1 0 The Runge approximation theorem 0 2 for generalized polynomial hulls n a J 2 Youssef ALAOUI and My Abdelhakim EL IDRISSI SAAD 2 ] V 1. Introduction C . h t We consider for a compact set K inCIn the polynomially convex hull a m Kˆ = {z ∈CIn : |P(z)| ≤ ||P|| for all polynomials P} [ K 1 Where ||P|| = sup{|P(z)| : z ∈ K}. v K 5 It is known from the Runge approximation theorem that every function 7 which is holomorphic in a neighborhood of compact sets K with K = Kˆ can 1 1 be approximated uniformly on K by analytic polynomials. 0 Our aim here is to prove the same result in the more general situation when 1 the rˆole of Kˆ is played by the generalized polynomial hull h (K) introduced 0 q / by Basener [1] and which can be defined, for each integer q ∈ {0,1,...,n−1}, h t by a m hq(K) = \ {z ∈CIn : |P(z)| ≤ δK(P,z)} v: P∈CI [z1,···,zn] i X Where δK(P,z) denotes the lowest value of ||P||K∩f−1(0) when f ranges in the r set of analytic polynomial mapsCIn →CIq vanishing at z. a Notice, however, that this result does not give anything new when q = 0, ˆ since h (K) = K. o 1 2. Proof of the theorem lemma 1 - Let K be a compact set in CIn with K = h (K) and let ω be a q neighborhood of K. Then one can find finitely many holomorphic polynomials P ,···,P and a closed polydisc ∆ ⊂CIn such that n+1 m m K ⊂ \ {z ∈ ∆ : |Pj(z)| ≤ δK(Pj,z)} = L ⊂ ω j=n+1 Proof. We may, of course, assume that ω is bounded. Then there exists a closed polydisc ∆(0,r) with center at 0 and r = (r ,···,r ) such that 1 n ω ⊂ ∆(0,r). It follows that ∆(0,r)\ω ⊂ ∆(0,r)\hq(K) ⊂ [ {z ∈CIn : P(z) > δK(P,z)} P∈CI [z1,···,zn] We shall first prove that {z ∈CIn : |P(z)| > δ (P,z)} is open. To see this, let K (z ) be a sequence of points in K = {z ∈CIn : |P(z)| ≤ δ (P,z)} which j j≥1 P K converges to a point z ∈CIn, and let f :CIn −→CIq be a polynomial map with f(z) = 0. Define f :CIn −→CIq by f = f −f(z ). Then j j j |P(zj)| ≤ ||P||K∩f−1(0). Let ζj be a point of K ∩fj−1(0) such that j |P(ζj)| = ||P||K∩fj−1(0). Then there is subsequence ζjk converging towards a point ζ. Since |P(z )| ≤ |P(ζ )| and f(ζ ) → 0 when k → +∞, j j j k k k then a passage to the limit shows that |P(z)| ≤ |P(ζ)| ≤ ||P||K∩f−1(0). Hence |P(z)| ≤ δ (P,z), and z ∈ K . K P Because ∆(0,r)\ω is compact there are P ,···,P such that n+1 m m ∆\ω ⊂ [ {z ∈CIn : |Pj(z)| > δK(Pj,z)} j=n+1 Let P (z) = zj j = 1,···,n . Since ||P || ≤ 1 then j rj j K m K ⊂ L = \ {z ∈ ∆ : |Pj(z)| ≤ δK(Pj,z)} ⊂ ω j=n+1 2 lemma 2 - Let L be a compact set in CIn and f :CIn −→CIq a holomorphic polynomial map with q in the range 1 ≤ q ≤ n−1 such that L∩f−1(0)∓∅. If ζ ∈ Lˆ ∩f−1(0) is nonsingular point of f−1(0), then ζ ∈ (L∩f−1(0))ˆ. Where (L∩f−1(0))ˆdenotes the polynomially convex hull of the set L∩f−1(0). Proof. Let F : CIn ×CI −→ CIq be the map defined by F(z,z ) = f(z)z . n+1 n+1 By identifyingCIn toCIn × {0}, L can be considered as a compact subset of the analytic variety F−1(0) = f−1(0)×CI ∪CIn×{0}. Since ζ lies in only one global