ETNA
Electronic Transactions on Numerical Analysis.
Volume 27, pp. 156-162, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
PERIODIC POINTS OF SOME ALGEBRAIC MAPS
VALERY G. ROMANOVSKI
Abstract. We study the local dynamics of maps
branch of the algebraic curve
where
is an irreducible
$% $ % !#"
*+
$ &%'()
We show that the center and cyclicity problems have simple solutions when
partial results are obtained.
,
is odd. For the case of even
,
some
Key words. discrete dynamical systems, polynomial maps, periodic points
AMS subject classifications. 37F10, 58F, 13P
1. Introduction. Consider a map of the form
(1.1)
@BFDCHG 4!IJLK
-/.103254768:9;4<9>? =
@ 4
@BADC(E
Denote by 0NMO2QPRIS6 the P -th iteration of the map (1.1).
D EFINITION 1.1. A singular point 4!.UT of the map (1.1) is called a center, if VXW<Y T
such that Z 4 [N\ 4
\7] W the equality 0^&25476_./4 holds, and a focus otherwise.
Clearly, if the right hand side of (1.1) is a polynomial, then 4 .1T is a center if and only
3
0
`
2
if 46_8a9;4bK
D EFINITION 1.2. A point 4Hc Y T is called a limit cycle of the map (1.1) if 4c is an
isolated root of the equation
0 ^ 254769d4X.eT(K
In the other words, a limit cycle is an isolated 2-periodic point of (1.1) [5].
Consider the equation
@
l
f 2`4 G -g6h.g-jik4gi ?
l m 4 - m .pT G
(1.2)
l FnmhA ^No
G G
where l m - 4IJ . Obviously, (1.2) has an analytic solution of the form (1.1),
o
-/.r03q 25476_.U9;4si/KKtKuK
(1.3)
D EFINITION 1.3.
G We say that the polynomial (1.2) defines (or has) a center at the origin
f
if the equation 254 -g6v.wT has a solution (1.3) such that the map 0 q has a center at the
origin, and we say that (1.2) defines a focus at the origin, if 0 q has a focus.
Thus, the problem arises to find in the space of coefficients x l mBy the manifold on which
o
the corresponding maps 0 q have a center at the origin and to investigate
the limit cycles bifurcations of such maps. For the first time this problem has been stated in [8]. As should be
z
Received May 23, 2003. Accepted for publication November 26, 2003. Recommended by A. Ruffing. This
work was supported by the Ministry of Education, Science and Sport of the Republic of Slovenia and the Nova
Kreditna
Banka Maribor.
CAMTP–Center for Applied Mathematics and Theoretical Physics University of Maribor. Maribor, Krekova 2,
SI-2000, Slovenia (valery.romanovsky@uni-mb.si).
156
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PERIODIC POINTS OF SOME ALGEBRAIC MAPS
157
remarked the center-focus problem and the problem of estimating the number of limit cycles
near 4X.eT (cyclicity) for the map 0 q are similar to the corresponding problems for the second
order system of differential equations{
{ .U9~}<ik2€| G }(6 G
|
}.p|Xij25| G }(6 G
where  and  are polynomials.
One possible way to investigate the behavior of trajectories of the map (1) near the origin
is a transformation to the normal form [1, 6]
4ƒ‚„…9;4N2‡†ˆiŠ‰ C 4 ^ Š
i ‰ ^ Œ4 ‹ˆieŽ6K
If the first coefficient which differs from zero, is ‰ and if ‰7 Y T then


0 ^ 2`46_./4gij‘B‰  4 ^ FDC ik’52 4 ^ FDC 6 G
which implies an unstable focus at 4ƒ.eT , otherwise, if ‰  ]jT the focus is stable.
Another possible way, suggested by Żoła̧dek [8], is based on making use of Lyapunov
functions. Namely, for the map (1.1) it is possible to find a Lyapunov function of the form

“ 2`46_./4 ^ 2”†ˆi>? =
 4 6
 ADCN•
with the property
“ 2`0325476‡69 “ 25476_.p– &4 ‹ˆi– 4 —˜ie™iš–
^'› F ^ iptK
^
^'› 4
‹
It is shown in [7] that if – ^  .œT for all  IeS then
the map (1.1) has a center in the
G
. T then 4X./T is a stable focus,
origin (with 0
^&2`46_814 ), and if – ^ .žKKtK.e– ^ HŸ ^ .1T – ^ ¡1
–
¢
]
T
–
T


when ^
, and an unstable focus, when ^ Y .
