MichelWaldschmidt Schanuel’sConjectureandCriteriaforAlgebraicIndependence Abstract DaleBrownawellandStephenSchanuel Schanuel’sConjecture

Conference“ModularFormsandFunctionFieldArithmetic”inhonorofProfessorJingYu’s60thbirthday.TaidaInstituteforMathematicalSciences(TIMS)NationalTaiwanUniversity,Taipei,Taiwan.http://www.tims.ntu.edu.tw/May19–22,2009

Schanuel’s Conjecture and Criteria fo r Algeb raic Indep endence

Michel W alds chmidt

Institut de Math ´ematiques de Jussieu & P aris VI

LecturegivenonMay22,2009;updated:May24,2009

1/55

Abs tract

One of the main op en problems in transcendental numb er theo ry is Schanuel ’s Conjecture which w as stated in the 1960’s : If x

,. .. ,x

are Q –linea rly indep endent complex numb ers, then among the 2 n numb ers x

,. .. ,x

, e

,. .. ,e

, at least n are algeb raically indep endent.

W e first consider some of the consequences of this conjecture ; next w e describ e the transcendental app roach which w as initiated by A.O. Gel’fond in the 40’s, and develop ed by a numb er of mathematicians including W.D. Bro wna w el l , G.V. Chudnovsky , P . Phi lipp on , Y u. Nesterenk o and mo re recently D. Ro y .

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Dale Bro wna w ell and Stephen Schanuel

3

Schanuel’s Conjecture

Let x

,. .. ,x

be Q -linea rly indep endent complex numb ers. Then at least n of the 2 n numb ers x

,. .. ,x

,e

,. .. ,e

are algeb raically indep endent.

In other terms, the conclusion is

tr deg

Q ! x

,. .. ,x

,e

,. .. ,e

" ≥ n.

Remark : F or almost all tuples (with resp ect to the Leb esgue measure) the transcendence degree is 2 n .

4

Origin of Schanuel’s Conjecture

Course given by Serge Lang (1927–2005) at Columbia in the 60’s

S. Lang – Intro duction to transcendental numb ers , Addison-W esley 1966. also attend ed by M. N agata (1927–2008) (14 th Problem of Hilb ert ). Nagata ’s Conjecture solved by E. Bombieri .

5/55

A.O. Gel’fond CRAS 1934

6/55

Statement by Gel’fond (1934)

Let β

,. .. , β

be Q -linea rly indep endent algeb raic numb ers and let log α

,. .. , log α

be Q -linea rly indep endent loga rithms of algeb raic numb ers. Then the numb ers

e

,. .. ,e

, log α

,. .. , log α

are algeb raically in dep endent over Q .

F urther statement by Gel’fond

Let β

,. .. , β

b e algeb raic numb ers with β

# =0 and let α

,. .. , α

b e algeb raic numb ers with α

# =0 , 1 , α

# =0 , 1 , α

# =0 . Then the numb ers

e

. . .

n−1eβn

and α

. . .

m

21

are transcendental, and there is no nontrivial algeb raic relation b et w een such numb ers.

Remark:Theconditiononα2shouldbethatitisirrational.

Easy consequence of Schanuel’s Conjecture

Acco rding to Schanuel ’s Conjecture, the follo wing numb ers are algeb raically indep endent :

e + π ,e π , π

,e

,e

,. .. , e

,. .. , π

, π

,. .. π

.. . log π , log (log 2) , π log 2 , (log 2)(log 3) , 2

, (log 2)

.. .

Pro of : Use Schanuel’ ’s Conjecture several times.

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Lang’s exercise

Define E

= Q . Inductively , fo r n ≥ 1 , define E

as the algeb raic closure of the field generated over E

by the numb ers exp( x )= e

, where x ranges over E

. Let E be the union of E

, n ≥ 0 . Then Schanuel ’s Conjecture implies that the numb er π do es not b elong to E .

Mo re precisely : Schanuel ’s Conjecture implies that the numb ers π , log π , log log π , log log log π ,. .. are algeb raically indep endent over E .

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A va riant of Lang’s exercise

Define L

= Q . Inductively , fo r n ≥ 1 , define L

as the algeb raic closure of the field generated over L

by the numb ers y , where y ranges over the set of complex numb ers such that e

∈ L

. Let L b e the union of L

, n ≥ 0 . Then Schanuel ’s Conjecture implies that the numb er e do es not b elong to L .

