3 / 36 References Hermite–Lindemann’s Theorem (continued) !Another alternative statement of Hermite–Lindemann’s Theorem : Letβbe a nonzero algebraic number

References

The role of complex conjugation in transcendental number theory

Michel Waldschmidt

Institut de Math´ematiques de Jussieu + CIMPA http://www.math.jussieu.fr/∼miw/

December 17, 2005

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References

DION 2005 TIFR December 17, 2005

Hermite–Lindemann’s Theorem as a consequence of Gel’fond–Schneider’s Theorem

The Six Exponentials Theorem and the Four Exponentials Conjecture

The strong Six Exponentials Theorem and the strong Four Exponentials Conjecture

Recent results

Product of logarithms of algebraic numbers References

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References

Hermite–Lindemann’s Theorem

!Letαbe a nonzero algebraic number and letlogαbe any nonzero logarithm ofα. Thenlogαis

transcendental.

!Notations.Denote byQthe field of algebraic numbers and byLtheQ-vector space of logarithms of algebraic numbers :

L={λ∈C;e^λ∈Q^×}= exp⁻¹(Q^×) ={logα;α∈Q^×}.

!Alternative statement of Hermite–Lindemann’s Theorem :

L∩Q={0}.

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References

Hermite–Lindemann’s Theorem (continued)

!Another alternative statement of

Hermite–Lindemann’s Theorem : Letβbe a nonzero algebraic number. Thene^βis transcendental.

!Question (G. Diaz) :Lettbe a non-zero real number andβa non-zero algebraic number. Is it true thate^tβ is transcendental ?

!Answer (G. Diaz) :No !

!First example :assumeβ∈R. Taket= (log 2)/β.

!Second example :assumeβ∈iR. Taket=iπ/β.

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References

Diaz’ Theorem

!Letβ∈Qandt∈R^×. Assumeβ%∈R∪iR. Thene^tβ is transcendental.

!Equivalently :forλ∈Lwithλ%∈R∪iR, Rλ∩Q={0}.

!Proof.Setα=e^tβ. The complex conjugateαofαis e^tβ=α^β/β. Sinceβ%∈R∪iR, the algebraic number β/βis not real (its modulus is 1 and it is not±1), hence not rational.Gel’fond-Schneider’s Theorem implies thatαandαcannot be both algebraic. Hence they are both transcendental.

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References

Gel’fond-Schneider implies Hermite-Lindemann (almost)

!Gel’fond-Schneider’s Theorem implies :there exists β0∈R∪iRsuch that

{β∈Q;e^β∈Q}=Qβ0.

!Remark.Hermite–Lindemann’s Theorem tells us that in factβ0= 0.

!Proof.From Gel’fond-Schneider’s Theorem one deduces that theQ-vector–space{β∈Q;e^β∈Q}has dimension≤1 and is contained inR∪iR.

!Schneider’s method : proof without derivatives.

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References

Reference

G. Diaz–«Utilisation de la conjugaison complexe dans l’´etude de la transcendance de valeurs de la fonction exponentielle»,J. Th´eor. Nombres Bordeaux16(2004), p. 535–553.

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References

The Six Exponentials Theorem

!Selberg, Siegel, Lang, Ramachandra.

!Theorem :Ifx1, x2areQ–linearly independent complex numbers andy1, y2, y3areQ–linearly independent complex numbers, then one at least of the six numbers

e^x1y1, e^x1y2, e^x1y3, e^x2y1, e^x2y2, e^x2y3 is transcendental.

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References

The Six Exponentials Theorem

References :

S. Lang–Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.

K. Ramachandra–«Contributions to the theory of transcendental numbers. I, II»,Acta Arith. 14 (1967/68), 65-72 ; ibid.14(1967/1968), p. 73–88.

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References

Corollary

!Example :Takex1= 1,x2=π,y1= log 2,y2=πlog 2, y3=π²log 2, the six exponentials are respectively

2,2^π,2^π2, 2^π,2^π2, 2^π3, hence one at least of the three numbers

2^π,2^π2, 2^π3 is transcendental

!Shorey :lower bound for

|2^π−α1|+|2^π2−α2|+|2^π3−α3| for algebraicα1,α2,α3. The estimate depends on the heights and degrees of these algebraic numbers.

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References

Relevant references

References :

T. N. Shorey–«On a theorem of Ramachandra», Acta Arith.20(1972), p. 215–221.

T. N. Shorey–«On the sum!3

k=12^πk−αk,αk

algebraic numbers»,J. Number Theory6(1974), p. 248–260.

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References

S. Srinivasan contributions

Further references :

S. Srinivasan–«On algebraic approximations to 2^πk(k= 1,2,3, . . .)»,Indian J. Pure Appl. Math.5 (1974), no. 6, p. 513–523.

S. Srinivasan–«On algebraic approximations to

2^πk(k= 1,2,3,· · ·). II»,J. Indian Math. Soc. (N.S.)

