Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field
In earlier papers the author studied some classes of equations with Carlitz derivatives for $\mathbb{F}_q$ -linear functions, which are the natural function field counterparts of linear ordinary differential equations. Here we consider equations containing self-compositions $u \circ u ... \circ u$...
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2005
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509754511589376 |
|---|---|
| author | Kochubei, A. N. Кочубей, А. Н. |
| author_facet | Kochubei, A. N. Кочубей, А. Н. |
| author_sort | Kochubei, A. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:32Z |
| description | In earlier papers the author studied some classes of equations with Carlitz derivatives for $\mathbb{F}_q$ -linear functions, which are the natural function field counterparts of linear ordinary differential equations.
Here we consider equations containing self-compositions $u \circ u ... \circ u$ of the unknown function. As an
algebraic background, imbeddings of the composition ring of $\mathbb{F}_q$ -linear holomorphic functions into
skew fields are considered. |
| first_indexed | 2026-03-24T02:46:08Z |
| format | Article |
| fulltext |
UDC 517.95
A. N. Kochubei (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
STRONGLY NONLINEAR DIFFERENTIAL EQUATIONS
WITH CARLITZ DERIVATIVES OVER A FUNCTION FIELD*
SYL|NO NELINIJNI DYFERENCIAL|NI RIVNQNNQ Z
POXIDNYMY KARLICA NAD FUNKCIONAL|NYM POLEM
In earlier papers the author studied some classes of equations with Carlitz derivatives for Fq -linear
functions, which are the natural function field counterparts of linear ordinary differential equations.
Here we consider equations containing self-compositions u u u� …� of the unknown function. As an
algebraic background, imbeddings of the composition ring of Fq -linear holomorphic functions into
skew fields are considered.
U poperednix stattqx avtora vyvçeno deqki klasy rivnqn\ iz poxidnymy Karlica vidnosno Fq -
linijnyx funkcij nad funkcional\nym polem, qki [ pryrodnymy analohamy linijnyx zvyçajnyx
dyferencial\nyx rivnqn\. U cij roboti rozhlqdagt\sq rivnqnnq, wo mistqt\ kompozyci]
u u u� …� nevidomo] funkci]. Alhebra]çnog osnovog zastosovano] metodyky [ vkladennq kil\cq
Fq -linijnyx holomorfnyx funkcij u tilo.
1. Introduction. Let K be the set of formal Laurent series t xj
j
j N
= =
∞∑ ξ with
coefficients ξj from the Galois field Fq , ξN ≠ 0 if t ≠ 0, q = p v, v ∈ Z+ , where
p is a prime number. It is well known that K is a locally compact field of
characteristic p, with natural operations over power series, and the topology given by
the absolute value | t | = q
–
N , | 0 | = 0. The element x is a prime element of K.
Any nondiscrete locally compact field of characteristic p is isomorphic to such K.
Below we denote by Kc the completion of an algebraic closure K of K. The
absolute value | · | can be extended in a unique way onto Kc.
An important class of functions playing a significant part in the analysis over Kc
is the class of Fq -linear functions. A function f defined on a Fq -subspace K0 of K
(or Kc), with values in Kc, is called F q -linear if f t t f t f t( ) ( ) ( )1 2 1 2+ = + and
f t f t( ) ( )α α= for any t, t1 , t2 ∈ K0 , α ∈ F q . A typical example is a Fq -linear
polynomial c tk
qk
∑ or, more generally, a power series c tk
q
k
k
=
∞∑ 0
, where
c Kk c∈ and c Ck
qk
≤ , convergent on a neighbourhood of the origin.
In the theory of differential equations over K initiated in [1, 2] (which deals also
with some nonanalytic F q
-linear functions) the role of a derivative is played by the
operator
d
q= � ∆ , ( ∆u )( t ) = u( xt ) – xu( t ) ,
introduced by Carlitz [3] and used subsequently in various problems of analysis in
positive characteristic [4 – 8].
The differential equations considered so far were analogs of linear ordinary
differential equations, though the operator d is only Fq -linear and the meaning of a
polynomial coefficient in the function field case is not a usual multiplication by a
polynomial, but the action of a polynomial in the Fq
-linear operator τ, τu = uq. Note
that Fq-linear polynomials form a ring with respect to the composition u � v (the usual
* This research was supported in part by CRDF under Grants UM1-2421-KV-02 and UM1-2567-OD-03.