Mo re precisely : Schanuel ’s Conjecture implies that the numb ers e, e

,e

,e

.. . are algeb raically indep endent over L .

11

Arizona Wi nter Scho ol A WS2008, T ucson

Theo rem [ Jonathan Bob er , Chuangxun Cheng , Bri an Dietel , Mathilde Herblot , Jing jing Huan g , Holly Krieger , Diego Ma rques , Jonathan Mason , Ma rtin Mereb and Rob ert Wilson .] Schanuel ’s Conjecture implies that the fields E and L are linea rly disjoint over Q .

Definition Given a field extension F /K and tw o sub extensions F

,F

⊆ F , w e sa y F

,F

are linea rly disjoint over K when the follo win g holds : any set { x

,. .. ,x

} ⊆ F

of K – linea rly indep endent elements is linea rly indep endent over F

.

Reference : arXiv.0804.3550 [math.NT] 2008.

12

F ormal analogs

W.D. Bro wna w ell (w as a student of Schanuel ) J. Ax’s Theo rem (1968) : V ersion of Schanuel ’s Conjecture fo r p ow er series over C (and R. Coleman fo r p ow er series over Q )

W ork by W.D. Bro wna w ell and K. Kub ota on the elliptic analog of Ax ’s Theo rem.

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Conjectures by A. Grothendieck and Y. Andr ´e

Generalized Conjecture on P erio ds by Grothendieck : Dimension of the Mumfo rd–T ate group of a smo oth projective va riet y. Generalization by Y. Andr ´e to motives.

Case of 1 –motives : Elliptico-T oric Conjecture of C. Bertolin .

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Ubiquit y of Schanuel’s Conjecture

Other contexts : p –adic numb ers, Leop oldt ’s Conjecture on the p –adic rank of the units of an algeb ra ic numb er field Non-vanishing of Regulato rs Non–degenerescence of heights Conjecture of B. Mazur on rational p oints Diophantine app ro ximation on to ri

Dip endra Prasad Gopal Prasad

Preda Mih ˘ailescu

arXiv :0905 . 1274

Date : F ri, 8 Ma y 2009 14 :52 :57 GMT (16kb)

Title : On Leop oldt’s conjecture and some consequences

Autho rs : Preda Mih ˘ailescu

http ://a rxiv.o rg/abs /0905.1274

The conjecture of Leop oldt states that the p - adic regulato r of a numb er field do es not vanish. It w as proved fo r the ab elian case in 1967 by Brumer, using Bak er theo ry . If the Leop oldt conjecture is false fo r a galois field K , there is a phantom Z

- extension of K

arising. W e sho w that this is strictly co rrelated to some infinite Hilb ert class fields over K

, which are genera ted at intermediate levels by ro ots from units from the base fields. It turns out that the extensions of this typ e have b ounded degree. This implies the Leop oldt conjecture fo r arbitra ry finite numb er fields.

Preda Mih ˘ailescu

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Metho ds from logic

Ehud Hrushovs ki Bo ris Zilb er Jonathan Kirb y

Calculus of ”p redimension functions” ( E. Hrushovski ) Zilb er ’s construction of a ”pseudo exp onentiation” Also : A. Macint yre , D.E. Ma rk er , G. T erzo , A.J. Wilkie , D. Bertrand .. .

18/55

Metho ds from logic : Mo del theo ry

Exp onential algeb rai ci ty in exp onential fields by Jonathan Kirb y

The dimension of the exp onential algeb raic closure op erato r in an exp onential field satisfies a w eak Schanuel prop ert y.

A co rolla ry is that there are at most countably many essential counterexamples to Schanuel ’s Conjecture.

arXiv :0810.4285v2

19

Daniel Bertrand

Daniel Bertrand,

Schanuel’s conjecture fo r non-iso constant elliptic curves over function fields.

Mo del theo ry with applications to algeb ra and analysis. V ol. 1, 41–62, London Math. So c. Lecture Note Ser., 349 , Camb ridge Univ. Press, Camb ridge, 2008.

20

Schanuel’s Conjecture fo r n =1

For n =1 , Schanuel ’s Conjecture is the Hermite–Lindemann Theo rem : If x is a non–zero complex numb ers, then one at least of the 2 numb ers x , e

is trans cendental .

Equivalently , if x is a non–zero algeb raic numb er, then e

is a transcendental numb er.

Another equivalent statement is that if α is a non–zero algeb raic numb er and log α any non–zero loga rithm of α , then log α is a transcendental numb er.

