Proc. Nat!. Acad. Sci. USAVol. 81, pp. 1926-1930, March 1984Mathematics

On some applications of diophantine approximations(Padd approximations/linear differential equations/E-functions/measure of rational approximations)

G. V. CHUDNOVSKYDepartment of Mathematics, Columbia University, New York, NY 10027

Communicated by Herbert Robbins, October 12, 1983

ABSTRACT Siegel's results [Siegel, C. L. (1929) Abh.Preuss. Akad. Wiss. Phys.-Math. Ki. 1] on the transcendenceand algebraic independence of values of E-functions are re-fined to obtain the best possible bound for the measures ofirrationality and linear independence of values of arbitrary E-functions at rational points. Our results show that values of E-functions at rational points have measures of diophantine ap-proximations typical to "almost all" numbers. In particular,any such number has the "2 + e" exponent of irrationality: 10- p/qj > lqlj2 for relatively prime rational integers p,q,with q > qO (0, E). These results answer some problems posedby Lang. The methods used here are based on the introductionof graded Pade approximations to systems of functions satisfy-ing linear differential equations with rational function coeffi-cients. The constructions and proofs of this paper were used inthe functional (nonarithmetic case) in a previous paper [Chud-novsky, D. V. & Chudnovsky, G. V. (1983) Proc. NatW. Acad.Sci. USA 80, 5158-5162].

In §1 we define, study, and prove the nondegeneracy ofgraded Padd approximations (GPA) to solutions of linear dif-ferential equations with rational function coefficients. Wefollow Siegel's studies (1) in applying GPA to diophantineapproximations of values of the E-functions of Siegel at ra-tional points. In §2, where the definition of E-functions usedin this paper is presented, the methods of §1 are applied andthe main result of the paper is proved:THEOREM I. Let fi(x), ..., f,,(x) be E-functions satisfying

linear differential equations over 0(x). Then for any E > 0and an arbitrary rational number r, r :& 0, there exists a con-stant co = co(E, r, fl, ..., fQ) > 0 with the following property.For arbitrary nonzero rational integers H1, ..., Hn and H =max (IH11, ..., JHI), if Hjf1(r) + ... + Hnfn(r) #& 0, then

jHjf1(r) + --- + Hnfn(r)l > JH1 ... Hnl-"Hl-Eprovided that H 2 co.COROLLARY. Let fi(x), ..., fn(x) be E-functions satisfying

linear differential equations over 0(x). Then every element 6of the field Q(f1[01, .., fn([0]) = U{0(f1(r), ..., fn(r)): r E 0}has the "2 + E" exponent of irrationality: for any e > 0 andfor arbitrary relatively prime integers p and q,

60 - p/qj > lql-2-Eprovided that IqI 2 qo(o, e).THEOREM II. Let K be the field, obtained by the addition

to Q of all values f(r) of E-functions f(x), satisfying lineardifferential equations over 0(x), at rational r # 0. Then forany E > 0 for arbitrary elements 01, ..., On ofK such that 1,O1, ..., On are linearly independent over 0 and arbitrary ra-tional integers q, qj, ..., qn we have

1q, qnljl+6*10,ql + -.. + 0nqnq1 > 1

and

IqI" j ---~.l-.IIO6qII> 1

provided that Iq1 .q.,I > c' and IqI > c'. Here c' = c'(01,.On, E) > 0 and II is the distance to the nearest integer.The second inequality in Theorem II follows from the first

one by the transfer principle. Moreover, if in Definition 2.1of E-functions we can replace mem by cm, then E in the sec-ond inequality of Theorem II can be replaced by (logloglql)-Y, y > 0. Bounds of Theorem I and Theorem II wereproved in 1964 in ref. 2 by Baker, when fi(x) = erix and O6 =eri for rational numbers ri.The idea of constructing the approximating form (though

in a simpler case) was proposed and carried out by Siegel (1),who examined the important property of the normality ofsystems of functions employed in §1. Complete proofs arepresented for normal systems of functions. For E-functionsfor which the normality condition was established in ref. 3our results provide, in particular, a solution to Lang's prob-lem (4) on the "2 + E" exponent of irrationality for values ofBessel functions. For values of exponential functions (con-nected with the Lindemann-Weierstrass theorem) we referto ref. 5.