branch Z of f−1(0), then Z ×CI andCIn ×{0} are the unique global o o branches of F−1(0) containing the point (ζ,0). Set Z = Z ×CI ∪CIn ×{0}. o If (ζ,0) ∈/ (L∩Z)ˆ thenthereexists aholomorphicfunctiong onF−1(0)such that |g(ζ,0)| > 1 > ||g|| . Let h be a holomorphic function on F−1(0) such L∩Z that h(ζ,0) = 1 and h = 0 on F−1(0)\Z. Then for sufficiently large positive integer k, the holomorphic function g = gkh satisfies |g (ζ,0)| > ||g || , k k k L a contradiction. We conclude that (ζ,0) ∈ (L∩Z)ˆ. lemma 3 - Let K be a compact set in CIn with K = h (K), and q in the q range 1 ≤ q ≤ n−1. Let ω and L be as in lemma 1. Then L is polynomially convex, ie. L = Lˆ. Proof. The proof is by induction on m−n. Suppose at first that L = {z ∈ ∆ : |P(z)| ≤ δ (P,z)} K Given a point z ∈ Lˆ and an analytic polynomial map f : CIn −→ CIq with f(z) = 0, we shall prove that |P(z)| ≤ ||P||L∩f−1(0), if L∩f−1(0)∓∅ From lemma 2 we obtain the above inequality for any regular point z ∈ Lˆ ∩f−1(0) in f−1(0). Now, let z ∈ Lˆ ∩Sing(f−1(0)), and assume that |P(z)| > ||P||L∩f−1(0). Let V be a small neighborhood of L∩f−1(0) such that ||P|| < |P(z)|. V By perturbing slightly the coefficients of the components of f, we can get an analytic polynomial map g = (g ,···,g ) :CIn −→CIq with g(z) = 0, 1 q 3 z a regular point of g−1(0), and g−1(0)∩L ⊂ V. (See [3]). Which implies that |P(z)| > ||P||L∩g−1(0), a contradiction. Hence |P(z)| ≤ ||P||L∩f−1(0) for all z ∈ Lˆ ∩f−1(0). Let ζ be a point of L∩f−1(0) such that |P(ζ)| = ||P||L∩f−1(0). Then, for all z ∈ Lˆ ∩f−1(0),|P(z)| ≤ |P(ζ)| ≤ δK(P,ζ) ≤ ||P||K∩f−1(0). Hence |P(z)| ≤ δ (P,z), and z ∈ L. K We now assume that the statement of our lemma is already known for all positif integers ≤ m−n−1, and let L = {z ∈ L : |P (z)| ≤ δ (P ,z)}, m−1 m K m where L = {z ∈ ∆ : |P (z)| ≤ δ (P ,z),j = n+1,···,m−1}. m−1 j K j Since, by the induction hypothesis, L is polynomially convex, then m−1 Lˆ ⊂ Lˆ ∩{z ∈ ∆ : |P (z)| ≤ δ (P ,z)}ˆ= L m−1 m K m theorem 1 - Let f be an analytic function in a neighborhood of a compact set K ⊂CIn with K = h (K), and q in the range 1 ≤ q ≤ n−1. Then there q is a sequence f of analytic polynomials such that f −→ f uniformly on K j j Proof. By lemma 3 we can find a compact polynomially convex set L containing K so that f is analytic in a neighborhood of L. By the known version of the Runge approximation theorem [2], f can be approximated uni- formly by analytic polynomials on L. This completes the proof. DEPARTEMENT DE MATHEMATIQUES, INSTITUT AGRONOMIQUE ET VETERINAIRE HASSAN II, BP 6202 INSTITUTS (10101) RABAT, MAROC REFERENCES [1] Basener, R.: Several dimensional properties of the spectrum of a uniform algebra. Pacific J. math. 74, 297-306 (1978). [2] Hormander, Lars.: An introduction to complex analysis in several variables. Third, revised, edition, 1990. [3] Lupacciolu, G. and Stout.: Holomorphic hulls in termes of nonnegative divisors and generalized polynomial hulls. Manuscripta Math. 98, 321-331 (1999) 4