Consequence : transcendence of numb ers lik e

e, π , log 2 ,e

.

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Not kno wn

For n =2 Schanuel’s Conjecture is not yet kno wn : ? If x

,x

are Q –linea rly indep endent complex numb ers, then among the 4 numb ers x

,x

, e

,e

, at leas t 2 are algeb raically indep endent.

A few consequences : With x

=1 , x

= i π : algeb raic indep endence of e and π . With x

=1 , x

= e : algeb raic indep endence of e and e

. With x

= log 2 , x

= (log 2)

: algeb raic indep endence of log 2 and 2

. With x

= log 2 , x

= log 3 : algeb raic indep endence of log 2 and log 3 .

22/55

Not kno wn

It is not kno wn that there exist tw o loga rithms of algeb raic numb ers which are algeb raically indep endent.

Even the non–existence of non–trivial quadratic relations among loga rithms of algeb raic numb ers is not yet established.

Acco rding to the four exp onen ti als Conjecture , any qua dra tic relation (log α

)(log α

) = (log α

)(log α

) is trivial : either log α

and log α

are linea rly dep endent, or else log α

and log α

are linea rly dep endent.

Kno wn

Lindemann–W eierstraß Theo rem = case where x

,. .. ,x

are algeb raic.

Let β

,. .. , β

b e algeb raic numb ers which are linea rly indep endent over Q . Then the numb ers e

,. .. ,e

are algeb raically indep endent over Q .

Hilb ert’s seventh problem

A.O. Gel’fond and Th. Schn ei der (1934). Solution of Hilb ert ’s seventh problem : transcendence of α

and of (log α

) / (log α

) fo r algeb raic α , β , α

and α

. A. Bak er , 1968. Let log α

,. .. , log α

be Q –linea rly indep endent loga rithms of algeb raic numb ers . Then the numb ers 1 , log α

,. .. , log α

are linea rly indep endent over the field Q .

25/55

Problem of Gel’fond and Schneider

Raised by A.O. Gel’fond in 1948 and Th. Schneider in 1952. Conjecture : If α is an algeb raic numb er, α # =0 , α # =1 and if β is an irrational algeb raic numb er of degree d , then the d − 1 numb ers α

, α

, .. . , α

are algeb raically indep endent. Sp ecial case of Schanuel’s Conjecture : T ak e x

= β

log α , n = d , so that # x

,. .. ,x

,e

,. .. ,e

$ is # log α , β log α , .. ., β

log α , α , α

, .. ., α

$ .

The conclusion of Schanuel ’s Conjecture is

tr deg

Q ! log α , α

, α

, .. . , α

" = d.

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Algeb raic indep endence metho d of Gel’fond

A.O. Gel’fond (1948) The tw o numb ers 2

and 2

are algeb raically indep endent. Mo re generally , if α is an algeb raic numb er, α # =0 , α # =1 and if β is a algeb raic numb er of degree d ≥ 3 , then tw o at least of the nu mb ers α

, α

, .. . , α

are algeb raically indep endent.

27

T ools

T ranscendence criterion : Replaces Liouville ’s inequalit y in transcendence pro ofs. Liouville : A non–zero rational integer n ∈ Z satisfies | n | ≥ 1 . Gel’fond : Needs to give a lo w er b ound fo r | P ( θ ) | with P ∈ Z [ X ] \{ 0 } when θ is trans cendental .

Zero estimate fo r exp onential p olynomials : C. Hermite , P . T uran , K. Mahler , R. Tijdeman ,. ..

Small transcendence degree : A.O. Gel’fond , A.A. Smelev , R. Tijdeman , W.D. Bro wna w ell .. .

28

Analytic zero estimates fo r exp onential p olynomials

C. Hermite, P . T uran, K. Mahler

29/55

Sk etch of pro of Assume the transcendence degree over k := Q ( α , β ) of the field L = k ! α

, α

, .. . , α

"

is ≤ 1 . By the Theo rem of Gel’fond and Schneider (solution to Hilb ert’s seventh problem) w e kno w tha t the transcendence degree is 1 .

Consider the exp onential functions

e

,e

,. .. ,e

which are algeb raically indep endent and satisfy di ff erential equations with co e ffi cients in Q ( β ) ⊂ k ⊂ L . These functions tak e values in L when the va riable z is in

Γ = ! Z + Z β ·· · + Z β

" log α .