© A. N. KOCHUBEI, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5 669
670 A. N. KOCHUBEI
multiplication violates the F q -linearity), so that natural classes of equations with
stronger nonlinearities must contain expressions like u � u or, more generally,
u u u� …� . An investigation of such "strongly nonlinear" Carlitz differential equations
is the main aim of this paper.
However we have to begin with algebraic preliminaries of some independent
interest (so that not all the results are used in the subsequent sections) regarding the
ring R
K of locally convergent F q -linear holomorphic functions. The ring is
noncommutative, and the algebraic structures related to strongly nonlinear Carlitz
differential equations are much more complicated than their classical counterparts. So
far their understanding is only at its initial stage. Here we show that R K is imbedded
into a skew field of Fq
-linear "meromorphic" series containing terms like tq k−
. Note
that a deep investigation of bi-infinite series of this kind convergent on the whole of
Kc has been carried out by Poonen [9]. We also prove an appropriate version of the
implicit function theorem.
After the above preparations we consider general strongly nonlinear first order Fq
-linear differential equations (resolved with respect to the derivative of the unknown
function) and prove an analog of the classical Cauchy theorem on the existence and
uniqueness of a local holomorphic solution of the Cauchy problem. In our case the
classical majorant approach (see e.g. [10]) does not work, and the convergence is
proved by direct estimates. We also consider a class of Riccati-type equations
possessing Fq -linear solutions which are meromorphic in the above sense.
2. Skew fields of Fq-linear power series. Let R K be the set of all formal power
series a a tk
q
k
k
= =
∞∑ 0
where ak ∈ K , a Ak
qk
≤ , and A is a positive constant
depending on a . In fact each series a = a ( t ) from R
K converges on a
neighbourhood of the origin in K (and Kc).
R K is a ring with respect to the termwise addition and the composition
a b a b tn l n
q
n
l
l
qn l
� =
−
==
∞
∑∑
00
, b b tk
k
qk
=
=
∞
∑
0
,
as the operation of multiplication. Indeed, if b Bk
qk
≤ , then, by the ultrametric
property of the absolute value,
a b A B Cn l n
q
n
l
n l
q q
q
qn n l n
n
l
−
= ≤ ≤
∑ ≤ ( ) ≤
−
0 0
max
where C = B max ( A, 1) . The unit element in R K is a( t ) = t . It is easy to check
that R K has no zero divisors.
If a ∈ R
K , a a tk
q
k
k
= =
∞∑ 0
, is such that | a0 | ≤ 1 and a Ak
qk
≤ , | A | ≥ 1,
for all k, then we may write
a Ak
qk
≤ −
1
1, k = 0, 1, 2, … ,
if we take A Aq qk k
1
1≥ −/( ) for all k ≥ 1. If also b b tk
q
k
k
= =
∞∑ 0
, b Bk
qk
≤ −
1
1,
B1 1≥ , then for
a b c tl
q
l
l
� = =
∞∑ 0
we have
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
STRONGLY NONLINEAR DIFFERENTIAL EQUATIONS … 671
c A B Cl
i j l
q q
q
qi j
i
l
≤ ( ) ≤
+ =
− − −max 1
1
1
1
1
1
where C1 = max ( A1 , B1) . In particular, in this case the coefficients of the series for
an (the composition power) satisfy an estimate of this kind, with a constant
independent of n.
Proposition 1. The ring R K is a left Ore ring, thus it possesses a classical ring
of fractions.
Proof. By Ore’s theorem (see [11]) it suffices to show that for any elements a,
b ∈ R K there exist such elements a′, b′ ∈ R K that b′ ≠ 0 and
a′ � b = b′ � a. (1)
We may assume that a ≠ 0,
a a tk
k m
qk
=
=
∞
∑ , b b tk
k l
qk
=
=
∞
∑ ,
m, l ≥ 0, am ≠ 0, bl ≠ 0.
Without restricting generality we may assume that l = m (if we prove (1) for this
case and if, for example, l < m, we set b t bqm l
1 =
−
� , find a″, b′ in such a way that
a″ � b1 = b′ � a, and then set ′ = ′′
−
a a tqm l
� ), and that al = bl = α, so that
a t a tq
k
k l
ql k
= +
= +
∞
∑α
1
,
b t b tq
k
k l
ql k
= +
= +
∞
∑α
1
, α ≠ 0.
