数理解析研究所講究録 1219 巻 2001 年 185-194

185

On apaper of Hermite and Diophantine
Approximation of Abelian IntegralsbyMarc HUTTNER

UFR de Math\’ematiques

UMR 8524 AGAT CNRS
Universit\’e des Sciences et Technologies de Lille

F-59665 Villeneuve d’Ascq Cedex, France

0Introduction
Acareful study of alargely forgotten article of C. Hermite Sur quelques
equations differentielles lineaires [HER], published in Crelle’s Journal in 1875
gives us many interesting new results about simultaneous approximations ofAbelian integrals.
Hermite’s paper begins (as his title says) with alinear differential equation
which is aparticular case of the s0-called Tissot Pochhammer differential equation. Later it deals with simultaneous approximations to logarithmic functions,both in the classical and hyperelliptic case.
However many proofs are not complete and it is often difficult to restore acorrect version of them. Nevertheless many tools of modern mathematics are
hidden in this paper of Hermite :‘\Psi ad\’e approximation of the second kind, m0nodromy, Picard Lefchetz principle, computation of the determinants of periodsof integrals”, infortunately without arithmetic applications.
This paper was published in 1875, two years after the publication of the proofof the transcendence of . If we carefully study Hermite’s proof, we can see thatboth constructions are similar and uses Pad\’e approximation of the second kind.
It is curious and strange that Hermite uses linear differential equations to study
simultaneous rational approximations of Abelian integrals at distinct points but
one can recall for instance SiegeFs $E$ and $G$ functions are also related to linear
differential equations.
In the following we denote by$\langle\langle$$\rangle\rangle$$e$$\Phi(x)$$=$$X^{m+1}+a_{1}X^{m}+\cdots+a_{m+1}\in \mathbb{Q}[X]$

(0.1)

$=$

$\prod_{0\leq i\leq m}(X-e_{i})$withfield,

for $i\neq j$ , $0\leq i\leq m$ , $0\leq j\leq m$ . I denotes aquadratic imaginary
his ring of integers and for , $1\leq i\leq m$ . For $k\geq 2$ and does not$e_{i}\neq e_{j}$$\mathbb{Z}(I)$

$i$

$k$

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divide $m+1$ , we set(0.2)$f_{\dot{l}}(x)= \Phi(x)^{1/k}\int_{e_{\mathrm{O}}}^{x}\iota^{:-1}\Phi(t)^{-1/k}dt$).
(for asuitable choice of the branch of $\Phi(x)^{1/k}$ and for
$x\in
I$
paper
says
satisfying some arithmetic
that forThe main result of this
, $f_{m}(x)$ are linearly independent overconditions the numbers 1, $f_{1}(x)$ , $f_{2}(x)$ ,and satisfy alinear independence measure condition, (see the theorems 2.1
and 2.2).
$e_{0}\in \mathbb{Q}$$\cdots$$\mathbb{Z}(I)$

1Tissot Pochhammer equation, cycloelliptic curves
and Pad\’e approximation
Let us consider the -cyclic covering of the projective plane$\mathrm{f}\mathrm{c}$$\mathrm{P}_{1}=\mathrm{P}_{1}(\mathbb{C})$(1.1)$V_{k}(t, u)=\{(t, u)\in \mathrm{P}_{1}\mathrm{x}\mathrm{P}_{1}|u^{k}=\Phi(t)\}$(cycloelliptic curve).

Using the Riemann Hurwitz formula it is easy to compute the genus$V_{k}(t, u)$For.

$m:\in \mathrm{Z}^{+}$, $0<m:<k$ , we set$\delta_{\dot{1}}$

$g$

of

$=m:/k$ and(1.2)$v(t)= \prod_{0\leq\dot{|}\leq m}(t-e:)^{\delta:-1}$

the branches of $v(t)$ are solutions of the linear differential equation of the first
order
(1.3)

$v’(t)+ \{\sum_{0\leq\dot{|}\leq m}\frac{1-\delta_{1}}{t-e_{\dot{1}}}\cdot\}v(t)=0$

If k $=1$ (ie. g $=0)$ we shall set $v(t)\equiv 1$ . The differential forms(1.4)$\omega_{x}(t)=\frac{v(t)dt}{x-t}$

are differential forms of the third kind on$V_{k}(t,u)$.

