value distribution and the lemma of the logarithmic derivative on

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Jun 18, 1983 - (n+ l)-dimensional, oriented, real analytic manifold such that 8r is an orientation ..... PROOF. Define z x y. Then. T where the j coordinate of the vectors z, x or y is positive if ..... A meromorphic function f on D(b) is the quotient of two holomorphic functions g ... These properties will be ...... Consequently log(b x.) ...
Internat. J. Math. & Math. Sci. Vol. 6 No. 4 (1983) 617-669

617

VALUE DISTRIBUTION AND THE LEMMA OF THE LOGARITHMIC DERIVATIVE ON POLYDISCS WILHELM STOLL Department of Mathematics, University of Notre Dame Notre Dame, Indiana 46556 U.S.A.

(Received June 18, 1983)

ABSTRACT.

Value distribution is developed on polydiscs with the special emphasis

that the value distribution function depend on a vector variable.

A Lemma of the

logarithmic derivative for meromorphic functions on polydiscs is derived.

Here the

Bergman boundary of the polyd+/-scs is approached along cones of any dimension and exceptional sets for such an approach are defined.

KEY WORDS AND PHRASES.

Value distribution theory, valence function, Jensen formula,

characteristic function, counting, function, spherical image, compensation function, Lemma

of the logarithmic derivative, and approach

cone.

1980 AS MATHEMATICS SUBJECT CLASSIFICATION CODES. 32H30, 32A22. i.

INTRODUCTION. Value distribution for polydisc exhaustions has been studied by Ronkin [i],

Stoll [2

and others.

They emphasized the growth of entire holomorphic and

meromorphic functions and the representation of canonical functions to a given divisor in

Cn.

C. Ward Henson

For applications to mathematical logic, Lee A. Rubel and

[3], [4] inquired if the Lemma of

the logarithmic derivative could

be established for a meromorphic function on a fixed given polydisc.

In the classical one variable theory, the Lemma of the logarithmic derivative has been one of the basic tools for a long time.

analogous Lemma was proved only recently.

In several variables, the

For ball exhaustions of

En,

Vitter

5

proved the Lemma for differential operators with constant coefficients and derived the defect relation for meromorphic maps f

n

/

m

See also Stoll

6].

618

W. STOLL

Vitter’s Lemma extends easily to differential operators with polynomial or, with

proper modifications, to differential operators with entire coefficients. Earlier, a weak version of the Lemma was proved by Gauthler and Hengartner [7] for the special operator

D’

/z

z

I +

+

Zn /Zn,

extension to a general differential operator.

which does not permit the

D’

The differential operator

suffices

for meromorphlc functions, but imposes unnecessary restrictions for meromorphlc maps

f

cn

/p

m

Recently, Shlffman has shown how to derive Vitter’s Lemma from the

[8],

result of Griffiths and King

which also can be interpreted as a Lemma of the

Biancoflore and Stoll [9] gave an elementary proof of

logarithmic derivative.

The same method will be used to obtain the Lemma of the

Vitter’s result.

logarithmic derivative for polydiscs.

A theory should reflect the intrinsic algebraic, geometric and analytic Our development of

structures of the mathematical landscape under consideration.

value distribution theory on polydiscs will adhere to this principle.

This is

an important feature of this paper, which is mostly self contained and requires only a minimal knowledge of several complex variables.

Let us outline the main result. by its coordinates define

II

(Iz

Denote by

II

z

nl).

llrll

The euclidean space the length of r in

/

Rn

partially ordered

For z

(z

z

I

n)

e

n define

For 0 < h e

RnO,)-- (r

n is

n

o 0, then

Fn.

(r,F)

If 0 < r

#

0, then o

e

T(p,n) with p

e

(2.17)

El n]

Take 0 e

Cn-p

Abbreviate

Then

(r,F)

(F

m< If r

() >

(2.18)

0, then

(0,F) Take p e lq[ l,n] and T e

Hence F

o0)flP

TZ

is defined on

exists for almost all

z

T(p n).

).

D

F(0).

(2.19)

Take z e m

then

By Fubini’s theorem

Tz0D)

M(r(r)

F

and we have

M(r,F)

M((r),H).

