The Runge approximation theorem for generalized polynomial hulls

4 Pages·0.07 MB·English

by  Youssef Alaoui

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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

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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
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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
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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
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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)
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