The family of integrals$\varphi_{j}(x)=\int_{\gamma_{\mathrm{j}}}\omega_{x}(t)dt=\int_{\gamma_{j}}\frac{v(t)dt}{x-t}$,$1\leq j\leq m$(1.5)

on $V_{k}(t,u)$ or aand ending at
wheredenotes any path beginning atand .
Pochhammer double loop encircling
do not surround any point “abovFor the moment we assume that the paths$e_{0}$$\gamma_{j}$$e_{0}$$e_{j}$$\gamma_{j}$$e_{j}$

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u).
on the Riemann surface
For this choice of path, we can write for |z| $>1$

$x^{\ovalbox{\tt\small REJECT}}$$\mathrm{X}\ovalbox{\tt\small REJECT} 7_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}(?,$$\varphi_{j}(x)=\sum_{k=0}^{\infty}(\int_{\gamma_{\mathrm{j}}}v(t)t^{k}dt)/x^{k+1}$(1.6)

ie. (fj(x) is holomorphic at infinity and has azero there. Under the above assumptions we set$R_{j}(x)= \int_{\gamma_{\mathrm{j}}}\frac{v(t)\Phi(t)^{n}dt}{(x-t)^{n+1}}$,$1\leq j\leq m$

(1.7)

then
$R_{j}(x)=1/x^{n+1} \sum_{k=0}^{\infty}(\int_{\gamma_{j}}v(t)\Phi(t)^{n}t^{k}dt)/x^{k}$(1.8)

This family of functions on $V_{k}(t, u)$ gives an answer to the following question :“Find a polynomial $Q(x)$ , not identically zero, of degree at most $N=nm$ suchthat for , $1\leq j\leq m$$j$$Q(x)\varphi_{j}(x)-[Q(x)\varphi_{j}(x)]_{N-1}=O(1/x^{n+1})$(1.9)(where the bracket square $[]_{N-1}$ denotes the truncated series of order $N-1$ )$R_{j}(x)$ has at the point at infinity azero of order at least $n+1$ .
This problem has asolution, since it reduces to asystem of $N=nm$ linear homogeneous equations in $N+1$ unknowns (the coefficients of $Q(x)$ ) andthe solution is called Hermite-Pad6 approximation of the second kind in theneighborhood of infinity for the set of functions.
Since for , $1\leq j\leq m$ , $R_{j}(x)=O(1/x^{n+1})$ , we can guess that for $1\leq j\leq m$
the solution of this problem will be explicit. As in the Hermite’s paper, we cangive two proofs.
$\varphi_{j}(x)$$j$First proof :A$n$fold integration by parts gives$R_{j}(x)=(-1)^{n}/n!

\int_{\gamma_{\mathrm{j}}}\frac{D_{t}^{(n)}\{v(t)\Phi(t)^{n}\}dt}{x-t}$(1.10)

where
denotes the derivation on the Riemann surface $V_{k}(t,u)$
If we set for $1\leq j\leq m$$D_{t}$$Q(t)= \frac{(-1)^{n}}{n!}v(t)^{-1}D_{t}^{(n)}\{v(t)\Phi(t)^{n}\}$$P_{j}(x)= \int_{\gamma_{j}}\frac{Q(x)-Q(t)}{x-t}v(t)dt$

,

(1.6)
(1.12)

