Table Of ContentLecture Notes in
Mathematics
Edited by A Dold and B. Eckmann
Series: Australian National University. Canberra
Advisers: L. G. Kovacs, B. H. Neumann, and M. F.N ewman
546
Kurt Mahler
Lectures on
Transcendental Numbers
Edited and Completed by B. Divi~ and W. J. Le Veque
Springer-Verlag
Berlin· Heidelberg· New York 1976
Author
Kurt Mahler
Department of Mathematics
Research School of Physical Sciences
Australian National University
Canberra, ACT 2600/A ustralia
Editors
B. Divi~ t
W. J. Le Veque
Claremont Graduate School
Claremont, California/USA
Library 01 Coalret. Cala",la, I. Pabll.atl• • Da.a
Mahler, Kurt.
Lectures on tl"8ollBcendentaJ. numbers.
(Lecture notes in ....t hematic. ; 5116)
Bibliography: p.
Inclucles index.
1. Nwnbers, TranscendentaJ.. I. Div:l.s, B.
II. LeVeque, William Judson. III. r1t1e. IV. Se
ries: Lecture notes in ....t bematics (Berlin) ; 546.
W.L2B no. 546 {QA247.51 510'.88 [512'.731 76-44348
AMS Subject Classifications (1970): 10F35
ISBN 3-540-07986-6 Springer·Veriag Berlin' Heidelberg· New York
ISBN 0-387·07986-6 Springer-Verlag New York' Heidelberg· Berlin
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© by Springer-Verlag Berlin' Heidelberg 1976
After the manuscript of this book had gone to
the printers, our collaborator, Professor B.
Divis, died suddenly at the early age of 32,
while attending a number theoretic meeting at
Illinois State University. This is a great
loss to his family, to mathematics, as well
as to us.
K. Mahler
W.J. Le Veque
PREFACE
The rather small list of modern books on transcendental numbers (Siegel 1949;
Gelfond 1952; Schneider 1957; Lang 1962; Ramachandra 1969) has recently been
enriched by four new ones (Stolarsky 1974; Waldschmidt 1974; Baker 1975; Masser
1975). Baker's book is particularly valuable because it contains an account of his
powerful new method and a review of some of its applications.
The present book is derived from lectures given by Mahler during the last
twenty years at different times and places, and edited and sometimes enlarged by Divis
and LeVeque. It aims at giving an account of some old and classical results and
methods on transcendency and in particular to present in all its details the
important Siegel-Shidlovski theory of the transcendency of a special class of entire
functions (the Siegel E-functions) which satisfy linear differential equations with
rational functions as coefficients. In the earlier books this theory was only
sketched.
Chapter 1 discusses the existence and gives first examples of transcendental
numbers and concludes with a rather general necessary and sufficient condition for
transcendency. As it will be needed later on, Chapter 2 gives a short account of the
theory of formal Laurent and power series in one variable, and it contains a simple
test for the transcendency of such series over the field of rational functions.
Chapter 3 begins then the study of the possible transcendency of the values of an
analytic function at an algebraic point. A number of claSsical results, of which
some go back to the time of Weierstrass, are proved in order to show that certain
very plausible conjectures are in fact false. Thus, (i) there exist entire
transcendental functions defined by power series with rational coefficients which,
with all their derivatives, assume algebraic values at all algebraic points, and
(ii) there also exist functions of the same kind which, say, are algebraic at the
algebraic point a = Y:2 , but are transcendental at the algebraically conjugate point
a = -v2. It follows that the problem whether a given analytic function f(a) is
algebraic or transcendental at a given algebraic point 8 = a may be very deep and
involved.
The next four chapters 4-7 deal then with Shidlovski's generalisation of
Siegel's method of 1929 aI,ld 1949. This theory was obtained by Shidlovski in a number
of paper,S that go back to 1954 and of which the most important ones date from 1959 to
1962. We try to bring out the algebraic basis of this theory by using both formal
series and simple facts from algebraic geometry.