(2.20)

More explicitly, this is written as

M(r F)

M(n(r)

M( (r)

F

Tz ))

(2.21)

If r > 0, then

m

(2.22)

zm m

Sometimes we shall write (r,F) as in (2.18) even if r has some zero coordinates. Then

(2.21) writes as

in

(2.22) which

is more instructive.

VALUE DISTRIBUTION ON POLYDISCS

3.

625

PLURISUBHARMONI C FUNCTIONS.

n.

Let B be a subset oflR

B +R u {_oo} is said to be increasing

A function g

+ if g(x) _
0

Naturally, if we consider divisors on complex manifolds and spaces the definition of divisors has to be localized.

+

Let V be a complex vector space of dimension m Then

,

operates on

V,

by multiplication.

> i.

Define

V,/,

The quotient space (V)

{0}.

V

V,

is a

connected, compact complex manifold of dimension m called the compl.e.x projective

(A)

(A n

{IP(z)

V,)

0

vector functions V

w(b)

vector functions v

O,(b,v)

V

^

W- 0.

V,

The residual map IP

s_pace of V.

+z

V.

/

A}.

Define

and w e

Let O(b,V) be the set of all holomorphic

O,(b,v)

..representation of f.

Two holomorphic

are called equivalent v

O,(b,v)

from)(b)

{0}.

O(b,v)

O,(b,V).

This defines an equivalence relation on

is said to be a meromorphic map

If A c_ V, define

/(V) is holomorphic.

into

w,

if

An equivalence class f

(V) and each

V in f is said to be a

The representation v is said to be reduced if for each

representation w of f there exists a holomorphic function g such that w

meromorphic map has a reduced representation and if v and

gv.

are reduced

representations, there are holomorphic functions h and h on )(b) such that v

v

hr.

g

and h e O (b) has no zeros.

Hence hh

Therefore the non-negative divisor

the choice of the reduced representation

Let v and

.

Moreover w

w g

Each

gv

ghv.

hv and

Hence

is well defined independent of

Moreover w is reduced if md only if

be reduced representations of the meromorphic map f, then the

in det e rmin a cy

If is well defined

[z

ID(b)

v(z)

analytic, with dim

f()

If

(v())

O}

{z

n- 2

(z)

]D(b)

If z

((z))

)(h)

IP(V)

O}

If

(3.16)

then

(3.17)

631

VALUE DISTRIBUTION ON POLYD%SCS

)(b)

is well defined and the map f

If

(V) is holomorphic.

+ O,

()

e )(h) with

representation of f and if

/

where

(V).

e

V,

The projective space (W)

O.

(V*) and if f()(h)

I, then (W)

Take a e (V*).

Then E[a]

If)

is said to be

ov depends on a and f only and

g

linearly

V,

then

is called the

The valence function of f for a is defined by

-

N

a(r’q)" f

(3.18)

If w is a representation of f, there is a holomorphic function g

w

P()

(ker ) is a hyperplane in

E[a], then f

Nf(r,q;a)

Hence

is called a

Then a

If v is a reduced representation of f and if

The divisor

a-divisor of f.

m-

E[a] is a bijective parameterization of all hyperplanes in (V).

non-degenerate for a. o v

V

is a linear map

The map a

If a

.

If p

Let V* be the dual vector space of V.

hyperplane.

IP(()).

then f()

Let W be a (p+ l)-dimensional linear subspace of V. is a project.ive, plane of dimension p in IP(V).

If m is any

v.

By definition

w

g

a

Take a positive hermitian form (i)

V

V

/zlz ).

llzll

gv.

Hen ce

m + f"

(om

The associated norm is defined by

0 with w

(3.19)

,

called a hermitian product on V.

If x

V and z

V, we have the

Schwarz inequality

(3.20) If

e V* and

B

V*, vectors

(z)

=

(zla)

e V and

B(z)

h

V exist uniquely such that

(zlh)

for all

e

V.

(3.21)

A dual hermitian product on V* is defined by

(IB)

(alb)

(3.22)

W. STOLL

6 32

I,

If

If z

E

then

I1=11.

1111

(V) and a

e

Therefore (3.20) and (3.21) imply the Schwarz inequality

V,

() with

(z) and a

(V*), then z

and

V,.