§1. Graded Pad6 Approximations

We study GPA to solutions of n linear differential equationswith rational function coefficients. Let fi = fi(x): i = 1, ....n, be a nonzero solution of matrix linear differential equationof the first order over C(x):

d ki

f ) = L Aj(,,fi):i =1, ...,ki [1.1]

for A() E C(x):jl = 1,..., ki and f )def fi: - 1,..., n.Equivalently, fi = fi(x) is a solution of a scalar linear differ-ential equation of the order ki over C(x)

dki dkilfi(x) + ak d fi(x) + ... + a'f&(x) = O. [1.1']

da(i) E +x): I = 0, ..., ki - 1; i = 1, ..., n. Any equation 1.1'can be reduced to the form 1.1 if one puts

fi 5J) = d i-'

A(') = S ill - 8,k, a(Y)? forj, l = 1, ..., ki; i = 1, ..., n.We assume that functions fi: i = 1, ..., n are linearly inde-

pendent (over C) and that all functions f (i)(x) are regular at x=0:j = 1, ..., ki; i = 1, ..., n.For number-theoretic applications we do not need a GPA

with the maximal possible order of zero at x = 0 for the re-mainder function. Instead, we use only E-GPA with an order

Abbreviation: GPA, graded Pad6 approximation(s).

1926

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of zero slightly smaller than the maximal possible one. Ourparameters are: (i) N-the level of the graded structure; (ii)E-"the defect in the order t of zero of the remainder func-tion at x = 0"; and (iii) Di: (i = 1, ..., n) the weights of the E-GPA (degrees of Pade approximants in x). The integer pa-rameters Di are sufficiently large with respect to N and e1and we put D = max(Di: i= 1, ..., n).We denote (for fixed kj, ...,k)

MN def (

defn (N+i)-MN Mi=l ki - 1 mdei=l

Pad6 approximants and remainder functions are enumerat-ed by multiindices J = (...; ai1,, ..., aik.; ...) from Zkl X *-- X

Zkn with nonnegative integers aij. To simplify notations weintroduce generating functions in n groups of variables ei =(Ciji, ..., Ck.)ii)= 1, ... n and we put c = (el, ..., en).

Definition 1.1 (of the E-GPA). Let Pi(xIe):i = 1, n bepolynomials in x and c = (el, ..., en) not all identically zero,

of degrees of at most D in x with zeroes at x = 0 of orders atleast D - Di and such that the remainder function

n ki

R(xIC) = P1(XIC)o ci1f n)(x4 [1.2]i=l j=1J

has a zero at x = 0 of an order of at least t for any choice of c

= (l, ..., en). Also let the polynomial Pi(xlc) be homoge-neous in each of the groups of variables C = (cjl,i-, Cj,k,): j+ i of degree N, and homogeneous in variables ei of degreeN- 1:i = 1,..., n. IfDi> ED, t2 -(2lMNjIDi -ED)/SNand t 2 Si 1 Di - ED, then Pi(xIc) are called Pade approxi-mants and R(xlc) is called a remainder function in the E-GPAproblem corresponding to parameters (N, t, Di, D).Remark. The total number of undetermined coefficients of

all polynomials P (xIc):i = 1,. n is =1l(Di + 1)MNi. Onthe other hand, R(xJe) is a homogeneous polynomial in eachgroup of the variables ei:i = 1, ..., n of degree N. Hence, thenumber of linear equations imposed on coefficients Pi(xjc)that define the remainder function in Eq. 1.2 is at most t SN.Thus, if t < EI4L(Di + l)MN i/SN and N is sufficiently large(so that 1 - E/n < MNi/SN:i = 1, ..., n), a nontrivial solutionto the e-GPA problem with parameters (N, t, Di, D) (and t 2XInlDi - ED) always exists.We can use differential equations 1.1 to construct, starting

from a given E-GPA and differentiating, other E-GPA withslightly different parameters (N, t', D;, D'). To prepare thenotations, let J = (...; aij, ..., ai,k,; ...) E Zk, X ... X Zk" be a

multiindex with nonnegative integers a1,j. We put IIJ Ij = N iffor any j =A i, Ik ajj = N while >;t11 aij = N - 1; and we

put IJI = N if for all i = 1, ..., n 11kl ai = N. In thesenotations polynomials Pi(xlc) are generating functions ofPi, (x) with J varying over J E Zki x