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we see that $Q(t|$ is apolynomial of degree mn anddegree mn-1 and according to (1.11) and (1.12)$P_{j}(x)$is apolynomial of(1.13)$R_{j}(x)=Q(x)\varphi_{j}(x)-P_{j}(x)$Remark 1.1 $Q(x)$ is ageneralisation of Jacobi’s polynomials.
Second proof:
beginsIn the remainder of this section, we assume that the path
$V_{k}(t,u)$, we can use
at and ends at , $(1 \leq j\leq m)$ . On the Riemann surface“Hermite’s trick” and deform this path to add to it aloop called a“vanishingcycle”.
in the positive
is composed of the path , aline segment
The new pathin the opposite sense.
sense, asmall circle $C(x)$ of center and the segment. With these notations
We set$\gamma_{j}=(e_{0}, e_{j})$$e_{0}$$e_{j}$$\gamma_{j}(x)$$\ell_{e_{j}x}$$\gamma_{j}$$x$$\ell_{e_{j}x}^{-1}$$\gamma_{x}=\ell_{e_{\mathrm{j}}x}^{-1}\circ \mathrm{C}(\mathrm{x})\circ\ell_{e_{j}x}$$\gamma_{j}(x)=\gamma_{j}+<\gamma_{j}|\gamma_{x}>\gamma_{x}$(Picard Lefchetz Principle)(114)

the intersection index, $<\gamma j|\gamma_{x}>=\pm 1$ . Now we
whereinstead of . It follows thatcan rewrite the integral 1.7 with$<\gamma_{j}|\gamma_{x}>\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{s}$$\gamma_{j}(x)$$\overline{R}_{j}(x)$$\gamma_{j}$$=$$\int_{\gamma_{j}(x)}\frac{v(t)\Phi(t)^{n}dt}{(x-t)^{n+1}}=\int_{\gamma_{j}}\frac{v(t)\Phi(t)^{n}dt}{(x-t)^{n+1}}$$+$$2i \pi \mathrm{r}\mathrm{e}\mathrm{s}\{\frac{v(t)\Phi(t)^{n}}{(x-t)^{n+1}}\}_{t=x}$,$(<\gamma_{j}|\gamma_{l}>=1)$Hence(1.15)$\overline{R}_{j}(x)=R_{j}(x)+2i\pi\frac{(-1)^{n}}{n!}D_{x}^{(n)}\{v(x)\Phi(x)^{n}\}$Similarly, writing (1.9) in the form$\overline{R}_{j}(x)=\overline{Q}(x)\int_{\gamma_{j}(x)}\frac{v(t)}{x-t}dt-\tilde{P}_{j}(x)$we get$\tilde{R}_{j}(x)=R_{j}(x)+2i\pi v(x)\tilde{Q}(x)$and by identification with (1.15) we obtain $Q(x)=\tilde{Q}(x)$ .
This second proof gives more. Namely that $R_{m+1}(x)=Q(x)v(x)$ is the$m+1$ -th solution of the Tissot-Pochhammer linear differential equation of order, $R_{m}(x)$ .
$m+1$ satisfied by the $m$ linear independent solutions $R_{1}(x)$ , $R_{2}(x)$ ,This equation, has the following form [INCE]$\cdots$$(T.P.E)$$\Phi(x)y^{(m+1)}-\mu\Phi’(x)y^{(m)}+$$\Phi_{0}(x)y^{(m)}-(\mu+1)\Phi_{0}’(x)y^{(m-1)}+$$(\begin{array}{ll}\mu +1 2\end{array})$$(\begin{array}{ll}\mu + 2\end{array})$$\Phi’(x)y^{(m-1)}-$$\Phi_{0}’(x)y^{(m-2)}$–.

..

(1.16)

189$\Phi_{0}(x)=-\Phi(x)\sum_{j=0}^{m}(1-\delta_{j})/(x-e_{j})$(1.17)$\mu=-(n+1)$(1.18)

This differential equation is Fuchsian with singularities
fication’s points of $V_{k}(t, u))$ .
We can associate to (1.6) the Riemann scheme$\mathrm{e}\mathrm{o}$,$e_{1}$,$\cdots$$\{\begin{array}{lllll}e_{0} e_{1} e_{m} \infty 0 0 0 n+11 \mathrm{l} 1 n+2\vdots \vdots \vdots \vdots m-\mathrm{l} m-1 m -1 n+m\delta_{0}-1 \delta_{1}-\mathrm{l} \delta_{m} -\mathrm{l} \mu’\end{array})$,$e_{m}$,$\infty$(rami-(1.19).
(exponents)of the indicial equations of the T.P.E. atrootswhich gives the
, .
,
, ,
, $R_{m}(x)$ belong to the exp0It is very useful to notice that $R_{1}(x)$ , $R_{2}(x)$ ,at infinity.
nents a1and that $R_{m+1}(x)$ belongs to the exponent
Furthermore, Cauchy’s formula yields$\mu’=\sum_{j=0}^{m}(\delta_{j}-1)+nm$$e_{0}$$e_{1}$$\cdots$$e_{m}$$\infty$$\cdots$$\mu’$$n$$R_{m+1}(x)= \frac{1}{2i\pi}\int_{\gamma_{m}+1}\frac{v(t)\phi(t)^{n}}{(x-t)^{n+1}}dt$(1.20)