The

projective distance from z to E[a] is defined by

(3.24)

and e.

independent of the choice of the representatives

+

The compensation function of f for a is defined for r e

0 -


0

or0(3( r (r))) O

If

and

then

(3.29)

-Tf(p,r)

Tf(r,p)

Tf(q,p)

Tf(r,q) The function

with

+

If r, p, q belong to

If.

q

n(b)

if 0 < q _< r < b.

(3.30)

r and decreasing in q

is logarithmic convex and increasing in

and continuous where 0 < q _< r < b.

Let m be a representation of f. holomorphic function g

$

Take a reduced representation v of f.

A

By definition

w

0 exists uniquely such that m

gv.

Dg"

Then

logllwll

)

]D

C

n

ii II lg"V’n

lglgiSn-

+

]D

]D 0 of

+

and

Fubini-Stud Kaehler

metric

We have

(3.33)

Denote the corresponding form on (V*) also by

.

Then

ram(a)

log

(3.34)

An exchange of integration implies

mf(r,a)m(a)

(3.35)

=i

Integration of the First Main Theorem yields

Nf(r,,a)n(a)

Tf(r,q)

(3.36)

a(V*) For V

m+l

ZlW + setting

write 1P

+

z w m m

m

1P(V) and 1P

m

IP(V*).

On

m+i

Identify the compactified plane

define (l:

{}

with IP

by

VALUE DISTRIBUTION ON POLYDISCS z

m(z,w)

W

if z

and 0

+

w e

identify

?I with

u

,

setting

and a

,

(Zl,Z2)

(I 0)

If e e

-

{oo} by

2

IP()

If z e

.

0 and

z

(z,l)

z

(2)*,

define

re(w,0)

(I,0)

re(l,0)

(3.37)

and (0,I)

el

2

and

setting

i#

if

2

az

Then (e)

2. i

IP()

and

0

then (z i)

(0 i)

635

with

(z,l) a and

if

0

I

(2)*

Also

z.

e(z,l)

+

z

a.

so_-

is defined by

Define

oo

by

Then

Ic(z,1) IIII ll

g

637

(=) there is a homogeneous polynomial P. of J

degree j such that

P. (z

g(z)

(4.1)

a)

J

j= (a)

g

for all z in a neighborhood of a in )(b) and such that

g

0 if and only if g(a)

(a)

0.

Ng+h(a) Therefore the map

Moreover Let on

)(b).

+

@,(b)

gh(a)

g(a)

(4.2)

Min(g(a),h(a)).

(4.3)

g(a)

if and only if h(a)

Then v

is a valuation of the ring

@(h).

0.

f

0 is a meromorphic function

where f

Let g and h be holomorphic functions on )(h) which are coprime at every

point of )(b) such that hf

meromorphic map f. u

Obviously

g(a) + Dh(a)

defined by g

be a divisor on ID(b).

0.

0 is a holomorphic function on B3(b) then

If h

gh(a)

P g ()

g.

Then (g,h) is a reduced representation of the

Let f, g and h be another choice.

@*(b) on )(b) without zeros exists such that f

reduced representation of f.

A holomorphic function uf.

Then

is another

O*(b) exists such that (g,h)

Hence v

Cons equent ly

9(a)

(ug,h)

(a) -D(a)= g(a) -h(a)

(vug,vh).

.

(4.4)

is well defined independent of the choice of f, g, and h and is called the

.

multiplicity of v a__t a. If

e

D(h) and p

(PVI

The map D(b)

defined by v

+

2

(=)

P)l(a)

+

2 (a)

The divisor v is non-negative if and only if (a)

v

(a) is a homomorphism:

7z, then

is the zero divisor if and only if

(a)

0(=)

_>

0 for all a

0 for all a e )(h).

identify a divisor v with its maltiplicity function a way of defining divisors on complex manifolds.

(a) and

o. )(h).

(4.5)

The divisor

Therefore we can in fact this is one

W. STOLL

638

Let f be a meromorphic function on ]D(h). a

f

is defined.

divisor

a

f()

Then

f

The function f

f(=)

f()

0 is holomorphic if and only if

-f().

0 and

fl

If

f2

a__t

a, the a-divisor

.

