1
ON CONVERGENCE SETS OF FORMAL POWER
SERIES
Tahir Tuychiev
1
, Laziza Zokirova
2
1,2
National University of Uzbekistan, Universitetskaya street 4, Tashkent 100174, Uzbekistan
tahir1955@mail.ru, laziza1912@mail.ru
https://doi.org/10.5281/zenodo.10471214
Key words: Power series, series in homogenous polynomials, domains of convergence power series, pluripolar set.
Abstract:
This work is devoted to the study of sets of convergence of divergent formal power series of many
variables. In this
note we study the region of convergence of formal power series with holomorphic
coefficients.
1 INTRODUCTION
This work is devoted to the study of sets of
convergence of divergent formal power series of many
variables. In this note we study the region of
convergence of formal power series with holomorphic
coefficients.
A formal power series
1
(
,...
)
n
f z
z
is called
convergent if this series converges in some
neighborhood of the origin in
n
. The classical
result of Hartogs states (see [1]) that a series
f
converges if and only if it converges along all
sections
1
n
−
P
, i.e.
1
( )
(
,...
)
n
f
t
f
t
t
=
converges as a series in a variable
t
for all
1
n
−
P
. This result can be interpreted as a
formal analogue of Hartogs' theorem on separate
analyticity. Since a divergent power series can
converge in some sections, it is naturally of interest to
study and describe such sections. For example, a
series
1
2
1
2
1
2
0
(
,
)
!
k
k
k
f z
z
k z
z
=
=
is divergent, but on complex lines
1
0
z
=
and
2
0
z
=
it converges.
Let
E
a Borel set in
n
. A set
E
is called
pluripolar
if for each point
z
E
there is a non-
constant plurisubharmonic function
( )
u z
defined in
some neighborhood
n
U
of the point
z
such
that
u
= −
on
E
U
. A set E is called
globally pluripolar
if there exists a non-constant
plurisubharmonic
n
function
( )
u z
such that
{ : ( )
}
E
z u z
= −
. Josephson's theorem
answers the question posed by P. Lelon and states
that it is pluripolar if and only if it is globally
pluripolar (see, for example, [2,3]).
A set
E
is called
complete pluripolar
if there
exists a non-constant plurisubharmonic
n
function
u
such that
{ :
( )
}.
E
z u z
=
= −
2 Materials and Methods
Consider a formal power series of the form
0
( , )
( )
,
j
j
j
f z t
P z t
=
=
(1)
where
1
( )
(
,...,
)
j
j
n
P z
P z
z
=
are polynomials of
n
variables. Define
in
( ) {
: ( , )
}.
n
converges as a power series
Conv f
z
f z t
t
=
Let A and B be nonnegative integers such that A> 0.
Series (1) is said to belong to the class Class(A, B)
if
deg
.
j
P
A j
B
+
Clearly, Class (1, 0) is a subclass of Class(A, B).
Suppose that
(
)
E
Conv
f
=
that for some
(
,
)
f
Class A B
. Let
0
( , )
( )
j
j
j
f z t
P z t
=
=
and
( , )
( ,
)
N
N
g z t
t
f
z t
=
, where N=A+B. Then
(1, 0)
g
Class
and
( )
( )
Conv g
Conv f
=
.
Therefore, the convergence set Class(A, B) coincides
with the convergence set Class(1,0).
Suppose that a series
0
( , )
( )
,
j
j
j
f
z t
P
z
t
=
=
is from the class (1, 0)
and
( )
n
Conv f
=
. Then, according to Hartogs’
fundamental theorem, the series
( , )
f z t
converges
as a power series in the
1
n
+
variables
z
and
t
, i.e.
( , )
f z t
converges absolutely in the variables
( , )
z t
in some neighborhood of the origin in
n
. In
2
this case, we say that
f
it is a
convergent series.
On
the contrary, if
( )
n
Conv f
, then
( , )
f z t
diverges as a power series in the variables
z
and
t
,
i.e. does not converge absolutely in any
neighborhood of the origin in
1
n
+
, then we say
that
f
it is a
divergent series
.
Definition 2.1.
A set
n
E
is called a
set of
convergence
in
n
if
( )
E
Conv f
=
for some
divergent series of class (1, 0).
In the work [4] Daowei Ma and Tejinder S.
Neelon, the following statement was proven.
Theorem 2.1. Let
E
the set of convergence in
n
. Then
E
is a countable union of G – complete
pluripolar sets. Therefore,
E
it is a
F
pluripolar
set.
Let us present a diagram of the proof of this
theorem and show what
E
a
F
pluripolar set is.
Consider the divergent series
0
( , )
( )
j
j
j
f z t
P z t
=
=
class (1, 0) such that
(
)
E
Conv f
=
. Let's put
1
:
,
( )
1, 2,...
n
j
m
j
E
z
z
m P z
m
for
j
=
=
for
1, 2,....
m
=
(2)
Then
1
m
m
E
E
=
=
. Let us prove that
m
E
they are
pluripolar sets.