In Chapter 4, K is any field of characteristic 0, c is an element of K ,
VI
and a is an indeterminate; K[a]. K(a) • and K(z-c) are then the ring of poly
nomials. the field of rational functions. and the field of formal Laurent series
in z • all with coefficients in K; in I n may be any integer. We denote the
degree of a polynomial p ~ 0 in X[z] by alp) and if the Laurent series I is
distinct from 0 and so. without loss of generality, In # 0 • the order of I is
defined by
ord/=n.
We are concerned with the formal theory of systems of homogeneous linear
differential equations
(h 1.2 •••••m ) •
nere m ~ 1 • and the coefficients arbitrary elements of X(s) ; let
K be their least common denominator. The system Q is said to be regular (that is
at c) if K does not vanish at the point z = c •
~enote by VQ the set of all vector solutions
· j:]
of Q with components wl • ..•• wm in X{z-c) This set VQ is a linear vector
space over K{a-c) • say of dimension M where of course 0 ~ M ~ m As can be
proved. M is also the dimension of VQ over the constant field X If Q is
regular. then M = m .
Next let
f
be any special element of VQ distinct from the zero vector 0 The rank p of
f is defined as the largest nwnber of components 11' •••• 1m of f that are
linearly independent over X(s)
A first theorem by Shidlovski deals with the case when Q is regular and f is
of rank p < m . If. without loss of generality. the first p components
VII
fl , ••• , fp are linearly independent over K(a) , denote by f4 the p-vector
Then the following result holds.
(A) : The components of f* satisfy a regu~ar system of homogeneous linear
differentiaZ equations
=
(h l.2, ••• ,p)
of onZy p equations with coefficients qhk, in K(a) •
Assume next that the system Q is not necessarily regular, but that f is of
maximal rank m. Let· Pl = Pll, ••• , Pm = Plm be any m polynomials in K[a] not
all zero. For any element W of VQ we form the linear form
in its components and for such forms define a linear operator D by
m dph m m
DA(W) L K az "'h + L L Kq1..:.P.ulh .
h=l h=l j=l ''oJ J
Also DA(W) is a linear form in the components of w with coefficients in K[a] •
The formulae
(h l,2,3 •••• )
thus generate an infinite sequence of linear forms
(h 1,2,3 •••• )
in the components of W with coefficients in K[z] •
Shidlovski proves for these forms the following basic theorem.
(8): To every eZement f of vQ of rank m there e:x:ista a positive integel' C
with the fo~Wwing properly.
If the poZynomiaZs Pl' •••• Pm satisfy the inequality
ord A(f) - (m-l}max(iHPl>• •••• iHPm» ~ C •
then the determinan t
VIII
is not identiaaz:ly aero.
Usini a method of Siegel, we can deduce from this the following third theorem.
(C): Let a be an element of K Buch that a 'F a and K(a) 'F 0; 'let I/l be
a aonstant satisfying 0 < I/l < 1; and let f be an element of vQ of
rank m. There exist wo positive integer'S no and nl with the
fo'l'louring pr'oper'ty.
If n:: no • and if the POZynomia'lB Pl' ... , Pm in A(w) ar>e not aU
zero and such that
then m SUffixes hl , ... , hm satisfying
can be ahosen such that the detemrinant
pea) say,
does not vanish.
These purely algebraic theorems are in Chapter 5 combined with the theory of
Diophantine approximations; the method is that of Siegel's paper of 1929. No longer
are we considering formal Laurent series in an indeterminate z, but entire
functions in the complex variable z. Further a is put equal to 0
The field K is specialised to be an algebraic number field, say of degree N
over the rational field Q. Every number a in K has 8 algebraic conjugates
(0) (1) (8-1)
a = a, a , .•• , a
and as usual we put
I I
Ii.C:"Il I=max('l0a )1 ,CI. (1)1 , ••• ,(1( N-l)l) ,
so that for any two elements a and 6 of K,
IX
An E-function is defined as a power series
\)
f(z) = L f\) ~!
V=O
with the following two properties of which the first one implies that f(3) is an
entire function.
(1) : The coefficients fv lie in K, and for every constant e: > 0 , as
v+oo ,
(2): Let d\) be the smallest positive integer such that all the products
are algebraic integers. Then for every constant e: > 0 ,as v'" 00 ,
From this definition, Dirichlet's principle (the Schubfachprinzip) leads to the
following existence result.
