisubharmonic function u on Cn . 2) z ∈ K The following lemma is due also from [6]. ˆ Lemma 2.3. Let K ⊂ Cn be compact and let U ⊂ Cn be an open neighborhood of K. Assume that there is a plurisubharmonic function u on U such that u = 0 on K. Then ˆ u = 0 on K. Here an appropriate tool is the following version of Kallin lemma (see [16, 18]): Lemma 2.4. (Kallin’s Lemma) Suppose X1 and X2 are polynomially convex subsets of Cn , suppose there is polynomial p mapping X1 and X2 into two polynomially convex subsets Y1 and Y2 of the complex plane such that 0 is a boundary point of both Y1 and Y2 and with Y1 ∩ Y2 = {0}. If p−1 (0) ∩ (X1 ∪ X2 ) is polynomially convex, then X1 ∪ X2 is polynomially convex. Let X ⊂ Cn be a compact set and P (K) be the uniform closure in C(K) of all (restrictions to K) polynomials. The following lemma is due from [17]. 5 Lemma 2.5. Let X be a compact subset of C2 , and let π : C2 → C2 be difined by π(z, w) = (z n , wm ). Let π −1 (X) = X11 ∪ X12 ∪ ... ∪ Xnm with X11 compact, and Xkl = {exp( 2π(k−1) z, exp( 2π(l−1) w : (z, w) ∈ X11 } for 1 ≤ k ≤ m, 1 ≤ l ≤ n. If m n −1 −1 P (π (X)) = C(π (X)) then P (X) = C(X). 3. Local polynomial convexity of graphs in Cn Now we come to the main results of this work. Theorem 3.1. Let U be a open neighborhood of 0 in Cn , let the functions f ∈ C 2 (U ). Assume that there exists C 2 smooth functions g defined on U satisfying the following conditions: (i) |f |2 ≤ Re(f g) on U ; (ii) |g(z)| ≤ |z|p , z ∈ U for some p ∈ N and ψ(z, w) = |w|2 − Re(wg(z)) is plurisubharmonic near the origin of Cn+1 ; 2 ∂f (iii) (z)w + Re f (z)Lf (z)w > |g(z)Lf (z)(w)| for every w = (w1 , ..., wn ) ∈ ∂z Cn , w1 w2 ...wn = 0 and z ∈ U \ {0}. Then Γf is locally polynomially convex at the origin in Cn+1 . Furthermore, there exists an r > 0 small enough such that a continuous function on Xr := Γf ∩ B(0, r) can be approximated uniformly by polynomials. Remark 3.2. 1) It follows from i) and ii) that |f | ≤ |g| and g(0) = 0. Combining this fact with iii) we arrive at z∈U : ∂f (0) = 0, k = 1, ..., n ∂z k consists at most the origin. 2) If g is holomorphic on U then ψ(w, z) = |w|2 − Re(wg(z)) is plurisubharmonic near the origin of Cn+1 . Before taking up the proof of Theorem 3.1, we give some examples and corollaries, which mentions Theorem 3.1 is effective. n 2 Example 3.3. X = {(z1 , z2 , z m 1 +z 2 ) : (z1 , z2 ) ∈ C }, m, n ∈ N and Y = {(z1 , z2 , z 1 z 2 ) : (z1 , z2 ) ∈ C2 } are locally polynomial convex at 0 ∈ C3 . n 2 m n Proof. Consider f (z1 , z2 ) = z m 1 + z 2 ,for all (z1 , z2 ) ∈ C . Let g(z1 , z2 ) = z1 + z2 for 2 all (z1 , z2 ) ∈ C . It is easy to check that i) and ii) are satisfied for g. It follows from f (z1 , z2 ) = z1m + z2n that Lf (z1 , z2 )(w1 , w2 ) = 0 for all (z1 , z2 ) ∈ C2 and (w1 , w2 ) ∈ C2 . Hence iii) is implied from the fact 2 2 2 ∂f ∂f ∂f (z)w = (z1 , z2 )w1 + (z1 , z2 )w2 = m2 |z1 |2(m−1) |w1 |2 +n2 |z2 |2(n−1) |w2 |2 > 0 ∂z ∂z 1 ∂z 2 2 for all w = (w1 , w2 ) ∈ C with w1 w2 = 0. Applying Theorem 3.1, we can deduce that X is locally polynomial convex at 0 ∈ C3 . For Y , we can conclude the same fact by considering f (z1 , z2 ) = z 1 z 2 ,for all (z1 , z2 ) ∈ C2 and g(z1 , z2 ) = z1 z2 for all (z1 , z2 ) ∈ C2 . 