ci+1 = Âµi â j+l=i l=0. â. â .... In each of them the right-hand side depends only on c1,...,ciâ1, so that the relations (9) determine .... K-...

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Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field ANATOLY N. KOCHUBEI∗ Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, Kiev, 01601 Ukraine

Abstract 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.

1

INTRODUCTION

Let K be the set of formal Laurent series t =

∞ P

ξj xj with coefficients ξj from the Galois field

j=N

Fq , ξN 6= 0 if t 6= 0, q = pυ , υ ∈ 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 non-discrete locally compact field of characteristic p is isomorphic to such K. Below we denote by K c the completion of an algebraic closure K of K. The absolute value | · | can be extended in a unique way onto K c . An important class of functions playing a significant part in the analysis over K c is the class of Fq -linear functions. A function f defined on a Fq -subspace K0 of K (or K c ), with values in K c , is called Fq -linear if f (t1 + t2 ) = f (t1 ) + f (t2 ) and f (αt) = αf (t) for any t, t1 , t2 ∈ K0 , P qk α ∈ Fq . A typical example is a Fq -linear polynomial ck t or, more generally, a power series ∞ P k k ck tq , where ck ∈ K c and |ck | ≤ C q , convergent on a neighbourhood of the origin. k=0

In the theory of differential equations over K initiated in [7, 8] (which deals also with some non-analytic Fq -linear functions) the role of a derivative is played by the operator d=

√ q

◦ ∆,

(∆u)(t) = u(xt) − xu(t),

introduced by Carlitz [1] and used subsequently in various problems of analysis in positive characteristic [2, 3, 6, 11, 12]. ∗

This research was supported in part by CRDF under Grants UM1-2421-KV-02 and UM1-2567-OD-03.

1

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 , τ u = uq . Note that Fq -linear polynomials form a ring with respect to the composition u ◦ v (the usual multiplication violates the Fq -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 RK of locally convergent Fq -linear holomorphic functions. The ring is non-commutative, 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 RK is imbedded into a skew field of Fq -linear “meromorphic” series containing −k terms like tq . Note that a deep investigation of bi-infinite series of this kind convergent on the whole of K c has been carried out by Poonen [10]. 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. [5]) 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 RK be the set of all formal power series a =

∞ P

k=0

k

k

ak tq where ak ∈ K, |ak | ≤ Aq , and A

is a positive constant depending on a. In fact each series a = a(t) from RK converges on a neighbourhood of the origin in K (and K c ). RK is a ring with respect to the termwise addition and the composition ! l ∞ ∞ X X X k qn ql bk tq , an bl−n t , b = a◦b= l=0

n=0

k=0

k

as the operation of multiplication. Indeed, if |bk | ≤ B q , then, by the ultra-metric property of the absolute value, l X l−n qn n l q qn an bl−n ≤ max A ≤ Cq Bq 0≤n≤l n=0

where C = B max(A, 1). The unit element in RK is a(t) = t. It is easy to check that RK has no zero divisors.

2

If a ∈ RK , a = may write if we take A1 ≥ Aq a◦b=

∞ P

ql

∞ P

k=0

k

k −1

|ak | ≤ Aq1

k /(q k −1)

cl t we have

k

ak tq , is such that |a0 | ≤ 1 and |ak | ≤ Aq , |A| ≥ 1, for all k, then we k = 0, 1, 2, . . . , ∞ P k k for all k ≥ 1. If also b = bk tq , |bk | ≤ B1q −1 , B1 ≥ 1, then for ,

k=0

l=0

j q i l i ≤ C1q −1 |cl | ≤ max Aq1 −1 B1q −1 i+j=l

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 RK is a left Ore ring, thus it possesses a classical ring of fractions. Proof. By Ore’s theorem (see [4]) it suffices to show that for any elements a, b ∈ RK there exist such elements a′ , b′ ∈ RK that b′ 6= 0 and a′ ◦ b = b′ ◦ a.

(1)

We may assume that a 6= 0, ∞ X

a=

qk

b=

ak t ,

k=m

∞ X

k

bk tq ,

k=l

m, l ≥ 0, am 6= 0, bl 6= 0. Without restricting generality we may assume that l = m (if we prove (1) for this case and m−l if, for example, l < m, we set b1 = tq ◦ b, find a′′ , b′ in such a way that a′′ ◦ b1 = b′ ◦ a, and m−l then set a′ = a′′ ◦ tq ), and that al = bl = α, so that ql

a = αt +

∞ X

qk

ql

b = αt +

ak t ,

∞ X

k

bk tq ,

k=l+1

k=l+1

α 6= 0. We seek a′ , b′ in the form ′

a =

∞ X

j a′j tq ,

′

b =

j=l

∞ X

j

b′j tq .