(D): Let m;:: 2; let fl(3), ••• , fm(z) be E-functions over K; let e:
and <jl be aonatantB 8ati8fying 0 < e: < 1, 0 < <jl < 1; 'let n be a
sUfficientLy Zarge positive integer; and 'let p = Im-[<jln]-l. Then
there exist m po'lynomia'l8
n-l
L
G all (h = 1,2, ••••m )
ll=O hjJ
with integra'l coefficients in K not aU zero, 8uch that the Unear form
can be expressed as a power 8eries
A(f<a»)
where, firstLy,
a a o ,
O 1
and, seaond'ly, as \I'" 00 •
max 1l""h(,;,11 -- O( n (l+e:)n) and
h,ll II
x
Now again let
(h l,2, ••• ,m)
be a system of m (::: 2) homogeneous linear dLfferential equations where the
coefficients qhk = qhk(z) lie in K(z) and their least common denominator
K = K(Z) is in K[2] •
Assume that the vector
f(z)
rCa)
m
derived from the mE-functions in CD) is an element of VQ . Then. with the
polynomials PI(2) • ...• Pm(z) in (D), put
m
I
ACW) Pk(2)Wk .
k=l
On applying to ACW) repeatedly the differential operator D of Q , we obtain
again a sequence of linear forms
\(W) (h 1.2.3 •••• )
with coefficients Phk(2) in K[z]
Now let a be a number in K satisfying a # 0 and KCO) # O. As before.
denote by p the largest number of components of Hz) that are linearly
independent over KCz) , and similarly let pea) be the largest number of
components of fCa) that are linearly independent over K. It is obvious that
pea) ~ p , and the problem arises of also establishing a lower estimate for pea)
This can be done by means of Siegel's method and leads to the following results.
First assume that p has its maximum value m. By the construction CD). the
hypothesis of CC) is satisfied as soon as n is sufficiently large. Hence it is
again possible to select m distinct suffixes hl , ... , hm not greater than
[$n] + m + nl for which the determinant PCa) in (C) is distinct from zero. By
means of simple estimates, Siegel's method leads to the inequality
pCa) ::: crc~)m ,
where cr(Kl 1 if K is a real number field. and o(K) 2 if K is non-real.
XI
The restriction p = m under which this estimate has been proved can finally
also be removed by means of Shidlovski's theorem (A), and it follows from the last
formula that always
(j{;'P .
(E): pea) ~
In particular, pea) p when K is the rational number field or any imaginary
quadratic field.
The beginning of Chapter 6 is again algebraic and is independent of Chapters 4
and 5.
Let L be any field of characteristic 0 and L~ any extension field of L.
As usual, a finite set of elements of L* is called (algebraically) independent over
L if it does not satisfy any non-trivial algebraic equation with coefficients in
L and the set is said to be (algebraically) H-independent over L if it does not
satisfy any non-trivial homogeneous algebraic equations with coefficients in L.
Let then xl' Xn ,where n ~ 2 , be fixed elements of L* not all zero of
which certain D ~ 1 , but no more, are H-independent over L and let
~l' ••• , Wn be an equal number of independent indeterminates. For every positive
integer t denote by vet) the set of all homogeneous polynomials
P = P(WI, ••• , Wn) in L[Wl, .•• , Wn] of dimension t, and by Set) the subset of
those polynomials P in Vet) for which
A polynomial P in Vet) belongs to set) if and only if the set of its
coefficients satisfies a certain system L(t) of, say, exactly h(t) linearly
independent homogeneous linear equations with coefficients in L.
We show now that
(F): There exist ~o positive integral aonstants a and a suah that for all
t >
(This is the weakened form of a very special case of a theorem by Hilbert (1990) on
polynomial ideals.)
From here on, the hypothesis is again the same as for theorem (E). Denote by
D ~ 1 the maximum number of E-functions fl{z), •.• , fm(z) that are H-independent
over K(z) , and similarly by D(a) ~ 1 the maximum number of function values
fl(a), "', fm(al that are H-independent over K. Then the system Q of m
homogeneous linear differential equations for fl(z), ••• , fm(z) implies for every