We seek a′, b′ in the form
′ = ′
=
∞
∑a a tj
j l
q j
, ′ = ′
=
∞
∑b b tj
j l
q j
.
The coefficients ′ ′a bj j, can be defined inductively. Set ′ = ′ =a bl l 1. If ′ ′a bj j,
have been determined for l ≤ j ≤ k – 1, then ′ ′a bk k, are determined from the equality
of the ( k + l )-th terms of the composition products:
′ + ′ = ′ + ′
+ = +
≠
+ = +
≠
∑ ∑a a b b b ak
q
i j k l
j l
i j
q
k
q
i j k l
j l
i j
qk i k i
α α
(the above sums do not contain nontrivial terms with ′ ′ ≥a b i ki i, , , since aj = bj = 0
for j < l).
In particular, we may set ′ =bk 0 ,
′ = ′ − ′( )
−
+ = +
< ≠
∑a a b b ak
q
i j k l
i k j l
i j
q
i j
qk i i
α
,
.
If this choice is made for each k ≥ l + 1, then we have ′ =bi 0 for every i ≥ l + 1,
so that
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
672 A. N. KOCHUBEI
′ = ′−
+ = +
< ≠
∑a a bk
q
i j k l
i k j l
i j
qk i
α
,
. (2)
Denote C1 = | α |–1. We have b Cj
q j
≤ 2 for all j. Denote, further, C3 =
= max (1, C1 , C2) , C Cql
4 3
2
=
+
. Let us prove that
′ ≤a Ck
qk
4 .
Suppose that ′ ≤a Ci
qi
4 for all i, l ≤ i ≤ k – 1 (this is obvious for i = 1, since
′ =al 1). By (2),
′ ≤ ≤
+ = +
< ≠
+ − +
a C C C C C Ck
q
i j k l
i k j l
q q q q qk i i j k k k l
1 4 2 1 4 2
1
max
,
≤
≤ C C C Cq q q q q q q
q
qk k l k l l l k l
k
k
3 3
1
3 4
1 1 2+ + + ++ + + + +
= ≤ ( ) =( ) ,
as desired. Thus a′ ∈ R K .
The proposition is proved.
Every non-zero element of R K is invertible in the ring of fractions A K , which is
actually a skew field consisting of formal fractions c–1d, c, d ∈ R K .
Proposition 2. Each element a = c–1d ∈ A K can be represented in the form
a t aq m
= ′
−
where tq m−
is the inverse of tqm
, a′ ∈ R K .
Proof. It is sufficient to prove that any non-zero element c ∈ R K can be written
as c c tqm
= ′ � where c′ is invertible in R K .
Let c c tk
q
k m
k
= =
∞∑ , cm ≠ 0, c Ck
qk
≤ . Then
c c t c c t tm m m l
q
l
ql m
= +
−
+
=
∞
∑ 1
1
�
where c c Cm m l
ql−
+
−≤1
1
1 for all l ≥ 1, if C1 is sufficiently large. Denote
w c c tm m l
q
l
l
= −
+
=
∞
∑ 1
1
, c′ = cm( t + w ) .
The series
( ) ( )t w wn n
n
+ = −−
=
∞
∑1
0
1
converges in the standard non-Archimedean topology of formal power series (see [12],
Sect. 19.7) because the formal power series for wn begins from the term with tqm
;
recall that wn is the composition power, and t is the unit element. Moreover,
w a tn
j
n q
j n
j
= =
∞∑ ( ) where a Cj
n q j( ) ≤ −
1
1 for all j, with the same constant
independent of n. Using the ultrametric inequality we find that the coefficients of the
formal power series
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
STRONGLY NONLINEAR DIFFERENTIAL EQUATIONS … 673
( )t w a tj
q
j
j
+ =−
=
∞
∑1
0
(each of them is, up to a sign, a finite sum of the coefficients aj
n( )) satisfy the same
estimate. Therefore ( c′ )–1
∈ R K , as desired.
The skew field of fractions AK can be imbedded into wider skew fields where
operations are more explicit. Let Kperf be the perfection of the field K. Denote by
AKperf
∞ the composition ring of Fq -linear formal Laurent series a a tk
q
k m
k
= =
∞∑ ,
m ∈Z , a Kk ∈ perf , am ≠ 0 (if a ≠ 0). Since τ is an automorphism of Kperf,
AKperf
∞ is a special case of the well-known ring of twisted Laurent series [12].