j=l

The coefficients a′j , b′j can be defined inductively. Set a′l = b′l = 1. If a′j , b′j have been determined for l ≤ j ≤ k − 1, then a′k , b′k are determined from the equality of the (k + l)-th terms of the composition products: X X i i k k a′k αq + b′i aqj a′i bqj = b′k αq + i+j=k+l j6=l

i+j=k+l j6=l

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(the above sums do not contain non-trivial terms with a′i , b′i , i ≥ k, since aj = bj = 0 for j < l). In particular, we may set b′k = 0, X i i k q q ′ ′ ′ −q . ai bj − bi aj ak = α i+j=k+l i

If this choice is made for each k ≥ l + 1, then we have b′i = 0 for every i ≥ l + 1, so that X i k a′k = α−q a′i bqj .

(2)

i+j=k+l i

j

Denote C1 = |α|−1. We have |bj | ≤ C2q for all j. Denote, further, C3 = max(1, C1 , C2 ), l+2 C4 = C3q . Let us prove that k |a′k | ≤ C4q . i

Suppose that |a′i | ≤ C4q for all i, l ≤ i ≤ k − 1 (this is obvious for i = 1, since a′l = 1). By (2), |a′k | ≤ C1q

k

i

max C4q C2q

i+j

i+j=k+l i

k

≤ C1q C4q

k−1

C2q

k+l

≤ C3q as desired. Thus a′ ∈ RK .

k +q k+l+1 +q k+l

(1+q l +q l+1 )q k

= C3

l+2 qk k = C4q , ≤ C3q

Every non-zero element of RK is invertible in the ring of fractions AK , which is actually a skew field consisting of formal fractions c−1 d, c, d ∈ RK . Proposition 2. Each element a = c−1 d ∈ AK can be represented in the form a = tq m −m tq is the inverse of tq , a′ ∈ RK .

−m

a′ where

Proof. It is sufficient to prove that any non-zero element c ∈ RK can be written as c = c′ ◦tq where c is invertible in RK . ∞ P k k ck tq , cm 6= 0, |ck | ≤ C q . Then Let c =

m

k=m

c = cm

t+

∞ X

ql c−1 m cm+l t

l=1

!

◦ tq

m

l

q −1 where |c−1 for all l ≥ 1, if C1 is sufficiently large. Denote m cm+l | ≤ C1

w=

∞ X

l

q c−1 m cm+l t ,

c′ = cm (t + w).

l=1

The series (t + w)−1 =

∞ P

(−1)n w n converges in the standard non-Archimedean topology

n=0

of formal power series (see [9], Sect. 19.7) because the formal power series for w n begins 4

m

n from the term with tq ; recall that composition power, and t is the unit element. w is the ∞ P (n) (n) q j q j −1 n for all j, with the same constant independent Moreover, w = aj t where aj ≤ C1 j=n

of n. Using the ultra-metric inequality we find that the coefficients of the formal power series ∞ P j (n) (t + w)−1 = aj tq (each of them is, up to a sign, a finite sum of the coefficients aj ) satisfy j=0

the same estimate. Therefore (c′ )−1 ∈ RK .

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 A∞ Kperf the composition ring ∞ P k ak tq , m ∈ Z, ak ∈ Kperf , am 6= 0 (if a 6= 0). Since τ of Fq -linear formal Laurent series a = k=m

is an automorphism of Kperf , A∞ Kperf is a special case of the well-known ring of twisted Laurent ∞ series [9]. Therefore AKperf is a skew field. qk Let AKperf be a subring of A∞ for all k ≥ 0. Kperf consisting of formal series with |ak | ≤ A Just as in the proof of Proposition 2, we show that AKperf is actually a skew field. Its elements −m can be written in the form tq ◦ c where c is an invertible element of the ring RKperf ∈ AKperf ∞ P k ak tq . In contrast to the case of the skew field AK , in AKperf the of formal power series q −m

k=0

by c is indeed the composition of (locally defined) functions, so that multiplication of t AKperf consists of fractional power series understood in the classical sense. Of course, AKperf can be extended further, by considering K or K c instead of Kperf . The above reasoning carries over to these cases (we can also consider the ring RK c of locally convergent Fq -linear power series as the initial ring). In each of them the presence of a fractional −m composition factor tq 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 !qj+λ ∞ XX X n +···+n n 1 k−1 1 ci+1 = µi + ai , i = 1, 2, . . . , (3) Bjkl cn1 cqn2 · · · cqnk j+l=i l6=0

k=1

n1 +···+nk =l

(here and below n1 , . . . , nk ≥ 1 in the internal sum), with coefficients from K c , such that j |µi | ≤ M, M > 0, |Bjkl| ≤ B kq , 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 c1 ∈ K c , the sequence determined by the relation (3) n satisfies the estimate |cn | ≤ C q , n = 1, 2, . . ., with some constant C ≥ 1.