30/55

Gel’fond–Schneider Metho d

F ollo wing the app roach of Gel’fond and Schneider, one constructs a non–zero p olynomial P ∈ L [ X

,. .. ,X

] such that the exp onential p olynomial

F ( z )= P ! e

,e

,. .. ,e

"

vanishes with some multiplicit y at many p oints in Γ , sa y % d dz &

F ! m

log α + m

β log α + ·· · + m

β

log α " =0 fo r t , m

,. .. ,m

non–negative integers in a certain range. This is achieved by means of Dirichlet’s Bo x Principle , hence one cannot get mo re such equations than there are unkno wns (where unkno wns are the co e ffi cients of the auxilia ry p olynomial).

Pigeonhole principle (Dirichlet), Thue–Siegel Lemma

Lejeune-Dirichlet, C.L. Siegel

Extrap olation : Cauchy Schw arz

F rom Schw arz’s Lemma w e get a sha rp upp er b ound fo r the maximum mo dulus of the auxilia ry function F on some disc. Using Cauchy’s inequalities , w e ded uce that many mo re values % d dz &

F ! m

log α + m

β log α + ·· · + m

β

log α "

have a small mo dulus. A zero estimate sho ws that these numb ers cannot all vanish. W e endup with a non–zero numb er γ in L with a very small absolute va lue, fo r which w e can also b ound the size .

Size

Assume fo r simplicit y that there is a transcendental numb er θ such that all the numb ers β and α

fo r 0 ≤ j ≤ d − 1 b elong to Z [ θ ] . Then the numb er γ which is pro duced is just in Z [ θ ] , and the size of γ measures the degree and the height of this p olynomial.

F or a transcendence pro of, one reaches the conclusion by means of Liouville’s inequalit y . Here another argument is required. This is the transcendence criterion .

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T rans cendence criterion

Simple fo rm : Given a complex numb er ϑ , if there exists a sequence ( P

)

of non–zero p olynomials in Z [ X ] , with P

of degree ≤ n and height ≤ e

, such that

| P

( ϑ ) | ≤ e

fo r all n ≥ 1 , then ϑ is algeb raic and P

( ϑ ) = 0 fo r all n ≥ 1 .

Simplification due to R. Tijdeman , W.D. Bro wna w ell ,. ..in the 70’s and mo re recently M. Laurent and D. Ro y .

35

Rob Tijdeman

http://www.wiskundemeisjes.nl/20080830/ridder-tijdeman/

On the algeb raic indep endence of certain numb ers. Nederl. Ak ad. W etensch. Pro c. Ser. A 74 =Indag. Math. 33 (1971), 146–162.

36

Gel’fond’s trans cendence criterion

First extension : Replace the upp er b ound fo r the degree by d

, the upp er b ound fo r the height by e

, and the upp er b ound fo r | P

( ϑ ) | by e

.

Assumptions on the sequences ( d

)

, ( h

)

and ( ν

)

: d

≤ d

≤ κ d

,d

≤ h

≤ h

≤ κ h

,

with some constant κ > 0 indep endent of n , and

ν

/d

h

→∞ .

37/55

An equivalent statement Let m ≥ 1 and ( ϑ

,. .. , ϑ

) ∈ C

. Let ( d

)

, ( h

)

and ( ν

)

satisfy :

d

≤ d

≤ κ d

,d

≤ h

≤ h

≤ κ h

,

with some constant κ > 0 indep endent of n , and

ν

/d

h

→∞ . Assume tha t there exists a sequence ( P

)

of non–zero p olynomials in Z [ X

,. .. ,X

] , with P

of degree ≤ d

and height ≤ e

, such that

0 < | P

( ϑ

,. .. , ϑ

) | ≤ e

fo r all n ≥ 1 , Then tw o at least of the numb ers ϑ

,. .. , ϑ

are algeb raically indep endent. The conclusion is that the transcendence degree of the field Q ( ϑ

,. .. , ϑ

) is at least 2 .

38/55

Criterion fo r la rge trans cendence degree

It might seem natural to exp ect that the same statement with the stronger assumption ν

/d

h

→∞ in pla ce of ν

/d

h

→∞ w ould yield the conclusion that the transcendence degree of the field Q ( ϑ

,. .. , ϑ

) is at least t .

A counterexample due to Khinchine (a reference is in Cassel ’s b o ok on Diophantine App ro ximation) rules this out. Some further assumption is necessa ry .

Lang’s transcendence typ e

An inductive pro cess has b een suggested by S. Lang : at each step one pro duces a quantitative estima te (transcendence measure to sta rt with, next measures of algeb raic indep endence) which replace Liouville ’s inequalit y at the next stage.