... X Zkn with 11J11i = N.If J(i, j) denotes the (i, j)th component of J, we put CJIHI cllf1C for i = 1, ..., n and] = 1, ..., k. We denoteby eij the unit vector from Zkl X zkn with 1 on the(i,j)th place. Now let d(x) be a common denominator of allrational functions Afl A (x): j, l = 1, ..., ki; i = 1, ..., n, inEq. 1.1. We then define inductively new (generating func-tions of) Pade approximants

P(m) (xWe) = p(m)(X).,J,llJlli=N

where P(O) (xlI) def Pi(xl), and for m = 0, 1, ....

p(m+l) (x) = d(x) -{d prM) (X) - E A (.'5x)i~ x(Jil) + 1) jI1=1 J(ij)'()

x (JU~) + 1)' i,J+ei,-ei~j (x)J [1.3]

The properties of the new Pade approximants p (xje) andthe remainder function

n ki

R(m) (xIe) = s p(m) (XJC) E cijf(i) (X)1=1 j=l

are summarized in the following:LEMMA 1.1. Let d = max{deg(d(x)) - 1, deg(d(x)A(')(x)):

jl=1,.. ki; i = 1, ..,n}, and let Pi (xlc): i = 1,) ...,. n andR(xl) be Pade approximants and the remainder function inthe e-GPA with parameters (N., t, Di, D). Then for m - 0,p(m) (x|C) are polynomials in x, c, ofdegrees ofat most D +md in x, with zeroes at x = 0 of orders D - Di - m, and arehomogeneous in variables Cj ofdegree N - Nj: ij = 1, ..., n.Also, the order ofzero ofthe remainderfunction R(m) (xld) isat least t - m for any choice of c. Hence pjm) (XNO) andR(m) (xlj) are Pade approximants and the remainder func-tion in the e-GPA with parameters (N. t - m, Di + md + m,D + md):i = 1, ..., n.The proof of Lemma 1.1 is a direct consequence of the

definition 1.3 of Pade approximants p(m) (x|C) and linear dif-ferential equations 1.1. For example, we have the inductivedefinition of the coefficients R(m) (x) of the remainder (gen-erating) function R(m) (x4c) = Y. ,II=N R(m) (x)eC' similar to1.3:

R (m+l) (x) = d(x) d R (m) (X)-d (X)

x (I(il) + 1)R? ),.e j (x)}.

In particular, we obtain a system of Pade-type approxima-tions to f1(x), ..., fn(x) at x = 0 if we put I = Io, where Io =

,0, ..., 0; ...; N,, ...,.O).We get form = O,1, ..., R(m) (x) =in1 Pi)e- I fi dMI 1 P( ) (x)fi(x). If Pi (xkc) and R(xlc) are

the Pade approximants and the remainder function in the E-GPA with parameters (N. t, Di, D) then Rk, (x) is a remain-der function in the Pade-type approximation problem tofl(X), ... fn(x) at x = 0 with an order of zero at x = 0 of atleast t - m for m = 0,1, ..., according to Lemma 1.1.For number-theoretical applications we need to have n lin-

early independent remainder functions of the form R$mo) (x)[or linear forms approximating f1(x), ..., fn(x) in the sense ofSiegel (1)]. These n linearly independent forms always exist,as the following general result that we present shows.THEOREM 1.1. Let Pi (xlj) and R(xld) be the Pade approxi-

mants and the remainderfunction in the E-GPA at x = 0 withparameters (N. t, Di, D). Let P|m)(x)for IIJ11i = N, i = 1, ..., nand m 2 0 be defined as above in 1.3, and let I = Io - eij forIo = (N,0, ..., 0; ...) as above. Then for any xO # 0 such thatd(xo) :§ 0, and sufficiently large D 2 Do(f1...f, e, N), therank of the following matrix is n:

n

(P (k)(x0): i =1. n): m = 0, .Z..9 Di + ED - t.