Let us assume the opposite, let for some positive
integer
m
,
(
)
0
m
c E
. Now we will use the
following Bernstein lemma (see [2]):
Let
E
a compact set in
n
and
( )
0
c E
.
Then there is a positive constant
E
c
such that for
each polynomial
( )
d
P z
a z
=
the inequality
holds
d
E
E
a
c
P
, (3)
where
( ) ,
.
E
P z
P
Sup
z
E
=
By Bernstein’s inequality (3), the coefficients
j
b
of the polynomial
( )
j
j
P
z
b
z
=
satisfy
the inequality
(
) ,
m
j
j
E
b
c
where
m
E
c
is a constant depending only on
m
E
. It
follows that the series
( , )
f z t
converges. We have
obtained a contradiction to the fact that the series is
divergent. Therefore, for each
m
the set
m
E
is
pluripolar and
E
is a
F
pluripolar set.►
3
Main results
In this paper, we consider a more general
situation, namely, we consider a divergent formal
power series of the form
0
( , )
( )
j
j
j
f z t
c
z t
=
=
(4)
with holomorphic coefficients of the
n
variables and
the set of convergence of series of the form (4) is
studied.
Give let a formal power series (4), where the
coefficients
( )
1
( ,...,
)
j
j
n
c
z
c z
z
=
are
holomorphic functions in some complete circular
domain
n
D
with respect to the origin. The
main result of the work is the following theorem.
Theorem 3.1
.
If
E
is the convergence set of
some divergent series (5), then
E
the set is a
pluripolar set.
It should be noted that the method of proving
Theorem 2.1 does not work when proving Theorem
3.1, since the proof of Theorem 2.1 essentially uses
the fact that the coefficients of the series are
polynomials. In the case where the coefficients are
not polynomials, we cannot use Bernstein's
inequality. We will solve this problem using the
following theorem (see [5]):
Theorem 3.2.
Let a formal series be given
0
( )
s
s
z
=
by homogeneous polynomials
s
and
L
the
family of complex lines
0
. If for every complex
line
L
the series converges in the circle
(0, ( ))
B
r l
, then it converges uniformly
inside the region
3
{
:
exp
(
,
)
1}
n
z
z
z
V
E
z
=
(5)
where
(
(0, ( )))
l
E
l
S
r l
=
L
and
(0, ( ))
(0, ( ))
S
r l
B
r l
=
.
Since the generalized Green's function
( ,
)
V
z E
+
if and only if the set
E
is
pluripolar in
n
, then from equality (5) it is clear
that the theorem is meaningful only if the set
E
is
nonpluripolar in
n
, which we will assume. We
also exclude the trivial case when
n
=
.
Note that if we set in Theorem 3.2,
( )
1
r
then we come to the well-known result of A.
Sadullaev [4]. And if we set
( )
1
r
and
(0,1)
E
S
=
, then Forelli’s theorem [6] easily
follows from it.
It should be noted that region (5) contains some
neighborhood of the origin.
Proof of Theorem 3.1.
We prove Theorem 3.1 in
the standard way. To do this, let us assume the
opposite, i.e. let
E
be the convergence set of some
divergent series (4) and not be a pluripolar set. Then
there exists a nonpluripolar set
1
n
L
−
such that
for all complex lines
l
L
series (4) converges.
Then, according to Theorem 3.2, the series converges
on a certain set containing a certain neighborhood of
the origin. In addition, this contradicts the fact that
the series is divergent. A contradiction proves the
validity of the statement, i.e.
E
a bunch of . is
pluripolar. ►
References
[1]
HartogsF
.“ZurtheorieanalytischenFunktionenme
hrererVeranderlichen” // Math. Ann. 1906, V.62.
P. 1-88.
[2]
Sadullaev.
Plurissubharmonic functions,//
Current problems in mathematics. Fundamental
directions, M., VINITI, 1985, Vol. 8, p. 65–113
(in Russian).
[3]
Sadullaev.
Pluripotential theory. Applications //
Palmarium academik publishing, Saarbruchen
Deutschland, 2012 (in Russian).
[4]
Daowei Ma and TejinderS.Neelon.“
On
convergence sets of formal power
series
”
//Comlex Analysis and its Sinergies
(2015)1:4.P. 1-21.
[5]
Тuychiev T.T.
ON DOMAINS OF
CONVERGENCE OF MULTIDIMENSIONAL
LACUNARY SERIES. // Journal of Siberian
Federal University. Mathematics & Physics.
2019, 12(6), 736-746.
[6]
Forelli F.
Pluriharmonicity in terme of harmonic
slices // Math.Scand. 1977, v.41, p. 358-364.