Therefore AKperf
∞ is a skew field.
Let
AKperf
be a subring of
AKperf
∞ consisting of formal series with a Ak
qk
≤
for all k ≥ 0. Just as in the proof of Proposition 2, we show that
AKperf
is actually a
skew field. Its elements can be written in the form t cq m−
� where c is an invertible
element of the ring
R AK Kperf perf
⊂ of formal power series a tk
q
k
k
=
∞∑ 0
. In contrast
to the case of the skew field AK , in
AKperf
the multiplication of tq m−
by c is
indeed the composition of (locally defined) functions, so that AKperf
consists of
fractional power series understood in the classical sense.
Of course, AKperf
can be extended further, by considering K or Kc instead of
Kperf. The above reasoning carries over to these cases (we can also consider the ring
R Kc
of locally convergent Fq
-linear power series as the initial ring). In each of them
the presence of a fractional composition factor tq m−
is a Fq -linear counterpart of a
pole of the order m.
3. Recurrent relations. In our investigations of strongly nonlinear equations and
implicit functions we encounter recurrent relations of the same form
c B c c c ai i
j l i
l
jkl
k n n l
n n
q
n
q
q
i
k
n
k
n n
j
k
+
+ =
≠
=
∞
+ + =
=
+∑ ∑ ∑
+ +
+
−
1
0
1 1
1 2
1 1 1
µ
λ
…
…
…
, (3)
i = 1, 2, …
(here and below n1 , … , nk ≥ 1 in the internal sum), with coefficients from Kc, such
that | µi | ≤ M, M > 0, B Bjkl
kq j
≤ , B ≥ 1, | ai | ≤ M for all i, j, k, l; the number
λ is either equal to 1, or λ = 0, and in that case | B01l| ≤ 1.
Proposition 3. For an arbitrary element c Kc1 ∈ , the sequence determined by
the relation (3) satisfies the estimate c Cn
qn
≤ , n = 1, 2, … , with some constant
C ≥ 1.
Proof. Set cn = σdn , | σ | < 1, n = 1, 2, … , and substitute this into (3). We
have
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
674 A. N. KOCHUBEI
d Bi i
j l i
l
jkl
k n n l
q q
k
n n n qk
j
+
+ =
≠
=
∞
+ + =
+ + + −=
∑ ∑ ∑
+ + −
+
1
0
1
1 1
1
1 1 1
µ σ
λ
…
… …( ) ×
×
d d d an n
q
n
q
q
i
n
k
n n
j
k
1 2
1 1 1 1…
…+ +
+
−( )
+ −
λ
σ .
Here
σ σ
λ λ( )1 1 11 1 1+ + + − −+ + +−
+
≤q q kqn n n q jk
j
… …
,
and (under our assumptions) choosing such σ that | σ | is small enough we reduce (3)
to the relation
d b d d d ai i
j l i
l
jkl
k n n l
n n
q
n
q
q
i
k
n
k
n n
j
k
+
+ =
≠
=
∞
+ + =
−= ( ) +∑ ∑ ∑
+ +
+
−
1
0
1
1
1
1 2
1 1 1
µ σ
λ
…
…
…
, (4)
i = 1, 2, … ,
where | bjkl | ≤ 1.
It follows from (4) that
di+1 ≤
≤
M d d d M a
j l i
l
k n n l
n n
q
n
q
q
i
k
n
k
n n
j
k
max sup max max ,
+ =
≠
≥ + + =
− −+ +
+
−( )
0
1
1 1
1
1 2
1 1 1
…
…
…
λ
σ .
Let B M d M a
i
i=
− −max , , , sup1 1
1 1σ . Let us show that
d Bn
q qn n
≤
− −+ + +1 2 1… , i = 1, 2, … . (5)
This is obvious for n = 1. Suppose that we have proved (5) for n ≤ i. Then
d M B Bi
j l i k n n l
q q q q q
k
n n n n n n n
+
+ = ≥ + + =
+ + + + + +≤ ( − − + − + −
1
1
1
1
1 2 1 21 1 1 2 1 2 1
max sup max
…
… … …
…
… … … ……Bq q q
qn n n n n n n n
j
k k k k k1 1 1 1 2 1 11 1
1
+ + + − + + + + + + +
+
− − − −+ + + ) ≤
≤
M B B B B
j l i
q q q q q q qj l j i i i i
max
+ =
+ + + + + + + ++ + − −
≤ =… … …1 1 1 1,
and we have proved (5). Therefore
c B Cn
q q qn n
≤ ≤− −σ ( )/( )1 1
for some C, as desired.