This implies, in particular, that, for arbitrary numbers a1,an, not all zero, there exists an m,O c m c 2' Di + eD

- t such that 1;'=i aiP<j'7(xo) 7& 0.Remark. The upper bound for m cannot in general be sig-

Mathematics: Chudnovsky

N+ ki- 2 n N+k,- 1x F1

ki 1 j=l;j.i ki 1

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1928 Mathematics: Chudnovsky

nificantly improved [we note that t 1;'= Di - eD.] How-ever, as we see below, under Siegel's normality condition,this bound improves considerably:

ndm ' E MNi Di + - MN - {tSN - 12MN} + CO.

b-l 2

This bound is already the best possible (for a sufficientlylarge D), because in the definition of the E-GPA, t can betaken as any, integer less than Y.;1 MNi (Di + 1)/SN.The definition of the GPA to the system of functions fi(x)

can be represented in terms of the graded structure of sub-modules of Picard-Vessiot extensions generated by symmet-ric powers of solutions of linear differential equations ad-joint tp 1.1. For i = 1, ..., n the equation 1.1 in matrix form is

d 'gi=Ai)t, [1.4]dx

where Xi is a fundamental ki x ki matrix of solutions of 1.1,the first column being (f W:j = 1, ..., ki)'. Linear differentialequations adjoint to 1.4 have the form

d-Fi =-A (')'F. [1.5]dyI

for Fi(y) de (4,{y)'f1: i = 1, ..., n. Then for F(y) = diag(Fi(y), ..., Fn(y)) and A(y) = diag (A(') (y), ..., Ain) (y)) wehave

d-F =-A'-F. [1.5']dy

If fi satisfy a scalar linear differential equation 1.1' (so that4, is the Wronskian matrix of 1.1'), then equations 1.5 arescalar differential equations adjoint to 1.1'.We now need symmetric power of the space of solutions

of the matrix equation 1.5'. As usual, for a k x k matrix B wedenote by PN(B) the Nth induced matrix of B. For any i = 1,

n we denote by A(i, N) the matrix PN(Al) ) ... 0 PN-1(A,) 0 ... 09 PN(An), and similarly F(i, N) def PN(Fl) 0 .. 0PN(Fi-)OPN-l(Fi) ) ... )0 PN(Fn). Then F(i, N) satisfiesthe natural differential equation

- F(i, N) =-A(i, N)'F(i, N): i = 1, ..,n. [1.6]dy

Our approach follows Siegel's studies (1), and, followingSiegel, we introduce normality conditions on (symmetric)powers of solutions of linear differential equations 1.5.Along the lines of ref. 1, we call the system of functions fi(x)normal if the fundamental matrices Fi(y) of 1.5 and the cor-responding fundamental matrices F(i, N): i = 1, ..., n of 1.6for all N . 1 are linearly independent over C(y). The lipearindependence of Fi = (Fij,l)f,=1: i = 1, ..., n over C(y) meansthat any linear relation Y li=1 pij(y)C,,1Fij,1 (y) 0 withconstants Ci,1 and polynomials Pij(Y) implies that all prod-ucts pi,jCi,l are zero.We point to the criterion of ref. 6 (exercise 12 of section 6,

chapter VI) of normality of fi(x): i = 1, ..., n in terms ofalgebraic independence of elements of matrices Fi(y) only.THEOREM 1.2. Let; in the notations above, the system of

functions f1(x): i = 1, ..., n be normal and let Pi(xld): i - 1,n and R(xld) be the Pade' approximants and the remain-

derfunction in the E-GPA at x = 0 with parameters (N, t, Di,D). Let P0(jx) for IIJ1II = N: i = 1, ..., n and mn 20 be definedas above in 1.3. Then, if t > (viny= MN,iDi)/SN - Di/S2 + EDfor i = 1, ..., n and D - D1(fi ...f, E, N), the determinant

of the MN X MN matrix

M(X) = (P.<J(x): iiJII = N; i = 1, ..., n)m=O..MN-1

is not identically zero.COROLLARY 1.1. Let us preserve all the notations ofTheo-

rem 1.2. If t > (II'=lMNiDi)/SN - D N/SN + EDfori 1.n and D 2 Di(fl, , fn, E, N), and ifx0 #& 0 is not a singular-ity of the system ofequations 1.1 (i.e., d(xo) :& 0), then for M= In i MNjDj + (d/2) MN - {tSN - N/2M2} + CO, the rank ofthe matrix

(P~m"(xo): liJ1i = N; i = 1, ..., n)m=o..M

is exactly MN, with CO depending on fVj) and N.It follows from Corollary 1.1 that the rank of the matrix

(P~M) (xo): i = 1, , n)m=o,...,M is n, which is an improve-men't over Theorem 1.1 (see the remark after Theorem 1.1).ProofofTheorem 1.2. Let ci(y) = (cij, ..., cika) be a solu-

tion of 1.5 with some initial conditions fi:

ei(y)' = Fi(y) s{: i = 1, ..., n.

