transcendental numbers - universiteit utrechtbeuke106/transcendentalsheets.pdf · 2) e, proved...
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Transcendental Numbers
Basic Notions in Mathematics
Utrecht, March 31, 2005
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Definition
A complex number, which is a zero of a non-trivial polynomial with integral coefficients iscalled algebraic. The set of algebraic numbersis denoted by Q.
Examples:
1. 3/5 zero of 5x− 3
2.√
2 zero of x2 − 2
3.√
1 + 3√5 zero of x6 − 3x4 + 3x2 − 6
4. . . .
Definition A complex number which is not al-gebraic is called transcendental.
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First known transcendental numbers
Liouville numbers constructed by Liouville in
1844.
Example: a =∑∞
n=01
10n! is transcendental. In
decimals:
a = 1.1100010000000000000000010000 · · ·
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A transcendence proof
Suppose a satisfies P (a) = 0 where P 6≡ 0 is
polynomial with integer coefficients. Denote
by aN the N-th aproximation aN =∑N
n=01
10n!.
Note, aN has denominator 10N ! and |a− aN | ≈10−(N+1)!.
Analytic upper bound:
|P (aN)| ≤ C(P )
10(N+1)!.
On the other hand: P (aN) is a non-zero ra-
tional number (if N sufficiently large) with de-
nominator dividing (10N !)d where d = degree(P ).
Hence (arithmetic lower bound):
1
10dN !≤ |P (aN)|.
Contradiction when N is sufficiently large. Hence
a transcendental.
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Cantor’s ideas
Theorem The algebraic numbers form a count-able set.
Proof
• It suffices to show that the polynomialswith integer coeffcients form a countableset.
• For a polynomialP = anxn + an−1xn−1 + · · · a1x + a0with an 6= 0 we defineH(P ) = n + |a0|+ |a1|+ · · ·+ |an|.
• Enumerate the polynomials according tothe size of H(P ).
Cantor (1874): The set of (real) transcenden-tal numbers is not enumerable.
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Famous transcendental numbers
1) π (Ludolphsche Zahl), proved transcenden-tal in 1882 by Lindemann. More generally:a ∈ Q and a 6= 0, then ea transcendental.
2) e, proved transcendental in 1873 by Her-mite.
3) 2√
2 (formerly Hilbert’s 7th problem) provedtranscendental by Gel’fond and Schneiderin 1934.
More generally, if a, b algebraic, a 6= 0,1and b irrational, then ab is transcendental.Example: i−2i = eπ.
4) 0.123456789101112131415161718 · · · (Cham-pernowne’s number) proven transcenden-tal by Mahler in 1936.
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Famous transcendental numbers
5) Let a algebraic and 0 < |a| < 1. Then∑∞n=0 an2
is transcendental (Nesterenko, 1997).
6) Let {an}∞n=1 = 0,0,1,0,0,1,1,0, . . . be the
paper folding sequence and
f(x) = 1 +∑
n≥1
anxn.
Then f(a) is transcendental for any alge-
braic number a with 0 < |a| < 1 (K.Mahler).
Note:f(x2)− f(x) = x3/(x4 − 1).
7) Γ(1/3),Γ(1/4) shown transcendental by
G.Chudnovski (1980’s).
8) Zeros of the Bessel function J0 are tran-
scendental (Siegel, 1929).
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Infamous transcendental numbers
1) γ (Euler’s constant), not known to be ir-
rational.
2) ζ(3), known to be irrational (Apery, 1978)
but not known to be transcendental.
3) e + π, not known to be irrational.
4) Γ(1/5) not known to be irrational.
5) log(2) log(3) not known to be irrational.
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Main streams, I
Gel’fond-Schneider (1934):
a, b ∈ Q, a 6= 0,1, b 6∈ Q⇒ ab 6∈ QEquivalent: Let α1, α2, β1, β2 ∈ Q∗ and sup-pose that α1, α2 are mulitiplicatively indepen-dent. Then
β1 logα1 + β2 logα2 6= 0.
Theorem (A.Baker, 1967) Let α1, . . . , αn ∈ Q∗and suppose α1, . . . , αn are multiplicatively in-dependent . Then logα1, . . . , logαn are linearlyindependent over Q.
Quantitative version: Let α1, . . . , αn be as above.Then, there exists C > 0 such that for anyb1, . . . , bn ∈ Z, not all zero, we have
|b1 logα1 + · · ·+ bn logαn| > B−C
where B = maxi |bi|. Moreover, C can be com-puted explicitly in terms of the αi.