6 Clearly, X, Y are not totally real at (0, 0, 0) ∈ C3 . We can not apply Wermer’s result which mention in the introduction for locally polynomial convexity of X and Y . Now, we come to some consequence for the function in one variable. Corollary 3.4. Let U be a open neighborhood of 0 in C, let the functions f ∈ C 2 (U ). Assume that there exist C 2 smooth function g defined on U satisfying the following conditions: 1) |f |2 ≤ Re(f g) on U ; 2) |g(z)| ≤ |z|p for some p ∈ N and ψ(z, w) = |w|2 − Re wg(z) is plurisubharmonic near the origin of C2 ; ∂f 2 ∂ 2f ∂ 2f 3) > |g | for all z ∈ U and z = 0. + Ref ∂z ∂z∂z ∂z∂z Then Γf is locally polynomially convex at the origin in C2 . Furthermore, there exists an r > 0 small enough such that a continuous function on Xr := Γf ∩ B(0, r) can be approximated uniformly by polynomials. The following corollary is main result of [6]. Corollary 3.5. [6] Let U be a open neighborhood of 0 in C, let the functions f ∈ C 2 (U ). Assume that there exist holomorphic function g on U and λ ∈ (0, 1) satisfying the following conditions: (a) |f |2 ≤ Re(f g) on U ; (b) g(0) = 0; ∂f 2 ∂ 2f ∂ 2f (c)λ + Ref > |g | for all z ∈ U and z = 0. ∂z ∂z∂z ∂z∂z Then Γf is locally polynomially convex at the origin in C 2 . Furthermore, there exists an r > 0 small enough such that a continuous function on Xr := Γf ∩ B(0, r) can be approximated uniformly by polynomials. Example 3.6. Let α be a complex number with |α| < 1. If n is a sufficiently enough integer then S = {(z, z n + αzz n−1 ) : z ∈ C} is locally polynomially convex at the origin in C2 . Proof. We shall show that S satisfies Corollary 3.4. Consider the functions f (z) = (1 + |α|)2 z n + αzz n−1 and g(z) = az n , with real number a > . Clearly, 2) holds. In 1 − |α| view of 1), we have (5) |f (z)|2 = |z n + αzz n−1 |2 |z|2n (1 + |α|)2 , ∀z ∈ C. and (6) Re f (z)g(z) = Re az n z n + αzz n−1 Since (5) and (6) we obtain |f |2 Re(f g). 7 ≥ a 1 − |α| |z|2n . ∂ 2f According to Corollary 3.4, it suffices checking the condition 3). Set ∆f = . We ∂z∂z have f (z) = z n + αzz n−1 v ∆f (z) = α(n − 1)z n−2 , ∀z ∈ C. Therefore |g(z)∆f (z)| = a|α|(n − 1)|z|2n−2 , ∀z ∈ C (7) and (8) Re f (z)∆f (z) = Re α(n − 1)z n−2 z n + αzz n−1 ≥ |α|(n − 1)(|α| − 1)|z|2n−2 , ∀z ∈ C. On the other hand ∂f (z) ∂z (9) 2 = |nz n−1 + αz n−1 |2 ≥ (n − |α|)2 |z|2n−2 ≥ (n − 1)2 |z|2n−2 , ∀z ∈ C. From (7), (8) and (9), we infer that the inequality |g∆f | < | ∂f 2 | + Re(f ∆f ) ∂z for every z = 0 holds if n > 1 + |α|(1 + a − |α|). Therefore, the conditions of Corollary 3.4 are fulfilled with n > 1 + |α|(1 + a − |α|). In particular, S = {(z, z n + αzz n−1 ) : z ∈ C} is locally polynomially convex at the origin in C2 . √ 3 √ then S is not Remark 3.7. 