To investigate bifurcations of limit cycles of the map (1.1) one can also find the return
£
(Poincaré) map
(1.4)
25476_./0 ^ 25476_.e4gi¤P ^ Œ4 ¥˜i¤P 4 ‹ ietuK
¥
We call the coefficient P › of the return map (1.4) the ¦ th focus quantity. All focus quantities
are polynomials in coefficients of (1.1).
The case of the cubic polynomial
f 254 G -g6_./4sik-jiŠ§g4 ^ ik¨ƒ4Œ-kik©ª- ^ ik«v4 ¥ ik¬­4 ^ -jik®¯4&- ^ iŠ°<- ¥ G
G G
G
where § ¨ KKK °±I² , was considered in [8, 7].
In this paper we consider maps defined by (1.2) in the form of the sum of the homogeneous linear polynomial -jik4 and a homogeneous polynomial of the degree ³ , that is,
@
@ Ÿ m - m .pTbK
fƒ´ @Bµ 254 G -g6˜.e-jik4si ?
@ Ÿ m'¶ m 4
(1.5)
mhA c o
Here and below the superscript 2 ³ 6 denotes the degree of the polynomial in (1.5) and indicates
the focus quantities relevant to (1.5), so
fƒ´ ^ µ 254 G -g6ˆ.¢-·ik4si
^
^ G
(1.6)
^ c4 i C'C 4Œ-ki c ^ o
o
o
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158
VALERY G. ROMANOVSKI
fƒ´ ¥ µ 254 G -ª6˜.¢-·ik4si
C ^ Œ4 - ^ i c ¥ - ¥ K
o
o
o
´ ^ µ G ´ ^ µ G KKK
´ ¥ µ G o ´ ¥ µ G KKtK
Correspondingly, P ^ P ¥
, are the focus quantities of (1.6), P ^ P ¥
, are the focus
quantities of (1.7),
etc.
To
compute
focus
quantities
of
(1.5)
we
first
look
for
the
branch of the
f ´ @Bµ 254 G -g6
-1./032`46 of the
algebraic curve
passing
through
the
origin
(that
is,
for
a
function
f ´ @Bµ 254 G 032`46h638/T
form (1.1) such that
). Then, the coefficients of the Taylor expansion of the
´ @Bµ G P ´ @&µ G KKtKuK
second iteration of 0325476 are the focus quantities P ^
For example, (1.6) implicitly
¥
defines the function
-/.103254763.a9;4<9¢2 c ^ 9 C'C i ^ c6‡4 ^ 9¢2`‘ c ^ 9 ChC 6u2 c ^ 9 ChC i ^ cH6”4 ¥ ipKtKKK
o
o
o
o
o
o
o
o
The second iteration of 0 is
(1.7)
0 ^ .e4<9‘(2 ^ c;9 c ^ 6u2
o
o
fo
so the focus quantities of
P ´ ^ µ .U9s‘(2 ^ c;9 c ^ 6u2
^
o
o
o
and so on.
¥
c 4 ¥ i
^
^ C 4 -ji
c ^ 9 C'C i ^ cH6”4 ¥ ·
9 2 ^;
c 9
o
o
o
´^ µ o
are
G P ´ ^ µ .a­
9 2
^ c;9 ChC i c ^ 6
¥
o
o
o
c ^ u6 2 c ^ 9 ChC i ^ cH6 ^ 4&‹ˆieKKtK G
o
o
o
^ c~9
c ^ u6 2 c ^ 9 ChC i ^ c™6 ^ G
o
o
o
o
2. The cyclicity of maps defined by (1.5). Denote
G the real
G space
G
polynomial (1.5) by ¸ , the ¹ -ball centered at º .»2 @¶ c @ Ÿ C¼¶ C KtKK c ¶ @
o º .À2 @7¶ c o G @
and let 0™q ¿ be the map (1.3) corresponding to a giveno point
o
o
the parameter space, that is,
(2.1)
of coefficients of
6ªI ¸ by ½_¾ 2 º 6 ,
Ÿ C¶ C G KKtK G c ¶ @ 6 of
o

0 ¿ .a9;4N2‡†_i ? =
 2 @7¶ c G @ Ÿ C¶ C G KKK G c ¶ @ 6”4 6K
 ADC E o
o
o
D EFINITION 2.1. Let ³ ¿ ¶ Á be the number of limit cycles of the map 0&¿ in the interval
T!]j4v] W K We say that a singular point 4X.pT of the map 0&¿&Â has the cyclicity ¦ with respect
to the space ¸ if there exist ¹ c Y T and W c Y T , such that for every T!] W ] W c and T ] ¹ ] ¹ c
ÃęÅ
¿ . ¦ K
¿ÆÇ7È ´ ¿ Â µ ³ ¶ Á
G
G
G
InË order to simplify notations we denote
byG 2 6 G the ³ -tuple 2 @7¶ c @ Ÿ C¼¶ C KtKK c ¶ @ 6 and
Gc
@ Ÿ C¼¶ C KtKo K c ¶ @ over the o field o  . Also we denote
o
by ÊÉ the ring of polynomials in @7¶
by
´ @Bµ 2 6
® ´ @Bµ 2 o 6 the map (1.1) defined by o the polynomial
o
o (more precisely, ®
(1.5)
is a family
o depending upon the parameter 2 6 ).