Results pro duced by the metho d are rather w eak and do not go further than small tra nscendence degree .

La rge trans cendence degree

G.V. Chudnovsky (1976) Among the numb ers

α

, α

, .. . , α

at least [log

d ] are algeb raically indep endent.

G.V. Chudno vsky – On the path to Scha nuel ’s Conjecture. Algeb raic curves close to a p oint. I. General theo ry of colo red sequences. II. Fields of finite transcendence typ e and colo red sequences. Resultants . Studia Sci. Math. Hunga r. 12 (1977), 125–157 (1980).

41/55

P artial res ult on the problem of Gel’fond and Schneider

A.O.Gel’fond,G.V.Chudnovskii,P.Philippon,Yu.V.Nesterenko.

G. Diaz (1989) : If α is an algeb raic numb er, α # =0 , α # =1 and if β is an irrational algeb raic numb er of degree d , then

tr deg

Q ! α

, α

, .. . , α

" ≥ ' d +1 2 ( .

42/55

Ho w could w e attack Schanuel’s Conjecture ?

Let x

,. .. ,x

be Q –linea rly indep endent complex numb ers. F ollo wing the transcendence metho ds of Hermite , Gel’fond , Schneider .. ., one ma y sta rt by intro ducing an auxilia ry function F ( z )= P ( z ,e

) where P ∈ Z [ X

,X

] is a non–zero p olynomial. One considers the derivatives of F

F

= % d dz &

F

at the p oints m

x

+ ·· · + m

x

fo r va rious values of ( m

,. .. ,m

) ∈ Z

.

43

The derivation

Let D denote the derivation

D = ∂ ∂ X

+ X

∂ ∂ X

over the ring C [ X

,X

] , so that fo r P ∈ C [ X

,X

] the derivatives of the function

F ( z )= P ( z ,e

) are given by % d dz &

F =( D

P )( z, e

) .

44

Auxilia ry function

Recall that x

,. .. ,x

are Q –linea rly indep endent complex numb ers. Let α

,. .. , α

b e non–zero complex numb ers. The transcendence machinery pro duces a sequence ( P

)

of p olynomials with integer co e ffi cients satisfying ) ) ) ) ) ! D

P

" *

n

+

j=1

m

x

,

,

j=1

α

- ) ) ) ) ) ≤ exp( − N

) fo r any non-negative integers k , m

,. .. ,m

with k ≤ N

and max { m

,. .. ,m

} ≤ N

.

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Ro y’s app roach to Schanuel’s Conjecture (1999)

If the numb er of equations w e pro duce is to o small, such a set of relations do es not contain any info rmation : the existence of a sequence of non–trivial p olynomials ( P

)

follo ws from linea r algeb ra.

On the other hand, foll owing D. Ro y , one ma y exp ect that the existence of a sequence ( P

)

pro ducing su ffi ciently many such equations will yield the conclusion :

tr deg

Q ! x

,. .. ,x

, α

,. .. , α

" ≥ n.

A rema rquable res ult of D. Ro y is that such equations imply α

= e

fo r some p ositive integer d , and this enables him to sho w that Schanuel ’s Conjecture is equivalent to the existence of su ffi ciently many small values.

46/55

Ro y’s Conjecture (1999) Let s

,s

,t

,t

,u p ositive real numb ers satisfying

max { 1 ,t

, 2 t

} < min { s

, 2 s

} and max { s

,s

+ t

} <u< 1 2 (1 + t

+ t

) . Assume that, fo r any su ffi ciently la rge p ositive integer N , there exists a non–zero p olynomial P

∈ Z [ X

,X

] with pa rtial degree ≤ N

in X

, pa rtial degree ≤ N

in X

and height ≤ e

which sati sfies ) ) ) ) ) ! D

P

" *

n

+

j=1

m

x

,

,

j=1

α

- ) ) ) ) ) ≤ exp( − N

) fo r any non-negative integers k , m

,. .. ,m

with k ≤ N

and max { m

,. .. ,m

} ≤ N

. Then

tr deg

Q ( x

,. .. ,x

, α

,. .. , α

) ≥ n.

Ro y’s Theo rem (1999)

Ro y ’s Conjecture is equivalent to Schanuel ’s Conjecture.

Mo re precisely , if Schanuel ’s Conjecture is true, then Ro y ’s Conjecture holds fo r any set of pa rameters s

,s

,t

,t

,u satisfying max { 1 ,t

, 2 t

} < min { s

, 2 s

}

and max { s

,s

+ t

} <u< 1 2 (1 + t

+ t

) .