43(1979), no. 1-4, p. 53–60 (1980).

K. Ramachandra&S. Srinivasan–«A note to a paper : “Contributions to the theory of transcendental numbers. I, II” by Ramachandra on transcendental numbers»,Hardy-Ramanujan J.6(1983), p. 37–44.

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References

Conjectures

!Remark :It is unknown whether one of the two numbers

2^π,2^π2

is transcendental. Oneconjectures(Schanuel) that each of the three numbers 2^π,2^π2,2^π3is transcendental

!and that the numbers

π, log 2, 2^π,2^π2, 2^π3 are algebraically independent.

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References

The Four Exponentials Conjecture

!Selberg, Siegel, Schneider, Lang, Ramachandra.

!Conjecture.Ifx1, x2areQ–linearly independent complex numbers andy1, y2areQ–linearly independent complex numbers, then one at least of the four numbers

e^x1y1, e^x1y2, e^x2y1, e^x2y2 is transcendental.

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References

Ramachandra’s trick

!Remark :Letxandybe tworealnumbers.

The following properties are equivalent : (i) one at least of the two numbersx,yis transcendental.

(ii) thecomplexnumberx+iyis transcendental.

!Example :(H.W. Lenstra) ifγis Euler’s constant, then the numberγ+ie^γis transcendental.

!Proof :checkγ%= 0 and use Hermite–Lindemann’s Theorem.

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References

Ramachandra’s trick

Other example.

!Letx1, x2be two elements inR∪iRwhich are Q–linearly independent. Lety1, y2be two complex numbers. Assume that the three numbersy1, y2, y2are Q–linearly independent. Then one at least of the four numbers

e^x1y1, e^x1y2, e^x2y1, e^x2y2 is transcendental.

!Proof :Sety3=y2. Thene^xjy3=e^±xjy2forj= 1,2 andQis stable under complex conjugation.

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References

Logarithms of algebraic numbers

Rank of matrices.An alternate form of the Six Exponentials Theorem (resp. the Four Exponentials Conjecture) is the fact thata2×3(resp.2×2) matrix with entries inL

"

λ11 λ12 λ13

λ21 λ22 λ23

#

(resp.

"

λ11 λ12

λ21 λ22

# ), the rows of which are linearly independent overQand the columns of which are also linearly independent overQ, has maximal rank2.

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References

A lemma on the rank of matrices

Remark.Ad×&matrixMhas rank≤1 if and only if there existx1, . . . , xdandy1, . . . , y%such that

M=







x1y1 x1y2 . . . x1y%

x2y1 x2y2 . . . x2y%

... ... . .. ...

xdy1 xdy2 . . . xdy%





.

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References

Linear combinations of logarithms of algebraic numbers

Denote byL*theQ-vector space spanned by 1 andL: henceL*is the set of linear combinations with algebraic coefficients of logarithms of algebraic numbers :

L*={β0+β1λ1+· · ·+βnλn;n≥0,βi∈Q,λi∈L}.

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References

The strong Six Exponentials Theorem

Theorem (D.Roy).Ifx1, x2areQ–linearly independent complex numbers andy1, y2, y3areQ–linearly independent complex numbers, then one at least of the six numbers

x1y1, x1y2, x1y3, x2y1, x2y2, x2y3

is not inL*.

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References

The strong Four Exponentials Conjecture

Conjecture.Ifx1, x2areQ–linearly independent complex numbers andy1, y2areQ–linearly independent complex numbers, then one at least of the four numbers

x1y1, x1y2, x2y1, x2y2

is not inL*.

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References

Lower bound for the rank of matrices

!Rank of matrices.An alternate form of the strong Six Exponentials Theorem (resp. the strong Four Exponentials Conjecture) is the fact thata2×3(resp.

2×2) matrix with entries inL*

"

Λ11 Λ12 Λ13

Λ21 Λ22 Λ23

#

(resp.

"

Λ11 Λ12

Λ21 Λ22

# ), the rows of which are linearly independent overQand the columns of which are also linearly independent over Q, has maximal rank2.

!Remark :Under suitable conditions one can show that a d×!matrix with entries inL*has rank≥d!/(d+!). This is a consequence of theLinear Subgroup Theorem.

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References

The strong Six Exponentials Theorem

References :

D. Roy–«Matrices whose coefficients are linear forms in logarithms»,J. Number Theory41(1992), no. 1, p. 22–47.

M. Waldschmidt–Diophantine approximation on linear algebraic groups, Grundlehren der

Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.326, Springer-Verlag, Berlin, 2000.

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References

Alternate form of the strong Four Exponentials Conjecture

!Conjecture.LetΛ1,Λ2,Λ3be nonzero elements inL*. Assume the numbersΛ2/Λ1andΛ3/Λ1are both transcendental. Then the numberΛ2Λ3/Λ1is not inL.*

!Equivalence between both statements :the matrix

"

Λ1 Λ2

Λ3 Λ2Λ3/Λ1

#

has rank 1.