o
of maps
0 C G KKtK G 0BÌ;I É o Ë (  is a field) we denote by Í 0 C G KtKK G 0BÌÎ the ideal of
Given
polynomials
Ë
G
G
É generated by 0 C KKtK 0BÌ and by Ï 25o Ð76 the (affine) variety of the ideal Ð ,
o
@BFDC [032 6_.eT G 0¤IÑÐ K
y
Ï 25Ð76˜. x 2 6LI 
Z
o Ë
o
´ @Bµ . P ´ @Bµ G P ´ @Bµ G
D EFINITION 2.2. The ideal of ² É generated by all focus quantities, Ò
Í ^
B
@
µ
´
KKtK Î , is called the Bautin ideal of theo map ®
2 6 . The set ÓNÔbÕ×ÖtØ . Ï 2 Ò ´ @Bµ 6vÙ»² @&FDC ¥ is
´ @&µ 2 6
o
called the center variety of ®
.
o
Speaking about the center varieties we will assume that the coefficients of (1.5) are complex and speaking on the cyclicity we restrict ourself to maps (1.5) with real coefficients.
´ @Bµ G ´ @Bµ G
´ @Bµ G ´ @Bµ
G ´ @Bµ
´ @&µ
Let Ú . Í P uÛ P ¼Ü KKK P ¼Ý Î be a basis of Ò
such that for any P ¼Þ P 'ß from Ú
´ @&µ
 l ]  m if à ]aá , and for any  Ì such that  l ]  Ìg]  l FDC the polynomial P â belongs to
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PERIODIC POINTS OF SOME ALGEBRAIC MAPS
159
´ @Bµ G ´ @Bµ G
G ´ @Bµ
Í P uÛ P ¼Ü KKK P ¼Þ Î . Using the results of [2, 7] it is easy to conclude that the following
statement holds.
´ @Bµ
T HEOREM 2.3. If Ú is the basis of Ò
defined above then the cyclicity of the origin for
·
9
&
†
K
any map (2.1) is at most ¦
We have seen at the end of the previous section that for the map defined by (1.5) with
³ ./‘
P ´ ^ µ .U9s‘(2 ^ c 9 ChC i c ^ 6u2 ^ c 9 c ^ 6 G
^
o
o
o
o
f ´ µ Go
and, as it was shown in [8], ^ 254 -g6 defines a center in the origin if and only if one of
ChC i c ^ .±T or (ii) ^ cg9 c ^ .±T holds. According to [7] for (1.5)
conditions (i) ^ cª9
e
.
ã
o
o
o
o
o
with ³
P ´ ¥ µ ./‘(2 c 9 ^ C i C ^ 9 c 6
¥
¥
^
o
oµ
o
o
´¥
´^ µ G P ´¥ µ
and the map defines a center if and only if P ^ .pTbK Because P ^
polynomials
^ are linear
´
´¥ µ
µ
´
^µ G ´ µ
they generate the corresponding Bautin ideals, Ò ^ . Í P ^ Î Ò ¥ . Í P ^ Î . Thus the
cyclicity of the origin for every map 0 q defined by (1.5) with ³ .e‘ and ³ ./ã is equal to zero.
T HEOREM 2.4. 1) The polynomial (1.5) with ³ odd defines a center in the origin if and
only if
?
2”9¯†H6 m ä m .eTbK
(2.2)
ä FnmhA@
o
2) The cyclicity of the map defined by (1.5) with ³ odd is equal to zero.