The proposition is proved.
4. Implicit functions of algebraic type. In this section we look for F q
-linear
locally holomorphic solutions of equations of the form
P t P t z P t z z P t z z zN
N
0 1 2 0( ) ( ) ( ) ( ) ( ) ( )+ + + + =� � � … � � �…���� (6)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
STRONGLY NONLINEAR DIFFERENTIAL EQUATIONS … 675
where
P P PN Kc0 1, , ,… ∈R . Suppose that the coefficient P t a tk jk
q
j
j
( ) = ≥∑ 0
is
such that a00 = 0, a01 ≠ 0; these assumptions are similar to the ones guaranteeing
the existence and uniqueness of a solution in the classical complex analysis. Then (see
Sect. 2) P1 is invertible in
R Kc
, and we can rewrite (6) in the form
z Q t z z Q t z z z Q tN
N
+ + + =2 0( ) ( ) ( ) ( ) ( )� � … � � �…���� (7)
where
Q Q QN Kc0 2, , ,… ∈R , that is
Q t b tk
j
jk
q j
( ) =
=
∞
∑
0
, b Bjk k
q j
≤ ,
for some constants Bk > 0, b00 = 0.
Proposition 4. The equation (6) has a unique solution z Kc
∈R satisfying the
“initial condition”
z t
t
( ) → 0, t → 0.
Proof. Let us look for a solution of the transformed equation (7), of the form
z t c t
i
i
qi
( ) =
=
∞
∑
1
, c Ki c∈ ; (8)
our initial condition is automatically satisfied for a function (8).
Substituting (8) into (7) we come to the system of equalities
c b c c c bi
k
N
j l i
j l
jk
n n l
n
n n
q
n
q
q
i
k
j
n
k
n n
j
k
= −
+
= + =
≥ ≥
+ + =
≥
∑ ∑ ∑
+ + −
2
0 1 1
0
1
1 2
1 1 1
,
…
…
…
, i ≥ 1. (9)
In each of them the right-hand side depends only on c1 , … , ci – 1 , so that the relations
(9) determine the coefficients of a solution (8) uniquely. By Proposition 3,
z Kc
∈R .
More generally, let
P t a tj
q
j
j
1 1( ) =
≥
∑
ν
, ν ≥ 0, aν1 ≠ 0.
Then the equation (6) has a unique solution in R Kc
, of the form
z t c t
i
i
qi
( ) =
= +
∞
∑
ν 1
, c Ki c∈ .
The proof is similar.
5. Equations with Carlitz derivatives. Let us consider the equation
dz t a z z z t a t
j k
jk
j
k
j
j
q j
( ) ( ) ( )= +
=
∞
=
∞
=
∞
∑ ∑ ∑
0 1 0
0τ � �…���� (10)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
676 A. N. KOCHUBEI
where a Kjk c∈ , a Ajk
kq j
≤ , k ≥ 1, a Aj
q j
0 ≤ , A ≥ 1. We look for a solution
in the class of Fq
-linear locally holomorphic functions of the form
z t c t
k
k
qk
( ) =
=
∞
∑
1
, c Kk c∈ , (11)
thus assuming the initial condition t–1z( t ) → 0, as t → 0.
Theorem 1. A solution (11) of the equation (10) exists with a non-zero radius of
convergence, and is unique.
Proof. We may assume that
| aj0 | ≤ 1, aj0 → 0, as j → ∞. (12)
Indeed, if that is not satisfied, we can perform a time change t = γt1 obtaining an
equation of the same form, but with the coefficients aj
q j
0γ instead of a j0 , and it
remains to choose γ with | γ | small enough. Note that, in contrast with the case of
the usual derivatives, the operator d commutes with the above time change.