in the interior of adenotes aclosed Jordan curve containing
wheredomain where $v(t)$ is holomorphic.
homological to
Now if we choose the $m+1$ steepest descent paths$1\leqi\leq m+1$ are the,ofpointsintegrands
theofsuch that the criticalroots of the family of equations$x$$\gamma_{n+1}$$\tilde{\gamma}_{j}(x)$$\gamma_{j}$$R_{i}(x)$$E_{x}$If

Oj$\theta_{1}(x)$,$(t-x)\Phi’(t)-\Phi(t)=0$ .

(1.21)

denote the roots of (1.21), then for $1\leq j\leq m- l$. With these choices, the new integrals$\theta_{2}(x)$$(\mathrm{x})\in \mathrm{O}\mathrm{j}(\mathrm{x})$:,$\cdots\theta_{m}(x)$$\tilde{R}_{j}(x)=\int_{\overline{\gamma}_{j}(x)}\frac{v(t)\Phi(t)^{n}dt}{(x-t)^{n+1}}$, $1\leq j\leq m+1$$1$,(1.22)

satisfy the equation (1.16). From this we can obtain the asymptotic behavioursof these integrals, namely$\lim_{narrow+\propto}1/n\log|\tilde{R}j(x)|=\log|\Phi’(\theta j(x))|$(1.20)

190
Remark 1.2 The relations
give$7_{\mathrm{i}}^{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}}(\mathrm{r})\ovalbox{\tt\small REJECT}$$O(1/x^{\mathrm{n}+[]})$$P\mathrm{y}(\mathrm{z})$,1$\ovalbox{\tt\small REJECT}$k$\ovalbox{\tt\small REJECT}$m and$Q(x)\ovalbox{\tt\small REJECT}$$\mathit{0}(x^{mn})$$\ovalbox{\tt\small REJECT}$$p_{i}(x)$oo$Q(\mathrm{r})$and if $|x|arrow\infty$(1.25)$Q(x)\infty\sim R_{j}(x)^{-m}$

2Arithmetic applications
2.1

The logarithmic caseIn the section, we assume that $v(t)\equiv 1$ and we obtain simultaneous rational
approximations of the integrals$\int_{e_{0}}^{e_{j}}\frac{dt}{x-t}$, $1\leq j\leq m$ .

For instance if we set $\Phi(t)=t^{3}-t$ , we can prove that forthree numbers
1,,$\log(1-\frac{1}{q})$(2.1)$q\in \mathbb{Z}^{+}$, $q\geq 4$ the$\log(1+\frac{1}{q})$are linearly independent over I and we find avery precise measure of linear

independence for these numbers.
Following amethod of M. Hata [HA] which uses Pad6-type approximations,
we can state the following estimate.
Theorem 2.1 For rational integers , ,with $H= \max(|b_{1}|, |b_{2}|)\geq 2$$b_{0}$$b_{1}$$b_{2}$(2.2)$|b_{0}+b_{1}\log 2+b_{2}\log 3|\geq H^{-10,101}\ldots$