Proof. Assume that the first different from zero coefficient of the map (1.1) is  (with
E
åY † ), then the first different from zero coefficient in the Poincaré map is
P 
.
‘  Gçæéè  æêDëìŒëíÊG
(2.3)
P  FC .
‘ E  FDC Gçæéè  æêDîï(ï K
E
Note that when  .:† the series expansion of the Poincaré map starts from P ^ .1‘ ^ 9š‘ ^C ,
E
E
however below we will deal only with the cases åY ‘ .
The map (1.1) defined by the polynomial (1.5) has the expansion
-1.U9;4¯9 ?
2”9¯†H6 ä ä m 4 @ 9·KKtKK
ä FnmhA@
o
Therefore if ³ is odd, ³ .e‘ ¦ iц , then the first different from zero coefficient of the Poincaré
map is
?
P ^'› ./‘
2”9¯†H6 m ä m K
ä FnmhA ^'› FDC
o
We have to prove that if P ^h› .1T then P ^h› F  .1T for all positive integer  . To do so, it
is sufficiently to show that if P ^'› ./T then the polynomial (1.5) has a branch symmetric with
respect to the line -ž.U4 . We show that under the condition P ^h› .UT the line -pik4v.UT is
an irreducible branch of (1.5). Indeed, consider the equality
^'›
Ÿ m
?
ä
m
25-jiŠ476u2‡†ˆi ?
6˜.¢-jik4gi
äm - 4 m K
m - 4 ^'›
c
ä FnmhA ^'› FDC o
mhA •
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160
VALERY G. ROMANOVSKI
Equaling the coefficient of the same terms in the both sides we get the system
F C¶ cg. ^'›
^'› D
•
^'› o ¶ C . ^'› i ^h› Ÿ C
•
•
t
o
C¶ ^h› . C i c
o c ¶ ^'› FDC • . c • G
•
o
which has a solution if and only if the coefficients of (1.5) satisfy (2.2).
2) As we have shown if P ^h› .ðT then P ^h› F v.ðT for all positive integer  . Therefore
P ^'› F ¯.šP ^h›<ñ (with some polynomial ñ ) for all such  . Hence,
ò . P
Í ^h› FDC Î
and, due to Theorem 2.3, the cyclicity of the map defined by (1.5) is equal to zero.
Consider now the case of (1.5) with ³ even. Then in the map (1.1) defined by (1.5) the
two first different from zero coefficients are
E
@ Ÿ C .
?
2‡9¯†6 m ä m G
ä F
mhAÊ@
o
E
.ôóõ
^ ´ @ Ÿ C‡µ
?
2”9¯†6 m ä m ö÷
ä FnmhAÊ@
o
óõ
?
2”9¯†6 mø ä m ö÷
ä FnmhAÊ@
o
K
Thus according to (2.3), P @ .ç‘ @ .±T . It is easily seen that in this case the first different
E
from zero coefficient of the Poincaré
map is
P ´ @ Ÿ C”µ .1‘
^
9 ³
^ ´ @ Ÿ C‡µ
^@ Ÿ C K
E
E
´ µ
´ µ
P ROPOSITION 2.5. The the center varieties of the maps ® ‹ 2 6 and ® — 2 6 are, correo
o
spondingly,
µ
µ
´‹
´‹ G
´ µ
Ó ‹ . Ï 2úù C 6Êû Ï 2úù ^ 6
´‹ µ
´‹ µ
G
G
where ù C . Í c ‹ 9 C ¥ i
^'^ 9 ¥ C i ‹ c™Î , ù ^ . Í C ¥ 9 ¥ C c ‹ 9 ‹ cÎ and
o
o
o
o ´ µ
o´ µ G o
o
o
´o µ
—
—
Ó — . Ï 2úù C 6Êû Ï 2úù ^ 6
´— µ
´— µ
G
G
where ù C . Í c — 9 C‡ü i ^ ‹ 9 ¥h¥ i ‹ ^ 9 ü¼C i — cHÎ , ù ^ . Í ^ ‹ 9 ‹ ^ C”ü 9 ü¼C c — 9 — cHÎK
p
.