Assuming (12) we substitute (11) into (10) using the fact that d c tk
qk( ) =
= c k tk
q q qk1 1 1/ /[ ]
−
, k ≥ 1, where k x xqk
[ ] = − . Comparing the coefficients we come
to the recursion
c i a c c c ai
j l i
j l
k
jk
q
n n l
n n
q
n
q
q
i
k
n
k
n n
j
k
+
−
+ =
≥ ≥
=
∞
+ + =
= +[ ]
+∑ ∑ ∑
+ +
+
−
1
1
0 1
1
01
1
1 2
1
1 1 1
,
…
…
…
,
i ≥ 1,
where c aq
1
1
001= [ ]− . This already shows the uniqueness of a solution. The fact that
c Cqi
1 ≤ for some C follows from Proposition 3.
The theorem is proved.
Using Proposition 4 we can easily reduce to the form (10) some classes of equations
given in the form not resolved with respect to dz.
As in the classical case of equations over C (see [10]), some of equations (10) can
have also nonholomorphic solutions, in particular those which are meromorphic in the
sense of Sect. 2. As an example, we consider Riccati-type equations
dy t y y t P y t R t( ) ( )( ) ( ) ( ) ( )= + ( ) +λ τ� (13)
where λ ∈Kc , 0 1 2
< ≤ −λ q q/ ,
P y t p y t
k
k
qk
( ) ( ) ( )τ( ) =
=
∞
∑
1
, R t r t
k
k
qk
( ) =
=
∞
∑
0
,
p r Kk k c, ∈ , p qk
q≤ −1 2/ , r qk
q≤ −1 2/ for all k.
Theorem 2. Under the above assumptions, the equation (13) possesses solutions
of the form
y t ct a tq
n
n
qn
( ) /= +
=
∞
∑1
0
, c a Kn c, ∈ , c ≠ 0, (14)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
STRONGLY NONLINEAR DIFFERENTIAL EQUATIONS … 677
where the series converges on the open unit disk | t | < 1.
Proof. For the function (14) we have
dy t c t a n tq q q
n
n
q q qn
( ) / / / /= −[ ] + [ ]
− −
=
∞
∑1 1
1
1 11
2 1
, −[ ] =1 1x q/ ,
( )( ) /
/
/y y t c ct a t a ct a tq
n
n
q
q
n
n
q
m
m
q
q
n m
n
� = +
+ +
=
∞
=
∞
=
∞
∑ ∑ ∑1
0
1
0
1
0
=
= c t ca ca tq q q q1 1
0
1
0
2 1+ − −
+ +( )/ / +
+
n
n
q q
n
q
l
q
m n l
m n
n m
qca c a t t a a
n n l n
=
∞
+ +
=
∞
+ =
≥
∑ ∑ ∑+( ) +
+
0
1
1
1
0
0
1/
,
.
Finally,
P y t p c t t p a
k
k
q q
l
q
i j l
i j
i j
qk k l i
( ) ( )
,
τ( ) = +
=
∞
+
=
∞
+ =
≥ ≥
∑ ∑ ∑
+
0
1
0
1 0
1
.
Comparing the coefficients we find that
c q= −[ ]−λ 1 11 / , a aq
0
1
0 0/ + = , (15)
a l c c al
q q q
l
l
+ ++[ ] −( ) − +
1
1 1
11
1/ / λ λ =
= λ
m n l
m n
n m
q
i j l
i j
i j
q
la a p a r
n i
+ =
≥
+ =
≥ ≥
∑ ∑+ +
, ,0 1 0
, l ≥ 0. (16)
By (15), we have | c | ≥ 1, and either a0 = 0, or | a0 | = 1. Next, (16) is a
recurrence relation (with an algebraic equation to be solved at each step) giving values
of al for all l ≥ 1. Let us prove that | aj | ≤ 1 for all j. Suppose we have proved
that for j ≤ l. It follows from (16) that
a l c a c a ql
q q
l
q q
l
q ql
+ + +
−+[ ] − − ≤
+
1 1 1
11
2
λ λ / . (17)
Suppose that | al + 1 | > 1. We have λq qc = −[ ]1 , so that λq q qc q= −1/ , and
since l q+[ ] = −1 1 and | c | ≥ 1, we find that
a l c a c al
q q
l
q q
l
ql
+ + ++[ ] < <
+
1 1 11
2
λ λ .
Therefore the left-hand side of (17) equals
λq q q
l
q qc c a q
l+
+
−>
1
1
1/ ,
and we have come to a contradiction.