We outline only the proof.
One chooses $\phi(t)=t(t-1/2)(t-1/3)$ ,, such that for$i,j$ , , $0\leq i\leq 2n$ , $0\leq j\leq 2n$ , $0\leq k\leq 2n$ , $i+j+k-3n\neq 0$ and$D_{n}\in \mathbb{Z}^{+}$$k$$D_{n}$$(\begin{array}{l}2ni\end{array})(\begin{array}{l}2nj\end{array})(\begin{array}{l}2nk\end{array})/(i+j+k-3n)\in \mathrm{Z}$,then it is easy to verify that we have simultaneous approximations of $\log(2/3)$
and $\log(3/4)$ , namely$\{6^{3n}D_{n}6^{3n}D_{n}\int \mathrm{o}dt=a_{n}\mathrm{l}\mathrm{o}\mathrm{g}(2/3)-b_{n}^{1}\int_{1/3}^{1/4}\frac{\frac{\phi(t)^{2\mathfrak{n}}}{(t-1\phi(t)\mathrm{t}_{\mathfrak{n}}^{Sn}}}{(t-1)^{S\mathrm{n}}}dt=a_{n}\mathrm{l}\mathrm{o}\mathrm{g}(3/4)1/3-b_{n}^{2}$with$a_{n}\in \mathbb{Z}$,$b_{n}^{1}\in \mathbb{Z}$,$b_{n}^{2}\in \mathbb{Z}$

.

(2.3)

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Abelian integrals

2.2

Prom now on we suppose that $g\geq 1$ , $k\geq 2$ and does not divide $m+1$( , )). We have to introduce some(i.e the point at infinity is ramified on
notations$v(t)=\Phi(t)^{-1+1/k}=u^{1-k}(t)$ (For asuitable determination of $u(t)$ ), $1\leq j\leq$, $a_{m+1}$ ,is such thatadenominator of , ,$k$$V_{k}$$t$$u$$\bullet$$d\sim$$\bullet$$a_{1}$$m+1$ , resp.

$\tilde{d}e_{0}\in \mathbb{Z}$$a_{2}$$e_{0}\mathrm{i}\mathrm{e}.\tilde{d}$$\cdots$$\tilde{d}aj\in \mathbb{Z}$.

$\bullet$$E_{k}=\{\begin{array}{l}e^{m}\prod_{p|k}p^{[\frac{m}{\mathrm{p}-1}]}\mathrm{i}\mathrm{f}a_{m+1}\neq 0p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}exp\{k/_{\varphi(k)}(s,k)=1\sum_{s=1}^{k}1/s\}p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\prod_{p|k}p^{[\frac{m}{\mathrm{p}-1}]}\mathrm{i}\mathrm{f}a_{m+1}=0\end{array}$where$\varphi$denotes Euler’s$\varphi$function.

$\bullet\tilde{D}_{k}=kbE_{k}\tilde{d}$In the sequel the roots$\theta_{i}(x)$, $1\leq i\leq m+1$ of (1.21) are chosen such that$|\Phi’(\theta_{1}(X))|\leq|\Phi’(\theta_{2}(x)|\leq\cdots|\Phi’(\theta_{m}(x)|\leq|\Phi’(\theta_{m+1}(x)|$and we set $R(x)=|\Phi’(\theta_{m}(x)|$ ; $E(x)=|\Phi’(\theta_{m+1}(x))$ . Then we have the main
result of this paper.
Theorem 2.2 If $x=a/b\in I$ satisfies$\tilde{D}_{k}^{m}R(x)<1$

.

(2.4)

, $f_{m}(x)$ are linearly independent over .
Then the numbers 1, $f_{1}(x)$ , $f_{2}(x)$ ,, , a2Moreover, for arbitrary rational integers
of I with$H= \max(|a_{1}|, \cdots, |a_{m}|)\geq 2$ , the linear
form
$I$

$\cdots$$a_{1}$$a_{0}$$\cdots$$a_{m}$$L=a_{0}+ \sum_{\dot{\iota}=1}^{m}a_{i}f_{\dot{l}}(x)$admits the lower estimate $|L|\geq \mathrm{c}H^{-C(x)}$ where
positive constant and$C(x)=\log(\tilde{D}_{k}^{m}R(x))/\log(\tilde{D}_{k}^{m}E(x))$$c$

is an effectively computable(Measure of linear independence)(2.5)