ý
o case
o ³ o theo calculation
o
o of the
o return mapo yields
o
o
o
o
o
Proof. In the
P ´ ‹ µ .e‘(2þ‘ c 9 C i
9‘ c6u2 c 9 C i ^h^ 9
i
c6 G P ÿ ´ ‹ µ 8pTà îï Í P Î G
‹
‹
‹
‹
—
¥
¥ C
¥
¥C
—
o
o
o
o P ´ ‹ µ 8 o † 2 o 9 o 6 o
o
C^
¥C
ý C¥
o
o
2 C 9š‘ ^'^ ikã C 9dý c 6 ^ 2 c 9 C i ^h^ 9
i
c 67à îï Í P ÎuK
‹
‹
‹
—
¥
¥
¥
¥C
o
o
o
o
o
o
o
o
o
§ ° we found that the minimal
With Singular [4]
making use of the routine ¦Ñà ³ by
´‹ µ G ù ´‹ µ
GP Î
P
ù
associate primes of Í — C ^ are the ideals C
^ given above. This yields that
µ
´
´‹ µ
´ µ
‹
Ó ‹
Ï 2úù C 6Êû Ï 2úù ^ 6K
To see that the opposite inclusion holds,
Ó
´ ‹ µ
´‹ µ
´‹ µ G
Ï ú2 ù C 6Êû Ï ú2 ù ^ 6
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PERIODIC POINTS OF SOME ALGEBRAIC MAPS
161
´ µ
f ´ µ G
C ‹ 6 the curve ‹ 254 -g6¡. T has the branch
´‹ µ
corresponding to the points from Ï ú2 ù ^ 6 are
´ µ
Similar reasoning applies also to the case ³ . , however in this case the variety Ó —
µ
µ
µ
µ
´
´
´
´
G
G
G
´— µ .
—
—
—
—
is defined by three focus quantities, Ó
Ï 2 Í P C c P ^ c P ¥ c Îh6 where P C c .À‘b25ã c — 9
‘ C‡ü i
9
:
i
‘
9
ã
u
6
2
9
i
9
i
9
i
c
c
c 6 G P o´ —c µ 8
üC
C‡ü
ü¼C
^‹
^‹
‹^
—
—
‹^
—
¥'¥
^
^ o 2 C‡ü o 9 ý o iŠý o 9 ü¼o C 62 o C‡ü 9‘ o ^ i ã o
9do ý
i üC o 9 o c 6^ 2 c 9
o
^
^
^
‹
‹
‹
‹
—
—
h
¥
¥
^
îï P cHÎ o G P ´ — c µ . o 2`‘b2 o 9
9
i
9
i
c
L
6
Ã
u
6
2
œ
9
‘
o
o
o
o
o
o
o
Cӟ io
ü
C
C
‡
C
ü
Í
^‹
^‹
^‹ i
‹^
—
‹^
¥'¥
¥
ão
9jý o ^ i o üC 9 o c6”^Œo 2 c 9 o C‡ü i
9
i
9
i
H
c
6
`
2
‘
9
üo C
o^ ‹
o ¥'¥ i
‹^
—o
¥'¥
o ¥'¥ 9 †tTo ‹ ü¼C iUo †T ct6”^Ho 6 —†&†‘ 3o à — îï oÍ P C´ — c µ G P o ´ ^ — c ‹ µ Î and
o for the
o polynomial
o
o reduction
o
weo used
‹^
—
^
c
Œ
c
K
o lexicographic
o
o with — Y C‡ü Y ^ ‹ Y ¥'¥ Y ‹ ^ Y üC Y —
the
order
´ µ
´ µ
o
o ® ‹ o 2 6 and
o ® — o 2 6 with a focus in the
P ROPOSITION 2.6. The o cyclicities
of o the maps
o
o
origin are equal, correspondingly, to 1 and 2.
. is similar). The variety Ï 2 Ò ´ ‹ µ 6 is
Proof. Consider the case ³ . ý (the case ³ ´‹ µ G ´‹ µ
´ µ
defined by P — P C ^ . Therefore the return map of ® ‹ 2 6 with a focus in the origin has the
o
£
expansion
c
2 476_.e4si¤P ´ ‹ µ 2 6‡4 i¤P ÿ ´ ‹ µ 2 6”4 C ieKKtK
—
o
o
o
£
or
2 476_.e4si¤P C´ £ ‹ µ 2 6‡4 C ¥ iRP C”´ ‹ ü µ 2 6‡4 C — ieKKKtK
^
o
o o
Obviously, in the first case the equation 2 476B9X4¯.eT has no roots if 9
is sufficiently
o o . Therefore, the
o has at most one root for such
small, and in the second case this equation
´ µ
o
cyclicity
of ® ‹ 2 6 with a focus at the origin is at most one.