The theorem is proved.
1. Kochubei A. N. Differential equations for Fq -linear functions // J. Number Theory. – 2000. – 83.
– P. 137 – 154.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
678 A. N. KOCHUBEI
2. Kochubei A. N. Differential equations for Fq -linear functions, II: Reqular singularity // Finite
Fields Appl. – 2003. – 9. – P. 250 – 266.
3. Carlitz L. On certain functions connected with polynomials in a Galois field // Duke Math. J. –
1935. – 1. – P. 137 – 168.
4. Carlitz L. Some special functions over GF q x( , ) // Ibid. – 1960. – 27. – P. 139 – 158.
5. Goss D. Fourier series, measures, and divided power series in the theory of function fields // K-
Theory. – 1989. – 1. – P. 533 – 555.
6. Kochubei A. N. Fq -linear calculus over function fields // J. Number Theory. – 1999. – 76. –
P. 281 – 300.
7. Thakur D. Hypergeometric functions for function fields, II // J. Ramanujan Math. Soc. – 2000. –
15. – P. 43 – 52.
8. Wagner C. G. Linear operators in local fields of prime characteristic // J. reine und angew. Math.
– 1971. – 251. – P. 153 – 160.
9. Poonen B. Fractional power series and pairings on Drinfeld modules // J. Amer. Math. Soc. –
1996. – 9. – P. 783 – 812.
10. Hille E. Lectures on ordinary differential equations. – Reading: Addison-Wesley, 1969. – XI +
723 p.
11. Herstein I. N. Noncommutative rings // Carus Math. Monogr. – Math. Assoc. Amer., J. Wiley and
Sons, 1968. – # 15. – XII + 199 p.
12. Pierce R. S. Associative algebras. – New York: Springer, 1982. – XII + 436 p.
Received 02.12.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
|
| id | umjimathkievua-article-3633 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:46:08Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0b/48d82006f549dbe97d693a8cd97d670b.pdf |
| spelling | umjimathkievua-article-36332020-03-18T20:00:32Z Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field Сильно нелінійні диференціальні рівняння з похідними Карліца над функціональним полем Kochubei, A. N. Кочубей, А. Н. In earlier papers the author studied some classes of equations with Carlitz derivatives for $\mathbb{F}_q$ -linear functions, which are the natural function field counterparts of linear ordinary differential equations. Here we consider equations containing self-compositions $u \circ u ... \circ u$ of the unknown function. As an algebraic background, imbeddings of the composition ring of $\mathbb{F}_q$ -linear holomorphic functions into skew fields are considered. У попередніх статтях автора вивчено деякі класи рівнянь із похідними Карліца відносно $\mathbb{F}_q$ -лінійних функцій над функціональним полем, які є природними аналогами лінійних звичайних диференціальних рівнянь. У цій роботі розглядаються рівняння, що містять композиції $u \circ u ... \circ u$ невідомої функції. Алгебраїчною основою застосованої методики є вкладення кільця $\mathbb{F}_q$ - лінійних голоморфних функцій у тіло. Institute of Mathematics, NAS of Ukraine 2005-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3633 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 5 (2005); 669–678 Український математичний журнал; Том 57 № 5 (2005); 669–678 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3633/3996 https://umj.imath.kiev.ua/index.php/umj/article/view/3633/3997 Copyright (c) 2005 Kochubei A. N. |
| spellingShingle | Kochubei, A. N. Кочубей, А. Н. Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field |
| title | Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field |
| title_alt | Сильно нелінійні диференціальні рівняння з похідними Карліца над функціональним полем |
| title_full | Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field |
| title_fullStr | Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field |
| title_full_unstemmed | Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field |
| title_short | Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field |
| title_sort | strongly nonlinear differential equations with carlitz derivatives over a function field |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3633 |
| work_keys_str_mv | AT kochubeian stronglynonlineardifferentialequationswithcarlitzderivativesoverafunctionfield AT kočubejan stronglynonlineardifferentialequationswithcarlitzderivativesoverafunctionfield AT kochubeian silʹnonelíníjnídiferencíalʹnírívnânnâzpohídnimikarlícanadfunkcíonalʹnimpolem AT kočubejan silʹnonelíníjnídiferencíalʹnírívnânnâzpohídnimikarlícanadfunkcíonalʹnimpolem |