1) We would to emphasize that if 1 > |α| > 1+ 3 satisfying Theorem 1.1 and Theorem 1.2 in [2]. 2)If f (z) = z n + o(|z|n ) then f satisfies Corollary 3.4 in a small disk centered at 0. Let χ and F be mentioned as in the previous section. The following lemma is key of the proof of our main result. Lemma 3.8. Assume that the conditions of Theorem 3.1 are fulfilled. Then, for every ε > 0, there is δ(ε) > 0 such that the function ϕε (z) = χ ◦ F (z) + ε|z|2 is plurisubharmonic on a ball Bε = z = (z1 , ..., zn , zn+1 , ..., zn+m ) ∈ Cn+m : |z| < δ(ε) . 8 Proof. Let us ε > 0, we need show δ(ε) > 0 such that Lϕε is nonnegative form on Bε = z = (z1 , ..., zn , zn+1 , ..., zn+m ) ∈ Cn+m : |z| < δ(ε) . Since n+m ∂ 2ϕ wν w µ Lϕε (w) = ε |w| + Lϕ (w) = ε |w| + ∂zν ∂z µ µ,ν=1 2 2 2 2 we have that n+m 2 2 Lϕε (w) ≥ ε |w| − 2 (10) µ,ν=1 ∂ 2ϕ ||w|2 . ∂zν ∂z µ On the other hand, we have ∂ 2ϕ ∂ 2F ∂F ∂F = χ F + χ F . ∂zν ∂z µ ∂zν ∂z µ ∂zν ∂z µ (11) Since χ (x) = x2 χ (x) for all x ∈ R+ , we obtain ∂ 2ϕ =χ F ∂zν ∂z µ (12) F2 ∂F ∂F ∂ 2F . + ∂zν ∂z µ ∂zν ∂z µ It follows from i) and ii) that |f (z)| ≤ |g(z)| ≤ |z|p for all z ∈ U . By this fact, we obtain |zn+k − fk ( z)|2 F (z) = k=1 n+m |fk ( z)| ) = 2( k=n+1 k=1 k=n+1 |wk |2 + |f ( z)|2 ) 2 |wk | + ≤ 2( n+m m 2 n+m (13) |wk |2 + |g( z)|2 ) ≤ 2( k=n+1 n+m |wk |2 + | z|2p ) ≤ 2( k=n+1 n+m ≤ 2( k=n+1 n 2 | z|2 ) = 2|z|2 , |wk | + k=1 χ (x) = 0, we can choose x→0 x a > 0 such that χ (x) < x for all |x| < a. Combining this fact, (12) and (13), we can find a small neighborhood V of the origin of Cn such that where z = (z1 , ..., zn , wn+1 , ..., wn+m ) ∈ Cn+m . Since lim (14) ∂ 2ϕ | | ≤ bνµ |z|2 , z ∈ V ∂zν ∂z µ 9 where bij is a positive constant independent on z. Set b = max{bνµ }. Since (10) and (14), we obtain n+m 2 Lϕε (w) ≥ ε − 2 µ,ν=1 Now, if we choose δ 2 (ε) = ∂ 2ϕ | |w|2 ≥ ε2 − 4(m + n)2 b|z|2 |w|2 . ∂zν ∂z µ ε2 then Lϕε (w) ≥ 0 on the ball 4(m + n)2 b Bε = z = (z1 , ..., zn , zn+1 , ..., zn+m ) ∈ Cn+m : |z| < δ(ε) . Hence ϕε is plurisubharmonic on the ball Bε . Proof of Theorem 3.1 For r > 0, put Xr = Γf ∩ B(0, r). We claim that, for r > 0 small enough, (15) ˆ r ⊂ Kr = (z1 , ..., zn , zn+1 ) ∈ B(0, r) : |zn+1 | ≤ |g(z)| . X Indeed, consider the function ψ(z, w) = |w|2 − Re wg(z) for z ∈ Cn and w ∈ C. Clearly {(z, w) : z ∈ B(0, r), w ∈ C, ψ(z, w) ≤ 0} ⊂ Kr . By ii) we have that ψ is plurisubharmonic near the origin of Cn+1 . Moreover, by part i) of the theorem we have ψ ≤ 0 on Xr for r > 0 small enough. Thus by Lemma 2.3 ˆ r . This proves our claim. we have that ψ ≤ 0 on X Let χ be the function defined before in Lemma 2.1. Using Lemma 2.1, we can find r > 0 small enough such that the function ϕ = χ ◦ F is plurisubharmonic on the open set 2 ∂f (16) Ωr = (z1 , ..., zn , zn+1 ) ∈ B(0, r) : (z)w > Re(zn+1 − f (z))Lf (z)(w)) . ∂z Next, for (z, zn+1 ) ∈ Kr with z = (z1 , ..., zn ) = 0, by (iii) we have Re(zn+1 − f (z))Lf (z)w ≤ |zn+1 Lf (z)w| − Re(f (z)Lf (z)w) ≤ |g(z)Lf (z)w| − Re(f (z)Lf (z)w) 2 ∂f (z)w ∂z n for all w = (w1 , ..., wn ) ∈ C with w1 w2 ...wn = 0. This implies that Kr \ {0} ⊂ Ωr . It follows from Lemma 3.8 and this fact, for every ε > 0, there exists a δ(ε) ∈ (0, r) (independent of r) such that the function < ϕε (z, zn+1 ) := χ(|zn+1 − f (z)|2 ) + ε(|z|2 + |zn+1 |2 ) is plurisubharmonic on Ωr ∪ B(0, δ(ε)), an open neighborhood of Xr . Observe that ˆ r . By letting ϕε ≤ εr2 on Xr . Therefore, applying Lemma 2.3 yields ϕε ≤ εr2 on X 10 ˆ r = Xr . Finally, we note that Γf is locally contained in a totally ε → 0 we infer that X real manifold outside the origin Cn+1 . Using the main theorem in [11], we conclude that continuous functions on Xr are uniformly approximated by polynomials. The proof is there by completed. 4. Approximation on small disks We will use the notation C 2 ({0}) for the class of C 2 functions defined near 0 ∈ C and vanishing at 0. As mentioned in introduction, in this section we give some result about approximation on disks. It is based on the results of previous section. We can now state our theorem. i n−i Theorem 4.1. Let m ≥ 3 and n ≥ 2 be integers. Let f (z) = z n + n−1 + i=0 ai z z n 2 o(|z| ) be C smooth functions near 0. Suppose that f satisfies the conditions of Corollary 3.4 and n−1 (17) |ai | < 0< i=0 π tan m π . 1 + tan m m Moreover [z m , f (z); D] = C(D) for Then z and f (z) separate points near 0. sufficiently small disks D. Proof. We show that z m and f (z) separate points near 0. Indeed, the function z m separates the points a, b such that b = a exp( 2πki ), ∀k = 0, 1, ..., m − 1. We need show m 2πli that f (z) separates the points a exp( 2πki ), a exp( ) with k = l and k, l = 1, ..., m − 1. m m First, we assume that a > 0. Then f a exp( 2πsni 2πki ) = an exp( ) 1+ m m n−1 ai θi + g a exp( i=0 2πki ) , m n where |θi | = 1, ∀i, s + k = m and g(z) = o(|z| ). We have n−1 tan(arg(1 + ai θi )) = i=0 Im 1 + Re n−1 i=0 ai θi n−1 i=0 ai θi ) ≤ By the condition n−1 |ai | < i=0 π tan m π , 1 + tan m we obtain n−1 tan arg(1 + ai θi ) < tan i=0 π . m Since n−1 ai θ i ) ≥ 1 − Re(1 + i=0 |ai | > 0, i=1 11 1− n−1 i=0 |ai | . n−1 i=0 |ai | we infer that n−1 ai θi ∈ (− arg 1 + i=0 π π , ). m m So, if a is sufficiently small then f a exp( 2πki ) is contained in the open sector with m vertex at 0 and the argument π 2πsn π 2πsn ϕk ∈ (− + , + ). m m m m Similarly, f a exp( 2πli ) is contained in the open sector with vertex at 0 and the m argument π 2πtn π 2πtn ϕl ∈ (− + , + ), m m m m where t + l = m, 1 ≤ t ≤ m − 1. It is easy to check that 2πtn π 2πtn π 2πsn π 2πsn π , + ) ∩ (− + , + ) = ∅. (− + m m m m m m m m It follows that 2πki 2πli f a exp( ) = f a exp( ) , m m for a > 0 small enough. Writing a = |a| exp(iθ) if a is arbitrary. By the same computing, we get that f a exp( 2πki ) is contained in the open sector with vertex at 0 and the argument m ϕk ∈ (−nθ − π 2πsn π 2πsn + , −nθ − + ). m m m m It implies that 2πki 2πli ) = f a exp( ) , m m for every a small enough. This yields z m and f (z) separate points near 0. f a exp( ˜ which is the