We consider the following symmetric product of vectorsei(y), ..., en(y): C(i, N) def Cj(y) *N0 0 1(y) *(N-W00en(y) *N. where ei(y) *N - CO(y) K. .*ei(y) is the Nth sym-metric power of ei(y): i = 1, ..., n. The vector C(i, N) enu-merates all monomials ej in P(ijc). We substitute C(i, N) =f(i, N)(y) into the generating function Pi(xlc) and put x = y:Pi(y) dd Pi(yI(i, N)): i = 1, ..., n. Then Pi(y) = YJIIJiIJ=NPijj(y).c(y)J [is a scalar product of C(i, N) and of the vector(P;,j(y): IlJ1i = N)]. Since c(i, N)(y) satisfies a matrix first-order system [1.6], any differentiation of Pi(y) is again a lin-ear combination of e(y)J (with rational function coeffi-cients). Moreover, the comparison of 1.5, 1.6, with 1.3 im-mediately implies

(d(y) ) Pi(y) =I P(')(Y) ) [1.7]

i=1,...,nform=0,1,....We use the properties of normal systems of functions giv-

en by Siegel (1). As in ref. 1, we have r systems of lineardifferential equations d( Yk,,)/dy = 1lm Y,,, (k = 1,

nm,; t = 1, ..., r) regular at y = 0, and T = T(y) is acommon denominator of rational functions Qk,l,,(y). Forfixed polynomials Pk,,(y) and an arbitrary linear com-bination R of functions Pk,t(Y)Yk,t(y), R = Y4=, lk-lCk,tPkY(y) Yk,,(y) we define, using the differential equa-tion for Yk,,: R ~m)= (T(y)d/dy)mR, R(m) = r=

-k1Ck,tPk,(Y)Ykt(y). Let q = max{deg(T(y)), deg-(T(Y)Qkl,(y))} and max{deg(Pk,,(y)): k = 1, ..., m,; t = 1,r} < D. In these notations we haveLEMMA 1.2. If there are s linearly independent functions

of theform R, l-th ofwhich has a zero at y = 0 oforder ofatleast ul: 1 = 1, ..., s, then always 1u1- tl mt}2/2 < p +Xt= mt{D + q( t[1 mt)/2}. Now if I=1u - {tf mt-1}2/2> p + { mt - 1} {D + q(1 m -1)/2}, then the deter-minant A(y) = det(Pkm)(y): k = 1, ..., Mt; t = 1, ..., r) m =0,1, ... 1m1 - 1, is not identically zero and has a zero at y= 0 of order at least Is u, - {E4=1 mt}2/2 - p.Here p depends only on the maximal order of zeroes of

Yk,t at y = 0. Proof of Lemma 1.2 for normal systems offunctions follows from ref. 2. In the general case the proof isa direct consequence of results of refs. 5 and 7 (and proofspresented there).We apply Lemma 1.2 to the following (normal) systems of

functions. They are components of various symmetric prod-ucts C(i, N) = ..(y)*N0 0C(y) *(N1)0.: i - 1 n.

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Normality of systems of functions c(i, N) and Lemma 1.2imply that various remainder functions R that are linear com-binations of Pi, j(y)e(y)J are linearly independent (by the as-sumption of E-GPA and Theorem 1.2). Then it follows fromLemma 1.2 that det M(x) # 0. Theorem 1.2 is proved.Proof of Corollary 1.1. According to Theorem 1.2, det

M(x) # 0. From the definition of M(x) and Lemma 1.1 itfollows that det M(x) is a polynomial in x of degree of at mostMND + (d/2)M2 since degx(Pi(j)(x)) c D + md.From the proof of Theorem 12 it follows that det M(x) has

a zero at x = 0 of order of at least tSN + Y. 1 MN,1(D - Di) -N- Po, where po depends only on f 1, ..., f,,. Thus we can

write det M(x) = Xa.AO(x), where Ao(x) is a nonzero polyno-mial of degree of at most MO C ;i=1 MNjDi - tSN + NMN (1+ d) + po. For a given xO 7 0, let a be the order of zero ofAo(x) at x = xO. In applications of Corollary 1.1 we use therepresentation 1.7 of Pi(y) with c having the following initialconditions: ci,j(y)ly=x, = 16 -fi(xo). Applying the differentialoperator d(y)d/dy to (d(y)d/dy)mPi(y) = YJIJI=N P,()(y)4(y)J a times we obtain, starting from M(xo), a matrix(PE(')(xo): IIJ11i = N: i = 1, ..., n) m = 0 ..., MN + a - 1having rank MN. Since a . MO, Corollary 1.1 is proved.