Remark 2.1 Using (1.25), we see that if $|x|arrow\infty$ , $C(x)arrow m$ .
With the same notations as in the previous theorem we have
Corollary 2.1 If $x=\mathfrak{P}(u)=a/b$ satisfies
denoting as$u-\omega_{1}/2$$\zeta(u)-\eta_{1}/2$,,
function)
numbersaretheusual the Weierstrasslinearly independent over I and$\overline{D}_{2}^{2}R(x)<1(\mathfrak{P}(u)$$\mathfrak{P}$$\varphi’(u)$$|b_{0}\mathfrak{P}’(u)+b_{1}(u-\omega_{1}/2)+b_{2}(\zeta(u)-\eta_{1}/2)|\geq \mathrm{C}\mathrm{J}\#^{-C(x)}$where$\omega_{1}$, resp.

$\eta_{1}$denotes a period (resp.) a quasi-period of the elliptic curve

192
We give here only the ideas of the proofs of the main lemmas. The detailed
proofs will appear in aforthcoming publication.

3Sketch of the proofs
Lemma 3.1 Interchange of the argument and the parameter.
If we consider the integrals$\int_{e_{0}}^{e_{j}}\frac{v(t)}{x-t}dt$with $v(t)=\phi(t)^{-1+1/k}$ , is called the argument andintegral satisfies the following relation
$t$

$x$

the parameter. Then this$\Phi(x)\eta(x)\int_{\gamma}\frac{v(t)}{x-t}dt=\sum_{j=0}^{m-1}\{\int_{\gamma j}(v(t)t^{j})dt\}\int_{\mathrm{e}_{0}}^{x}\lambda_{m-1-j}(t)\eta(t)dt$$\eta(t)$(3.1)

being a solution of the adjoint differential equation related to (3.1)$-\phi(t)\eta’(t)+(\Phi_{0}(t)-\Phi’(t))\eta(t)=0$

is a polynomial of degree $m-1-j$ that depends on the differential
equation T.P.E.

$\lambda_{m-1-j}(t)$We do not prove this relation here [HU]. It uses some tools about differential
equations [INCE] or [SCH]. The reader can nevertheless verify the relation(3.2)$\frac{\partial}{\partial x}\{\frac{\sqrt{\phi(x)}}{(t-x)\sqrt{\phi(t)}}\}-\frac{\partial}{\partial t}\{\frac{\sqrt{\phi(t)}}{(x-t)\sqrt{\phi(x)}}\}=\frac{U(x,t)}{\sqrt{\phi(x)}\sqrt{\phi(t)}}$which is aparticular case of (3.1) for the hyperelliptic case.
Where $U(x, t)$ denote the polynomial (anti-symmetric) of degree $m-1$ .
$U(x, t)= \frac{\frac{1}{2}(\phi’(x)+\phi’(t))(t-x)+\phi(x)-\phi(t)}{(t-x)^{2}}$.

To obtain arelation in the hyperelliptic case it suffices to integrate this relation
by respect to .
Lemma 3.2$t$$\int_{\gamma\cdot}.\frac{Q(x)-Q(t)}{x-t}v(t)dt=\sum_{s=0}^{m-1}(\int_{\gamma}t^{\epsilon}v(t)dt)Q_{m-\epsilon}(x)$

.

(3.3)

This lemma says that the module of differential forms $t^{k}v(t)dt$ withisgenerated modulo exact differentials by $t^{k}v(t)$ , $0\leq k\leq m-1$ and the proof
uses De Rham cohomology on $V_{k}(t, u)$ .
$k\in \mathbb{Z}^{+}$

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Having disposed of these preliminary steps, we find $m$ relations that we canwrite for $1\leq i\leq m$$R_{i}(x)-- \sum_{j=1}^{?n}(\int_{\gamma}.\cdot t^{j-1}v(t)dt)\{Q(x)(\Phi(x)\eta(x))^{-1}\int_{e0}^{x}\lambda_{m-1-j}(t)\eta(t)dt-Q_{m-1}(x)\}$(3.4)
Now, we see that we must invert the matrix$\Delta$

$=$

where

$(\alpha_{\dot{\iota}j})$$1\leq i\leq m1\leq j\leq n$$\alpha_{ij}=\int_{\gamma i}t^{j-1}v(t)dt$.