inverse of the Next, for small closed disk D we consider the set X compact X = z m , f (z) : z ∈ D under the map (z, w) → (z m , w). We have m−1 ˜= X Xk k=0 where 2πik )z, f (z)) : z ∈ D . m By Corollary 3.4, X0 is polynomially convex for sufficiently small disks D. Since the map Ψ(z, w) = exp( 2πik )z, w is biholomorphic for each k, it follows that Xk is m polynomially convex for D small enough, for every k. Now, as mentioned in the introduction we will use Kallin’s lemma to show polyno˜ It is enough to take a polynomial of the form mially convexity of X. Xk = exp( p(z, w) = z n w. 12 It implies that, for each k = 0, 1, ..., m − 1 p(Xk ) = 2πik ) |z|2n + exp( m n−1 ai z n+i z n−i + o(|z|2n ) : z ∈ D . i=0 We have n−1 2n tan arg(|z| ai z + n+i n−i z ) i=0 n−1 Im i=0 ai z n+i z n−i =| n+i n−i |z|2n + Re n−1 z i=0 ai z ≤ 1− ≤ tan n−1 i=0 |ai | n−1 i=0 |ai | π . m It implies that n 2n arg |z| ai z n+i z n−i ∈ (− + i=1 π π , ). m m So, if D is suficiently small then p(Xk ) will be contained in open sector vertex at 0 and the argument π 2πkn π 2πkn ψk ∈ (− + , + ). m m m m It follows that these sectors only meet at 0. Applying Kallin’s lemma (Lemma 2.4) ˜ = m−1 Xk is polynomially convex for D small enough. repeatedly shows that X k=0 On the other hand, we have ∂( n−1 i=0 ai z i−1 z n−i ) (η) = ∂z n−1 n−1 iai η i−1 η n−i ≤ ( n−1 i|ai |)|η|n−1 < n( i=0 i=1 |ai |)|η|n−1 i=0 π tan m ∂z n n−1 n−1 [...]... 149–152 (1966) [6] Nguyen Quang Dieu and Kieu Phuong Chi, Local polynomial convexity of certain graphs in C2 , Michigan Math J 58 (2), 479-488 (2009) [7] Nguyen Quang Dieu, Local polynomial convexity of tangentials union of totally real graphs in C2 , Indag Math 10, 349-355 (1999) [8] Nguyen Quang Dieu, Local hulls of union of totally real graphs lying in real hypersurfaces, Michigan Math Journal 47 (2),... Xr Finally, we note that Γf is locally contained in a totally ε → 0 we infer that X real manifold outside the origin Cn+1 Using the main theorem in [11], we conclude that continuous functions on Xr are uniformly approximated by polynomials The proof is there by completed 4 Approximation on small disks We will use the notation C 2 ({0}) for the class of C 2 functions defined near 0 ∈ C and vanishing... (2005), 429445 [3] G Bharali, Polynomial approximation, local polynomial convexity, and degenerate CR singularities, J Funct Anal 236 (2006), 351368 [4] L H¨ ormander, An Introduction to Complex Analysis in Several Variables, Norh-Holland Math Library, 7, North- Holland, Amsterdam, 1990 [5] E Kallin, Fat polynomially convex sets, Function Algebras, (Proc Inter Symp on Function Algebras, Tulane Univ, 1965),... Algebras of continuous functions on disks, Proc of the R Irish Acad., 96A, 85-90 (1996) [15] P.J de Paepe, Approximation on a disk I, Math Zeit 212, 145-152 (1993) [16] P.J de Paepe, Eva Kallin’s lemma on polynomial convexity, Bull of London Math Soc 33, 1-10 (2001) [17] K.J Preskenis Approximation on disks, Tran Amer Math Soc.171, 