0, the

If f

a__t =.

is said to be the _multiplicity of f

holomorphic and without zeros if and only if

I/f()

If f

is called the a-multiplicity of f

is defined and

f

?I"

Take a e

> 0 for

(), and f

all

0 for all t e ID(I).

llf(i)

Also

0 are meromorphic functions on ]D() then

flf2 () flCa) + f2Ca)" The map

K,(b)

defined by f

/

/

Let V be a divisor on )(b).

9(z)

+

f(a)

(4.6)

is a homomorphism.

0 is called the support of 9 and denoted by supp 9.

0, then the support of

)(b).

Let R(supp ) be the set of regular points of supp

.

a.

a

and c 6R

p.

divisor v on D(b) such that

)(b) such that

f.

If

f

supp

f +

lq

of is

If 9 > 0, then

If.

Take a e

I"

Then

O, then supp

Take p

9

Then the function

{z D(b) () > 0}. Let f be a meromorphic function on )(b) with indeterminacy

supp

If a

R(supp ).

0 then supp

If

pure dimension n

is an analytic subset of

constant on the connectivity components of

Assume that f

.

The closure in )(b) of the set of all z e D(b) with

If

supp

is

Of

u {Z e ID(b)

u supp

Let

(D(c))

Then f

f(z)

If

a}.

(4.7)

f.

)(c) /)(b) be a holomorphic map. supp

9.

Take a

Let f be a meromorphic function on

0 is a meromorphic function on D(c).

The

pullback diviso r

q*())

(4.8)

foq

is well defined and independent of the choice of f.

pj

and

j

are divisors onD(b) with

(ID(c))

{

If

supp

9

V.3

>- 0, then *(v) >- 0. for j

i, 2, then

If

639

VALUE DISTRIBUTION ON POLYDISCS

(m(=))

i supp(pl91

+

and

P22

+

*(Plgl

.

all u that

_

En

For 0

+

If 0

()(t0(r)))

)(h).

said to be restrictable

,

Z

direction

For z

If u e

is defined.

and 0 -< t


exists such

supp w.

.

us for

R (r), the pullback divisor

variation on each compact subinterval of [0

If

>

(u)

as the finite sum

The function t

If 0

t0(r)

Take

(D(t0(r)))

)

n[z,t]

for all 0

is defined by

be a divisor onD(h).

9

if and only if

real analytic subset of ).

%()

En

and 0 < r < h, a largest number

Let

__t

+

) such that w is not restrictable to

the set of all

V[]

z

an injective linear map

ID

(4.9)

pl#*(l) + p2*(92).

P292

and if p. e

for

W. STOLL

640

n

Let f

z

pI vl+p22

[z t]

p n

i

0 be a meromorphic function on )(b) such that

- Rv(r).

Then [z]

Df.

t0(r)

For 0 < q < r
0, then

B and t

+ t(h

and


0 such that for each

log(I/6)

e B we have

< oo.

dB

(6.15)

B

The closure B is compact and contained in

n[h].

If

e

h + (

B, then b

N)

B

+ which contradicts (c).

Hence

Define the associated approach cone by

M

If

{ + t(h

B, then x < h and EXAMPLE I. Take

)

+ t(h


I, with p

with k

_>

B()

< m.

B(B)

and with

ran((1

B

For each %

(0,i] define

)b).

+

Assume there are constants c > 0 and q > 0 such that %

657

VALUE DISTRIBbION ON POLYDISCS

(0, I].

B(B(%))

_


< I.

_< t

>

q0 b > q

and

q.e.d.

c > 0

T(s,q0h

T(sb,q)

T(r,q)

Now, the Lemma of the logarithmic derivative follows easily, which constitutes the main result of this paper.

THEOREM 6.18. function on 3(b).

Take b

+

+n

U

un(b)

and q

Let M be an approach cone.

log+Idzf/ fl D

n

Let f be a non-constant meromorphic

log+Tf(r,q)

-< 17

lq[l,n].

Take

+

Then

+

(6.42)

19 log

for r e M.

PROOF.

Then c > 0 and s e

Take q > 0 with qb < q.

r > q and T(r,q)

>_ c

for all r

6

(0,I)

n

M +IR

For T(r,q) define p

M[s,l).

exist such that

by (6.32) and

@(r) as the largest number such that r < O(r)r

_