£
o
It
is
easy
to
see
that
it
is
equal
to
one,
because
there
are
maps
such
that the equation
2 476~914/.ôT has a small positive real root. Indeed, let
.#2 c G C G ^'^ G ‘ c i
‹
‹
¥
´ µ
G
o when
o
o¹ . T ,
C¥ o 9p‘ ‹ c i ¹ c ‹ 6 . Then P — ‹ . ‘ ¹ 2 c ‹ 9 ^'^ i ‘ ¥ C 9pã o ‹ c i ¹ o 6 and
P o C´ ‹ µ .ž9so ‘(2 c 9 o ^'^ ij‘ C 9ã c6 ¥ 2 co 9
c6
P ´ ‹ µ .1
T o due to our assumption that
‹
‹
‹
‹ o (and o C ^
¥
^
® ´ ‹ µ 2 6 has
o a focus
o at theo origin).
o Obviously,
o
o we can choose ¹ such that \ P ´ ‹ µ !\ >\ P C´ ‹ ^ µ \ and
—
£
o of P ´ ‹ µ is opposite from that of P C´ ‹ ^ µ . That yields a small positive root of the function
the sign
—
2 4769š4 .
ä
ø#" †
o Recall that an ideal Ð is called radical if 0 I¤Ð for any integer
implies that 0 IRÐ .
Ð
Ð
The radical of an ideal is denoted by $ .
P ´ ‹ µ G P C´ ‹ µ Î and Í P C´ — c µ G P ´ — c µ G P ´ — c µ Î are not radical ideals in
P
ROPOSITION 2.7. The ideals Í —
^
^
¥
² É Ë.
o Proof. It is easy to check the statement of the lemma using any computer algebra system with an implemented routine for computing the radical of a polynomial ideal (Singu´‹ µ G ´‹ µ
lar, Macaulay, CALI etc.). Computing with Singular [4] we found that % Í P — P C ^ Î and
´‹ µ G ´‹ µ
´‹ µ G ´‹ µ
Í P — P C ^ Î have different reduced Gröbner bases. That means, that Í P — P C ^ Î is not a radical ideal. Similarly one can check that the second ideal is not radical as well.
To conclude, we have shown that the center and cyclicity problems for the map defined
by the polynomial (1.5) with odd ³ has a simple solution. The case of even ³ is more difficult.
´ µ
Basing on Proposition 2.5 we conjecture that the center variety of the maps ® ^h› 2 6 consists
o
of two components:
?
2”9¯†H6 m ä m .eT
ä FnmhA ^'›
o
one can check that for the points from Ï 2úù
- i/4k.ÀT and the curves f ´ ‹ µ 2`4 G -g6.ÀT
symmetric with respect to the line -1./4 .
ETNA
Kent State University
etna@mcs.kent.edu
162
VALERY G. ROMANOVSKI
and
^'› Ÿ l ¶ l 9
l ¶ ^h› Ÿ l .eT G
G G
G
à .eT † KKtK ¦
9·† G
o
o
4 6u2‡†i'&
where the first component corresponds to (1.5) of the form 2€-ši 7
G ƒ
1
„
‚
4
4 ‚„ and the second one to those invariant under the involution
µ
µ
µ
µ
µ
´
´
´
´
´
‹)C ( G P ‹ I P ‹ G P C ‹ Î
— c I P C´ — c µ G P ´ — c
P
P
Í —
Í
We have checked that
^‹
^ and ‹
^
probably,
l F
mhAÊ@ Ÿ C
lm 4 l- m 6
o
µ. G P ´ — c µ Î K
Therefore,
¥
´ ‹ µ G P ´ ‹ µ ÎÑÄ íbï
´— µ G ´— µ G ´— µ G
´ µ
C^
Ò — . ÍP C c P ^ c P ¥ c Î
´ µ
´ µ
yielding that the cyclicity of maps ® ‹ 2 6 and ® — 2 6 with a center in the origin are, respecµ
µ
´
´
´— µ G ´— µ G ´— µ
G
‹
‹
tively, 1 and 2 as well. If the ideals Í P — o P C ^ Î and Í o P C c P ^ c P ¥ c Î were radical ideals then
(2.4) would be true. However Proposition 2.7 shows that these ideals are not radical ones.
Thus there remains an open problem to find the center variety of map defined by (1.5) with ³
even and to investigate the cyclicity of this map.
(2.4)
´ µ
Ò ‹ .
ÍP —
REFERENCES
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