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References

Consequences of the strong Four Exponentials Conjecture

Assume the strong Four Exponentials Conjecture.

!IfΛis inL \* Qthen the quotient 1/Λis not inL*.

!IfΛ1andΛ2are inL \* Q, then the productΛ1Λ2is not inL*.

!IfΛ1andΛ2are inL*withΛ1andΛ2/Λ1

transcendental, then this quotientΛ2/Λ1is not inL.*

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References

Example where the strong Four Exponentials Conjecture is true

!Theorem(G. Diaz).Letx1andx2be two elements of R∪iRwhich areQ–linearly independent. Lety1, y2be two complex numbers such that the three numbersy1, y2,y2areQ-linearly independent. Then one at least of the four numbers

x1y1, x1y2, x2y1, x2y2

is not inL*.

!Proof :Sety3=y2. Thene^xjy3=e^±xjy2forj= 1,2 andL*is stable under complex conjugation.

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References

Example where the strong Four Exponentials Conjecture is true

!Corollaryof Diaz’ Theorem.LetΛ1,Λ2,Λ3be three elements inL*. Assume that the three numbersΛ1,Λ2, Λ2are linearly independent overQ. Further assume Λ3/Λ1∈(R∪iR)\Q. Then

Λ2Λ3/Λ1%∈Q.

!Proof :setx1= 1,x2=Λ3/Λ1,y1=Λ1,y2=Λ2.

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References

Example where the strong Four Exponentials Conjecture is true

Consequence :one deduces examples where one can actually prove that numbers like

1/Λ, Λ1Λ2, Λ2/Λ1

(withΛ,Λ1,Λ2inL*) are not inL*.

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References

Transcendence ofe^π2

!Open problem :is the numbere^π2transcendental ?

!More generally :forλ∈L \ {0}, is it true that λλ%∈L?

!More generally :forλ1andλ2inL \ {0}, is it true thatλ1λ2%∈L?

!Forλ1andλ2inL \ {0}, is it true thatλ1λ2%∈L*?

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References

Product of logarithms of algebraic numbers

!Theorem (Diaz).Letλ1andλ2be inL \ {0}. Assume λ1∈R∪iRandλ2%∈R∪iR. Thenλ1λ2%∈L*.

!Proof.Apply the strong Six Exponentials Theorem to

"

1 λ2 λ2

λ1 Λ Λ

#

withΛ∈L*.

!Diaz’ Conjecture.Letu∈C^×. Assume|u|is algebraic.

Thene^uis transcendental.

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References

Recent results

G. Diaz–«Utilisation de la conjugaison complexe dans l’´etude de la transcendance de valeurs de la fonction exponentielle»,J. Th´eor. Nombres Bordeaux16(2004), p. 535–553.

G. Diaz–«Produits et quotients de combinaisons lin´eaires de logarithmes de nombres alg´ebriques : conjectures et r´esultats partiels», Submitted (2005), 19 p.

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References

Recent results

M. Waldschmidt–«Transcendence results related to the six exponentials theorem»,Hindustan Book Agency (2005), p. 338–355.

Appendix by H. Shiga : Periods of the Kummer surface, p. 356–358.

M. Waldschmidt–«Further variations on the Six Exponentials Theorem», The Hardy-Ramanujan Journal, vol.28, to appear ondecember 22, 2005.

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References

Further result

LetMbe a 2×3 matrix with entries inL*: M=

"

Λ11 Λ12 Λ13

Λ21 Λ22 Λ23

#

· Assume that the five rows of the matrix

"

M I3

#

=









Λ11 Λ12 Λ13

Λ21 Λ22 Λ23

1 0 0

0 1 0

0 0 1









are linearly independent overQand that the five columns of the matrix

,I2, M-

=

"

1 0 Λ11 Λ12 Λ13

0 1 Λ21 Λ22 Λ23

#

are linearly independent overQ.

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References

Further result

Then one at least of the three numbers

∆1= ++ ++Λ12 Λ13

Λ22 Λ23

++ ++, ∆2=

++ ++Λ13 Λ11

Λ23 Λ21

++ ++, ∆3=

++ ++Λ11 Λ12

Λ21 Λ22

++ ++ is not inL.*

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References

Higher rank : an example

LetM= (Λij)1≤i≤m;1≤j≤%be am×&matrix with entries in L. Denote by* Imthe identitym×mmatrix and assume that them+&column vectors of the matrix (Im, M) are linearly independent overQ. LetΛ1, . . . ,Λmbe elements of L. Assume that the numbers 1,* Λ1, . . . ,ΛmareQ-linearly independent. Assume further&> m². Then one at least of the&numbers

Λ1Λ1j+· · ·+ΛmΛmj (j= 1, . . . ,&) is not inL*.

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References

DION 2005 TIFR December 17, 2005

Janam din diyan wadhayian, Tarlok !

http ://www.math.jussieu.fr/∼miw/

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