§2. E-Functions

Definition 2.1. Let f(x) be a solution of a linear differentialequation over 0(x), regular at x = 0 and such that f(x) =1;=o amxm/m! with am E 0. Let Iaml c mem and denom{ao,a,, ..., am} c maim for m 2 mo(e) for every E > 0. Then f(x) iscalled an E-function (of Siegel).To prove Theorem I, let f1 = f1(4, ..., fn = fn(x) be arbi-

trary E-functions satisfying linear differential equations over0(x) and linearly independent over C(x).

Let fi satisfy a scalar linear differential equation of theform 1.1' over 0(x) of order ki: i = 1, ..., n. In the notationsabove of 1.1, let f V) (x) = (d/dx)-1 fi(x): j = 1, ..., ki; i = 1,

n. We fix an E, 0 < E < 1/2, and a parameter N. Theweights Di: i = 1, ..., n of the Pade approximants below aresufficiently large integers, depending on the approximationof the linear form in f1(r), ..., fj(r) only. We put D = max(Di:i = 1, ..., n), Di > ED for i=1, ..,n.THEOREM 2.1. For any D D2(f1O ...f, E, N) there exist

polynomials Pij(x): J E Zk, X .. X Zkn; IJJIti = N: i = 1, ..., n,not all zero and with integer rational coefficients, such thatthe following conditions are satisfied.

(1) Polynomials Pi j(x) have degree in x at most D, zeroesat x = 0 of orders at least D - Di, height at most Di!DED,and are of the form

DD

PiJ(x) = Z Pj~mXM [2.1]m=D-Dm

IJII = N: i = 1, ... n. Here Pi,Jm are rational integers ofabsolute value at most DED (m = D - Di, ..., D). We alsoformally put Pi,J,m = 0 ifJ has a negative component or m <D - Di: i = 1, ...,n.

(2) For any I E Zk, X ... X Zkn, III = N, the (remainder)function

n kiR1(x) def Z Pi l-e j (X)f(j)(X) [2.2]

i=1 j=l

has a zero at x = 0 of order at least M, where f~j)(x) =(d/dx)-l fQ(x) and

M = [(MNiDi - eD)/SN]

Proof. Let fi(x) = Ym=o am,ixm/m! and fP)(x) =-'

am,i,jx n/m!: i = 1, ..., n;j = 0,1, ..., ki. We obtain an expan-sion ofRI(x) at x = 0 in terms of am ij and piJ,m. Namely, for

R(x) = I DIm=0 m!

we obtain the following representation for rl m:

n kirj,m,,e Z I

i=l j=1,J(ij)a1

min(m,D)

I=D-Di[2.3]

III = N, and I(i, j) denotes the (i, jDth component of I. Theonly conditions that determine polynomials Pi,j(x): IIJIII =N: i = 1, ..., n, are given by the following system of MSNlinear equations on coefficients pijm:

rlm= 0: m = 0,1, ...,M -1; II = N [2.4]

(in the notation of 2.3). The total number of unknowns Pi,jmin 2.4 is IIn'= MN,i(Di + 1). Finally, according to the defini-tion of E-functions and 2.3, the coefficients at pijm in thesystem 2.4 of linear equations are rational numbers whoseabsolute values and common denominator are uniformlybounded by DWD for any 8 > 0 and D . D3(f1, . f, 8 ).Thus we can use Siegel's lemma (1, 2) and find a nontrivialsolution pi,j,, of the system of equations 2.4 in rational inte-gers and such that max{Ipi,jrI: IIJIi = N: i = 1, ..., n; m = D- Di, ..., D} c bED provided that D 2 D2(f1, ..., fn E, N).Theorem is proved.As in §1 we construct generating functions of Pij(x) and

R1(x) we obtain polynomials in auxiliary variables cij: j = 1,k,; i = 1, ..., n: Pi(xIe) = YJIJII.=N PiJ(X)C j, R(xl) =