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Proof. Set cn = σdn , |σ| < 1, n = 1, 2, . . ., and substitute this into (3). We have di+1 = µi

(

∞ XX

j+l=i l6=0

X

Bjkl

k=1

n1 +···+nk−1 n1 ) σ (1+q +···+q

q j+λ

−1

n1 +···+nk =l

j+λ n1 +···+nk−1 q n × dn1 dqn21 · · · dqnk Here

)

+ σ −1 ai .

(1+qn1 +···+qn1 +···+nk−1 )qj+λ −1 σ ≤ |σ|kqj+λ−1 ,

and (under our assumptions) choosing such σ that |σ| is small enough we reduce (3) to the relation di+1 = µi

∞ XX

j+l=i l6=0

X

bjkl

k=1

n1 +···+nk =l

n1 dn1 dqn2

n1 +···+nk−1 · · · dqnk

qj+λ

+ σ −1 ai ,

i = 1, 2, . . . ,

(4)

where |bjkl | ≤ 1. It follows from (4) that |di+1 | ≤ M max sup j+l=i l6=0

max

k≥1 n1 +···+nk =l

) ( qj+λ n +···+n n 1 k−1 1 , M −1 σ −1 ai . max |dn1 | · |dqn2 | · · · |dnk |q

−1 −1 Let B = max 1, M, |d1 |, M sup |σ ai | . Let us show that i

|dn | ≤ B q

n−1 +q n−2 +···+1

,

n = 1, 2, . . . .

(5)

This is obvious for n = 1. Suppose that we have proved (5) for n ≤ i. Then |di+1 | ≤ M max sup

max

j+l=i k≥1 n1 +···+nk =l

× · · · Bq

n −1 n −2 n1 +n2 −1 +q n1 +n2 −2 +···+q n1 1 +q 1 +···+1 Bq · Bq ···

n1 +···+nk−1 +nk −1 +q n1 +···+nk−1 +nk −2 +···+1

+ q n1 +···+nk−1

qj+1

≤B·B and we have proved (5). Therefore |cn | ≤ |σ|B for some C, as desired.

q n −1 q−1

6

≤ Cq

n

≤ M max B q

j+l +···+q j+1

j+l=i

q i +q i−1 +···+q

i

= B q +q

i−1 +···+1

,

4

Implicit functions of algebraic type

In this section we look for Fq -linear locally holomorphic solutions of equations of the form P0 (t) + P1 (t) ◦ z + P2 (t) ◦ (z ◦ z) + · · · + PN (t) ◦ (z ◦ z ◦ · · · ◦ z) = 0 | {z }

(6)

N

where P0 , P1 , . . . PN ∈ RK c . Suppose that the coefficient Pk (t) =

P

j

ajk tq is such that a00 = 0,

j≥0

a01 6= 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 RK c , and we can rewrite (6) in the form z + Q2 (t) ◦ (z ◦ z) + · · · + QN (t) ◦ (z ◦ z ◦ · · · ◦ z) = Q0 (t) {z } |

(7)

N

where Q0 , Q2 , . . . , QN ∈ RK c , that is

Qk (t) =

∞ X

j

bjk tq ,

j=0

j

|bjk | ≤ Bkq ,

for some constants Bk > 0, and b00 = 0.

Proposition 4. The equation (6) has a unique solution z ∈ RK c satisfying the “initial condition” z(t) −→ 0, t → 0. t Proof. Let us look for a solution of the transformed equation (7), of the form z(t) =

∞ X

i

ci tq ,

i=1

ci ∈ K c ;

our initial condition is automatically satisfied for a function (8). Substituting (8) into (7) we come to the system of equalities q j N X X n1 +···+nk−1 n1 X ci = − bjk cn1 cqn2 · · · cqnk + bi0 , k=2

j+l=i j≥0,l≥1

n1 +···+nk =l nj ≥1

(8)

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 ∈ RK c . More generally, let

P1 (t) =

X

j

aj1 tq ,

ν ≥ 0,

j≥ν

aν1 6= 0.

Then the equation (6) has a unique solution in RK c , of the form z(t) =

∞ X

i

ci tq ,

i=ν+1

The proof is similar.

7

ci ∈ K c .