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Mordell’s equation
Baker’s quantitative theorem can be used tosolve diophantine equations of various forms,e.g
Mordell’s equation y2 = x3 + k.
Theorem (Mordell) Let k ∈ Z and supposek 6= 0. The the equation y2 = x3 + k has atmost finitely many solutions in x, y ∈ Z.
Example, y2 = x3+17 has the integer solutions
32 = (−2)3 + 17
42 = (−1)3 + 17
52 = 23 + 17
92 = 83 + 17
232 = 433 + 17
3876612 = 52343 + 17
Baker’s theory provides us with an effectivemethod of solution.
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Thue’s equation
Theorem (Thue,1909) Let F ∈ Z[x, y] homo-
geneous and k ∈ Z. If F has at least three
distinct (complex) zeros then the equation
F (x, y) = k
has at most finitely many solutions x, y ∈ Z.
Example, x3 + x2y − 2xy2 − y3 = 1 has the
solutions
(x, y) = (1,0), (0,−1), (−1,1)
(−1,−1)(2,−1)(−1,2)
(5,4), (4,−9), (−9,5)
Baker’s theory provides us with an effective
method of solution.
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Catalan’s problem
Catalan (1844) Je vous prie, Monsieur, de vouloirbien enoncer, dans votre receuil, le theoremesuivant, que je crois vrai, bien que je n’aie pasencore reussi a le demontrer completement:d’autres seront peut-etre plus heureux:
Consider the infinite sequence of perfect pow-ers
1,4,8,9,16,25,27,32,32,36,49,64,
81,100,121,125,128, . . .
The only two consecutive numbers in this se-quence are 8 and 9.
R.Tijdeman (1976), using linear forms in log-arithms: There at most finitely many con-secutive perfect powers. They all lie belowexp exp exp exp(730).
P.Mihailescu (2002), using algebraic numbertheory: Catalan’s conjecture is true.
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Main stream I, continued
Gel’fond-Schneider theory has been extended
to values of elliptic functions, period of elliptic
curves and abelian varieties, etc.
Theorem (Wustholz, 1982) Let G be a com-
mutative algebraic group defined over Q of di-
mension n. Let
expG : Cn → G(C)
be an exponential map defined over Q. Let
u = (u1, . . . , un) ∈ Cn be a point such that
u 6= 0 and expG(u) ∈ G(Q). Let Wu be the
smallest linear subspace of Cn, defined over
Q, which contains u. Then expG(Wu) is an
algebraic subgroup of G defined over Q.
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Algebraic independence
Definition The numbers a1, . . . , an are called
algebraically independent (over Q) if for every
non-trivial polynomial P in n variables and co-
efficients in Z we have P (a1, . . . , an) 6= 0.
The transcendence degree of a set b1, . . . , bn
is the maximal number of algebraically inde-
pendent elements among b1, . . . , bn. Notation:
degtrQ(b1, . . . , bn).
Conjecture (Schanuel). For any a1, . . . , an that
are linearly independent over Q we have
degtrQ(a1, . . . , an, ea1, . . . , ean) ≥ n.
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Exercises
Suppose Schanuel’s conjecture holds. Deduce
1. that π is transcendental
2. that if αi are Q-linear independent algebraic
numbers then eα1, . . . , eαn are algebraically
independent (Lindemann-Weierstrass the-
orem).
3. Gel’fond-Schneider theorem
4. Baker’s (qualitative) theorem on linear forms
in logarithms
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Main stream II, E-functions
Lindemann-Weierstrass theorem: let α1, . . . , αn
be distinct algebraic numbers. Then eα1, . . . , eαn
are Q-linear independent.
Alternatively: consider the functions
f1(z) = eα1z, . . . , fn(z) = eαnz
Then their values at z = 1 are Q-linear inde-
pendent.
C.L.Siegel extended the functions fi(z) to a
more general class, the (Siegel) E-functions.
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E-functions, definition
An entire function f(z) given by a powerseries
∞∑
n=0
ak
k!zk
is called a Siegel E-function if
1. a0, a1, a2, . . . ∈ Q
2. f(z) satisfies a linear differential equationwith coefficients in Q(z).
3. logH(a0, a1, . . . , aN) = O(N) for all N .
Here, H(α1, . . . , αn) denotes the logarithmic ab-solute height of the vector (α1, . . . , αn) ∈ Qn.
Example:
H(p1/q, p2/q, . . . , pn/q)
= max(|p1|, |p2|, . . . , |pn|, |q|)17
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E-function examples
1. exp(αz) =∑∞
k=0(αz)k
k! where α ∈ Q∗.