I,III=N R1(x) ec. Then according to conditions 1 and 2 ofTheorem 2.1, Pi(xlc) and R(xjc) are Pade approximants andthe remainder function in E-GPA at x = Q with parameters(N, M, Di, D) (see the definition ofM in Theorem 2.1). Weput P 1°j(x) def Pi j(x), R 0O(x) de-R1 (x). Then we can use Lem-ma 1.1 to obtain new approximating forms R~M)(x) = 1,7=Ej=1 I(i j);J<P,7e;,j (x)f Wj(x) with polynomials P(m)_ j i,j(x), IjJI1i = N: i = 1, ..., n defined inductively for m 2 0 in1.3. We need a simple upper bound on the sizes of polynomi-als P (j) (x) and upper bounds of Rim) (x).LEMMA 2.1. In the notation above, polynomials PIJ)(X)

have degrees at most Di + md, zeroes at x = 0 of orders atleast D - Di - m andheights at most Di!DED (D + d cl,where c1 = c1(f1, ..., f, N) > 0 depends only on the system oflinear differential equations satisfied by f'? and N: j = 1,.ki; IIJI1i = N: i = 1, ..., n, and m = 0,1,. For m = 0,1,and given x + 0 we also have

|R(m) (x)l . m3m.cM.cM M(-1+2E)M

with C2 > 0, C3 > 0 depending only on the system of lineardifferential equations satisfied by f~j)(x), on x and N: III =N.Proof of Lemma 2.1. The upper bounds for degrees of

Pm)(x) follow from Lemma 1.1, while the upper bounds ofheights ofP (mj)(x) are direct consequences of the recurrenceformula 1.3. To obtain the upper bounds of R(i)(x) we usethe recurrence definition 1.3, from which it follows that R<,(x) is a linear combination of functions (d/dx)' R1'(x) withcoefficients bounded in absolute value by m2m.cT. To esti-mate from above (d/dx)m' RI(x) at x :& 0, we use property 2of Theorem 2.1. We have, in the notation of the proof ofTheorem 2.1,

RrRI,(x) = E!

fl^MX .

Mathematics: Chudnovsky

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Using definition 2.3 and the upper bounds on heights ofpolynomials Pi j(x) in property 1 of Theorem 2.1, we imme-diately obtain

|d)Ml

(x) sD E m)2,E (~m-m')\dx/ M=M (m -

s in'! > (m! E+2e2mlxlm-m' < m.c3m(M!)-M=M

This proves Lemma 2.1.Following Siegel's method (1, 3), it remains only to con-

struct a system of n linearly independent approximatingforms to numbers f1(r), ..., f,(r), r :& 0. For this we useTheorem 1.1 on the nondegeneracy of the E-GPA, followingmethods of ref. 7.THEOREM 2.2. Let r + 0 be a rational number different

from singularities of the system 1.1. Then for any 8> 0 andsufficiently large integers Dj, D - D4 (f1, ..., fn, 8) there existrational integers Pij: i, j = 1, ..., n, such that the followingconditions are satisfied. The determinant of Pjj-det(Pij)!j=l-is nonzero; the absolute values of numbers Pijare bounded by Dj!D6 (i, J = 1, ..., n) and

n n

E Piif(j(r) 1 D1!1.D(1+AD: i - 1 n.j=1 j=1

J

Proof of Theorem 2.2. Let, as above, D be sufficientlylarge with respect to N and E- and Di > eD for all i = 1, ...

n. We use Lemma 2.1 for I = o= (N, 0, ..., 0; N, 0,..., 0;...;N 0, ..., 0) and 1, = I0 - ei,1: i = 1, ..., n. According toTheorem 1.1, the matrix of (P(,,?(r): i = 1, ..., n): m = 0,1,..,MO has rank n, where MO < ;in1 Di + ED - M and asufficiently large D 2 D5(f1, ..., fE,E, N). Then there are ndistinct integers 0 < m1 < ... < m, < Mo such that det(P(',f)(r)Xinj=I + 0. If r = a/b for rational integers a and b, thenaccording to Lemma 2.1, PIi) (a/b) bD+md is a rational in-teger, and we define Pi "= P<./i) (a/b) b D+mid: ij = 1, ..., n.Then Pij are rational integers' with det(Pij)i'j=1 0. FromLemma 2.1 and the upper bound MO 1E4'_ Di + ED - M itfollows that for D sufficiently large with respect to N, and Nsufficiently large with respect to 8-1, we obtain max{IlPi, i= ..., n} < Di!-D8D. From 1.3 it follows that

n

R(m)(x) = E p(m) (x)f (X).