The determinant of this matrix is the determinant of periods of integrals on
the curve $V_{k}(t, u)$ .
The proof that $\det(\Delta)\neq 0$ uses the properties of the differential equation (1.16)
and in the particular case of elliptic curves we find$\det\Delta=\omega_{1}\eta_{2}-\omega_{2}\eta_{1}=i\pi/2$(Legendre’s relation).

The other lemmas are more classical applications of the theory of diophantine
approximation [HU].
In the following, we set $v(x)=\phi(x)^{1-1/k}$ .
Lemma 3.3 Let us set$\tilde{E}_{k}=\tilde{E}_{k}(n, m)=\tilde{d}^{nm}k^{nm}\prod_{p|k}p^{[\frac{nm}{\mathrm{p}-1}]}\ell cm(1, 1+k, \cdots, 1fknm)$

if $a_{m+1}\neq 0$ and$\tilde{E}_{k}=\tilde{E}_{k}(n,m)=\tilde{d}^{nm}k^{nm}\prod_{p|k}p^{[\frac{\mathfrak{n}m}{p-1}]}\ell cm(1,2, \cdots,nm)$

if $a_{m+1}=0$ . Then$\tilde{E}_{k}Q(x)$resp.

$\tilde{E}_{k}Q_{m-i}(x)\in \mathbb{Z}[x]$, $1\leq i\leq m$ . Furthermore(see 2.5)$\tilde{E}_{k}\leq\tilde{D}_{k}^{nm}$To finish the proof of the theorem (2.2), we are obliged to compute the determinant of the following lemma and to show that it is nonzero.
For $s=n$ , $n+1$ , $\cdots n+m$ , we set$Q=Q_{s}$ ,$R_{i}=R_{s,i}$,$P_{i}=Q_{s,m-i}$ ,$0\leq i\leq m$

Lemma 3.4 The determinant$\delta_{m}(x)=|\begin{array}{lll}Q_{n}(x) P_{n,1}(x) P_{n,m}(x)Q_{n+1}(x) P_{n+1,1}(x) \vdots\vdots \vdots \vdots Q_{n+m}(x) P_{n+m,1}(x) P_{n+m,m}(x)\end{array}|$

where $w(x)$ is the wronskian of the Tissot-Pochhammei,}, $\Delta_{m}(x)\neq 0$ .
equation. In particular for $x\neq$ { , e2,This lemma shows that among the linear forms $b_{0}Q_{i}(x)+ \sum_{j=1}^{m}b{}_{ji,j}P(x)$ ,there exists at least one form which is nonzero.

$=\phi(x)^{n(m+1)/2}w(x)/v(x)$$e_{1}$$\cdots$$e_{m}$

194
Acknowledgments
The author wishes to thank the professors Leo Murata, Isao Wakabayashi and the R.I.M.S for the excellent organization of the Conference “Analytic
number theory”. He is also very thankful for professors Masayoshi Hata, Noriko
Hirata and Hironori Shiga for their welcome and their hospitality.

R\’ef\’erences[AB] ABEL N.H. “Sur une propri6t6 remarquable d’une classe tr6s Stendue
de fonctions transcendantes.”[INCE] INCE EL. “Ordinary differential equations.” Dover.
[HA] HATA M. “Rational approximations to and some other numbers.” Acta
Arithmetica LXIII, 4, 1993.
[HER] HERMITE C. Lettre de M. C. Hermite de Paris A M.L. Fuchs de
G\"ottingue “Sur quelques Equations diff6rentie11es linfeaires.” Journal deCrelle, t.79, 1875, 324-338.
[HU] HUTTNER M. On apaper of Hermite and Diophantine approximationsof Abelian integrals. Pub IRMA, Lille 2000, Vol
VII[SCH] SCHLESINGER L. “Theorie der linearen differential Gleichungen.”
Zweiten bandes erster theil, Leipzig 1897$\pi$

$49\mathrm{n}^{\mathrm{o}}$