2. J0(−z2) =∑∞
k=0z2k
k!k! =∑∞
k=0
(2kk
)z2k
(2k)!.
3. Confluent hypergeometric series qFp withrational parameters
∞∑
n=0
(µ1)n · · · (µp)n
(ν1)n · · · (νq)n
z(q+1−p)n
n!
Differential equations
y′ − αy = 0
zy′′ + y′ − 4zy = 0
Dq∏
j=1
(D + νj − 1)F = zp∏
i=1
(D + µi)F
where D = z ddz.
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The main theorem
Let f1(z), . . . , fn(z) be E-functions satisfying a
system of n differential equations
d
dz
y1...
yn
= A
y1...
yn
where A is an n×n-matrix with entries in Q(z).
We assume that the common denominator of
the entries is T (z).
Theorem (Siegel-Shidlovskii, 1929, 1956).
Let α ∈ Q and αT (α) 6= 0. Then
degtrQQ(f1(α), f2(α), . . . , fn(α)) =
degtrQC(z)(f1(z), f2(z), . . . , fn(z))
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A relation
f(z) =∞∑
k=0
((2k)!)2
(k!)2(6k)!(2916z)k
satisfies
FtQF = (z)
where
F =
f(z)
Df(z)
D2f(z)
D3f(z)
D4f(z)
, D = zd
dz
and
Q =
z − 324z2 −18z 198z −486z 324z
−18z −109
232 −28 18
198z 232 −120 297 −198
−486z −28 297 −729 486
324z 18 −198 486 −324
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The last problem?
Theorem (Nesterenko-Shidlovskii, 1996). Let
f1(z), . . . , fn(z) be E-functions which satisfy a
system of n first order equations. Then there
is a finite set S such that for every ξ ∈ Q, ξ 6∈ S
the following statement holds. Any relation of
the form
α1f1(ξ) + · · ·+ αnfn(ξ)) = 0
is obtained by specialisation of a polynomial
linear relation
p1(z)f1(z) + · · ·+ pn(z)fn(z) = 0
with pi(z) ∈ Q(z) at z = ξ.
A similar statement holds for polynomial rela-
tions.
Conjecture S = {α|αT (α) = 0}.
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Yves Andre’s work
Theorem (Y.Andre, 2000) Let f(z) be an E-function. Then f(z) satisfies a differential equa-tion of the form
zmy(m) +m−1∑
k=0
zkqk(z)y(k) = 0
where qk(z) ∈ Q[z] has degree ≤ m− k.
Corollary Let f(z) be an E-function with coef-ficients in Q and suppose that f(1) = 0. Then1 is an apparent singularity of the minimal dif-ferential equation satisfied by f .
Proof Consider f(z)/(1 − z). This is againE-function. So its minimal differential equa-tion has a basis of analytic solutions at z = 1.This means that the original minimal differen-tial equation for f(z) has a basis of analyticsolutions all vanishing at z = 1. So z = 1 isapparent singularity.
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Andre’s proof of transcendence of π
Suppose 2πi algebraic. Then the E-functione2πiz − 1 vanishes at z = 1. The product overall conjugate E-functions is an E-function withrational coefficients vanishing at z = 1. Sothe above corollary applies. However linearforms in exponential functions satisfy differen-tial equations with constant coefficients, con-tradicting existence of a singularity at z = 1.
By a combination of Andre’s Theorem and dif-ferential galois theory one can show more.
Theorem (FB, 2004) Let f(z) be an E-functionand suppose that f(ξ) = 0 for some ξ ∈ Q∗.Then ξ is an apparent singularity of the mini-mal differential equation satisfied by f .
Additional use of a home-made zero lemmagives us:
Theorem (FB, 2004) The Nesterenko-Shidlovskiiconjecture holds with S = singularities ∪ 0.
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Values at the singular points
Example f(z) = (z−1)ez. It satisfies (z−1)f ′ =zf and f(1) = 0.
More generally,
(z − ξ)k d
dz
f1(z)...
fn(z)
= A(z)
f1(z)...
fn(z)
where A(ξ) 6= O. Then,
A(ξ)
f1(ξ)...
fn(ξ)
= 0.
Theorem Let f(z) = (f1(z), . . . , fn(z)) be E-function solution of system of n first orderequations and suppose they are Q(z)-linear in-dependent. Then there exists an n×n- matrixB with entries in Q[z] and det(B) 6= 0 andE-functions e(z) = (e1(z), . . . , en(z) such thatf(z) = Be(z) and e(z) satisfies system of equa-tions with singularities in the set {0,∞}.
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