445-467, (1972) [18] E L Stout, Polynomial convexity Progress in Mathematics,... C 2 ({0}) in a small neighborhood of 0 and f (z) = o(|z|)n So [z m , g n ; D] = C(D) by Theorem 4.1 References [1] H Alexander and J Wermer, Several Complex Variables and Banach Algebras, Grad Texts in Math., 35, Springer- Verlag, New York, 1998 [2] G Bharali, Surfaces with degenerate CR singularities that are locally polynomially convex, Michigan Math J 53 (2005), 429445 [3] G Bharali, Polynomial. .. ∈ D m By Corollary 3.4, X0 is polynomially convex for sufficiently small disks D Since the map Ψ(z, w) = exp( 2πik )z, w is biholomorphic for each k, it follows that Xk is m polynomially convex for D small enough, for every k Now, as mentioned in the introduction we will use Kallin’s lemma to show polyno˜ It is enough to take a polynomial of the form mially convexity of X Xk = exp( p(z, w) = z n w... Birkh´auser Boston,2007 [19] J Wermer, Approximation on a disk, Math Ann.155 331-333, (1964) [20] J Wiegerinck, Locally polynomially convex hulls at degenerated CR singularities of surfaces in C2 , Indiana Univ Math J 44 (1995), 897915 14 Kieu Phuong Chi, Department of Mathematics, Vinh University,182 Le Duan, Vinh City, Vietnam E-mail address: chidhv@gmail.com 15 ... real (locally contained in a totally real manifold), by an approximation X theorem in [11] ( mentioned in introduction), we get that every continuous function ˜ can be uniformly approximated by polynomials, which is equivalent to the fact on X ˜ = C(X) ˜ By the Lemma 2.5, we conclude that P (X) = C(X), it means that P (X) m [z , g; D] = C(D) for D small enough The theorem is proved The following proposition... 0 ∈ C and vanishing at 0 As mentioned in introduction, in this section we give some result about approximation on disks It is based on the results of previous section We can now state our theorem i n−i Theorem 4.1 Let m ≥ 3 and n ≥ 2 be integers Let f (z) = z n + n−1 + i=0 ai z z n 2 o(|z| ) be C smooth functions near 0 Suppose that f satisfies the conditions of Corollary 3.4 and n−1 (17) |ai | < 0 0, i=1 11 1− n−1 i=0 |ai | n−1 i=0 |ai | we infer that n−1 ai θi ∈ (− arg 1 + i=0 π π , ) m m So, if a is sufficiently small then f a exp( 2πki ) is contained in the open sector with m vertex at 0 and the argument π 2πsn π 2πsn ϕk ∈ (− + , + ) m m m m Similarly, f a exp( 2πli ) is contained in the open ... Chi, Local polynomial convexity of certain graphs in C2 , Michigan Math J 58 (2), 479-488 (2009) [7] Nguyen Quang Dieu, Local polynomial convexity of tangentials union of totally real graphs in. .. (X)) then P (X) = C(X) Local polynomial convexity of graphs in Cn Now we come to the main results of this work Theorem 3.1 Let U be a open neighborhood of in Cn , let the functions f ∈ C (U ) Assume... following version of Kallin lemma (see [16, 18]): Lemma 2.4 (Kallin’s Lemma) Suppose X1 and X2 are polynomially convex subsets of Cn , suppose there is polynomial p mapping X1 and X2 into two polynomially