Substituting x = r and using the upper bound of R(i) (r) ofLemma 2.1, we obtain forD sufficiently large with respect toN and N sufficiently large with respect to 8',

n n

IEPi jfj (r)'-nf D>! D (1 + B)D

Theorem 2.2 is proved.According to Siegel (1), the existence of n linearly inde-

pendent approximating forms satisfying the conditions ofTheorem 2.2 gives the lower bound for the measure of linearindependence of numbers fl(r), ..., fO(r).Proof of Theorem I. Let H1, ..., Hn be rational nonzero

integers such that max(lHil: i = 1, ..., n) = H and such that

Hlfl(r) + * + Hnfn(r) = 1, 0 < Ill < 1.

We apply Theorem 2.2 and obtain for any 8 > 0 and for theintegers D1, D 2 D4(fl, ...,f, 8), n linearly independent

forms in f1(r), ..., f,(r) with rational integer coefficients, ofthe form

n

Li= Pijf1(r): i = 1, ..., n

such that max(jPi,1: i = 1,..., n) < DJ!D D and max(LiLl: i =1, ._ n) < llfn_1 Dj!-lD!D8D. The linear independence offorms L1, ..., Ln means that det(Pi j)i1j # 0. Hence we canalways find n - 1 forms, say, L1, ..., L,,1, that are linearlyindependent together with the form 1. We assume, withoutloss of generality, that D = D,. Let V be a matrix formedfrom coefficients of forms L1, ..., L,_1, l; let Al be a deter-minant of V, and Aij be the algebraic complement of the (i,j)th element of V. Then

n-1

fi (r)Al = Z L'j,&,i + lAni: i = 1, ..., n.j=1

[2.5]

From the definition of V it follows that jA,,jI S (n -1)!D( f-l)aDIH"DJ I max(1A1,i: I = 1, ..., n - 1) c (n1)!nD(-2)aD H Dj'D'! Y.=1lHjl/Dj!: i = 1, ..., n. Let i = 1, 2,

n be chosen in a way such that fM(r) #& 0, which is possi-ble since 1 # 0. Then

n-1 nl

If1(r)ilj c n!.D(n-1)8D. ( Dj!'Ill + Z 1Hjl/Dj!).Since A1 :& 0 and all elements of V are rational integers,

1A1 .- 1. We choose now an integer Dj to be the smallestinteger . D4 such that

Ifi(r)KH-1*Alj(2n*n!).D(n-)aD < D>! and Dj > SD + D4:j = 1,., n.

For this definition of Dj we get Ifi(r)l ' 2n! 111.112Dj!.D(n-')"D. This implies a lower bound on Ill:

n

I c41 7 IHil-KH.D-n28Dfor C4 = C4 (f1, ...f, 8, r). For a sufficiently small 8 thisproves Theorem I.Remark. The proof above of Theorem I corresponds to the

case when r is not a singularity of the system of differentialequations 1.1. In this case, as it is easily seen, numbers fI(r),

., (r) are linearly independent over Q. However, Theo-rem I holds with a similar proof, even when r is a singularityof the system 1.1, when f1(r), ..., fM(r) are not necessarilylinearly independent over Q [e.g., r can be a zero of f,{x)].For the proof in this case we use the same system of approxi-mating forms that was constructed in the proof of Theorem2.2, applying Lemma 2.1 and using the regularity of func-tions f1(x) at x = r. It is necessary to divide polynomialsp( J) (x) by the highest power of (x - r) by which they aredivisible.

This work was partially supported by the National Science Foun-dation under Grant MCS-82-10292 and the U.S. Air Force underGrant AFOSR-81-0190.1. Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl.

1. -

2. Baker, A. (1965) Can. J. Math. 17, 616-626.3. Siegel, C. L. (1949) Transcendental Numbers (Princeton Univ.

Press, Princeton, NJ).4. Lang, S. (1965) Bull. Soc. Math. Fr. 93, 177-192.5. Chudnovsky, G. V. (1983) Proc. Natl. Acad. Sci. USA 80,

3139-3141.6. kolchin, E. R. (1973) Differential Algebra and Algebraic

Groups (Academic Press, New York).7. Chudnovsky, D. V. & Chudnovsky, G. V. (1983) Proc. Nati.

Acad. Sci. USA 80, 5158-5162.

Proc. NatL Acad Sci. USA 81 (1984)

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