5

Equations with Carlitz derivatives

Let us consider the equation dz(t) =

∞ X ∞ X j=0 k=1

kq j

j

ajk τ (z ◦ z ◦ · · · ◦ z)(t) + {z } | k

∞ X

aj0 tq

j

(10)

j=0

qj

where ajk ∈ K c , |ajk | ≤ A (k ≥ 1), |aj0| ≤ A , A ≥ 1. We look for a solution in the class of Fq -linear locally holomorphic functions of the form z(t) =

∞ X

k

ck tq ,

k=1

−1

(11)

ck ∈ K c ,

thus assuming the initial condition t z(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 j the same form, but with the coefficients aj0 γ q instead of aj0 , 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. k−1 1/q qk = ck [k]1/q tq , Assuming (12) we substitute (11) into (10) using the fact that d ck t k

k ≥ 1, where [k] = xq − x. Comparing the coefficients we come to the recursion !qj+1 ∞ X X X n +···+n n 1 k−1 1 cn1 cqn2 · · · cqnk + ai0 , i ≥ 1, ci+1 = [i + 1]−1 aqjk j+l=i j≥0,l≥1

k=1

n1 +···+nk =l

where c1 = [1]−1 aq00 . This already shows the uniqueness of a solution. The fact that |ci | ≤ C q for some C follows from Proposition 3.

i

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 [5]), some of equations (10) can have also non-holomorphic solutions, in particular those which are meromorphic in the sense of Sect. 2. As an example, we consider Riccati-type equations dy(t) = λ(y ◦ y)(t) + (P (τ )y)(t) + R(t)

where λ ∈ K c , 0 < |λ| ≤ q

−1/q 2

,

(P (τ )y)(t) =

∞ X

k

pk y q (t),

k=1

pk , rk ∈ K c , |pk | ≤ q

−1/q 2

(13)

, |rk | ≤ q

−1/q 2

R(t) =

∞ X k=0

for all k. 8

k

rk tq ,

Theorem 2. Under the above assumptions, the equation (13) possesses solutions of the form 1/q

y(t) = ct

∞ X

+

n

an tq ,

c, an ∈ K c , c 6= 0,

n=0

(14)

where the series converges on the open unit disk |t| < 1. Proof. For the function (14) we have dy(t) = c

1/q

[−1]

1/q q −2

t

+

∞ X

1/q q a1/q t n [n]

n−1

[−1] = x1/q − x,

,

n=1

(y ◦ y)(t) = c ct1/q + 1

= c1+ q tq

−2

∞ X n=0

qn

an t

!1/q

+

∞ X

an ct1/q +

n=0

∞ X

qm

am t

m=0

!q n

∞ ∞ −1 X n X X n+1 l n 1/q 1/q + ca0 + ca0 tq + can+1 + cq an+1 tq + tq an aqm . n=0

Finally, (P (τ )y)(t) =

∞ X

pk+1 c

l=0

q k+1 q k

t +

k=0

∞ X

tq

l

l=0

X

m+n=l m,n≥0

i

pi aqj .

i+j=l i≥1,j≥0

Comparing the coefficients we find that 1/q

c = λ−1 [−1]1/q , 1/q

al+1 ([l + 1]1/q − λc) − λcq

l+1

al+1 = λ

a0 + a0 = 0, X

m+n=l m,n≥0

n

an aqm +

X

i+j=l i≥1,j≥0

(15) i

pi aqj + rl ,

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 q q q q l+2 q (17) al+1 [l + 1] − λ c al+1 − λ c al+1 ≤ q −1/q . Suppose that |al+1 | > 1. We have λq cq = [−1], so that |λq cq | = q −1/q , and since |[l+1]| = q −1 and |c| ≥ 1, we find that l+2 |al+1 [l + 1]| < |λq cq al+1 | < λq cq aql+1 .

ql+1 q Therefore the left-hand side of (17) equals |λ c | · c · al+1 > q −1/q , and we have come to a contradiction. q q

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References [1] L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), 137–168. [2] L. Carlitz, Some special functions over GF (q, x), Duke Math. J. 27 (1960), 139–158. [3] D. Goss, Fourier series, measures, and divided power series in the theory of function fields, K-Theory 1 (1989), 533–555. [4] I. N. Herstein, Noncommutative Rings, The Carus Math. Monograph No. 15, Math. Assoc. of America, J. Wiley and Sons, 1968. [5] E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, 1969. [6] A. N. Kochubei, Fq -linear calculus over function fields, J. Number Theory 76 (1999), 281–300. [7] A. N. Kochubei, Differential equations for Fq -linear functions, J. Number Theory 83 (2000), 137–154. [8] A. N. Kochubei, Differential equations for Fq -linear functions, II: Regular singularity, Finite Fields Appl. 9 (2003), 250–266. [9] R. S. Pierce, Associative Algebras, Springer, New York, 1982. [10] B. Poonen, Fractional power series and pairings on Drinfeld modules, J. Amer. Math. Soc. 9 (1996), 783–812. [11] D. Thakur, Hypergeometric functions for function fields II, J. Ramanujan Math. Soc. 15 (2000), 43–52. [12] C. G. Wagner, Linear operators in local fields of prime characteristic, J. Reine Angew. Math. 251 (1971), 153–160.

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