Conversely , if Ro y ’s Conjecture holds fo r one set of pa rameters s

,s

,t

,t

,u satisfying thes e conditions, then Schanuel ’s Conjecture is true.

Extending the range

Recently Nguy en Ngo c Ai V an succeeded to extend slightly the range of the admissible values of the pa rameters s

,s

,t

,t

,u .

Such an extension is interesting fo r b oth implications of the equivalence b et w een Schanuel ’s Conjecture and Ro y ’s Conjecture.

49/55

Equivalence b et w een Schanuel and Ro y

Let ( x, α ) ∈ C × C

, and let s

,s

,t

,t

,u b e p ositive real numb ers satisfying the inequa lities of Ro y ’s Conjecture. Then the follo wing conditions are equivalent : (a) The numb er α e

is a ro ot of unit y. (b) F or any su ffi ciently la rge p ositive integer N , there exists a non–zero p olynomia l Q

∈ Z [ X

,X

] with pa rtial degree ≤ N

in X

, pa rtial degree ≤ N

in X

and height H( Q

) ≤ e

such that ) ) ( D

Q

)( m x, α

) ) ) ≤ exp( − N

) .

fo r any k ,m ∈ N with k ≤ N

and m ≤ N

.

50/55

Ro y’s program to w ards Schanuel’ s Conjecture

In Gel’fond’s transcendence criterion,

• replace a single va riable by tw o va riables X , Y

• intro duce several p oints ( m

x

+ ·· · + m

x

, α

·· · , α

)

• intro duce mul ti plicit y involving the deriva tive

D =( ∂ / ∂ X )+ Y ( ∂ / ∂ Y ) ,

• get la rge transcendence degree .

51

T rans cendence criterion with multipliciti es With derivatives : Given a complex numb er ϑ , ass ume that there exists a sequence ( P

)

of non–zero p olynomials in Z [ X ] , with P

of degree ≤ d

and height ≤ e

, such that

max

nn

P ( ϑ ) ;0 ≤ j < t ≤ e )

#)) $ ) fo r all n ≥ 1 . Assume ν

t

/d

h

→∞ . Then ϑ is algeb raic. Due to M. Laurent and D. Ro y , appl ications to algeb raic indep endence with interp olation determinants .

52

Criterion with several p oints

Goal : Given a sequence of complex numb ers ( ϑ

)

, assu me that there exists a sequence ( P

)

of non–zero p olynomials in Z [ X ] , with P

of degree ≤ d

and height ≤ e

, such that

max

ninn

P ( ϑ ) ;0 ≤ j < t , 1 ≤ i ≤ s ≤ e )

#)) ) $ fo r all n ≥ 1 . Assume ν

t

s

/d

h

→∞ . W e wish to deduce that the numb ers ϑ

are algeb raic.

D. Ro y : Not true in general, but true in some sp ecial ca ses with a structure on the sequence ( ϑ

)

. Combines the elimination arguments used fo r criteria of algeb raic indep endence and fo r zero estimates.

53/55

Small value estimates fo r the additive group

D. Ro y . Small va lue estimates fo r the additive group . Intern. J. Numb er Theo ry , to app ea r.

Let ξ b e a transcendental complex numb er, let h , σ , τ and ν b e non-negative real numb ers, let n

b e a p ositive integer, and let ( P

)

b e a sequence of non–zero p olynomials in Z [ X ] satisfying deg ( P

) ≤ n and H ( P

) ≤ exp( n

) fo r each n ≥ n

. Supp ose that h> 1 , (3 / 4) σ + τ < 1 and ν > 1+ h − (3 / 4) σ − τ . Then fo r infinitely many n , w e have max # | P

( i ξ ) | ;0 ≤ i ≤ n

, 0 ≤ j ≤ n

$ > exp( − n

) .

54/55

Small value estimates fo r the multiplicative group

D. Ro y . Small va lue estimates fo r the multiplicative group . Acta Arith., to app ea r.

Let ξ

,. .. , ξ

b e multiplicatively indep endent complex numb ers in a field of transcendence degree 1 . Under suitable assumptions on the pa rameters h, σ , τ , ν , fo r infi nitel y many p ositive integers n , there exists no non–zero p olynomial P ∈ Z [ Y ] satisfying deg ( P ) ≤ n , H ( P ) ≤ exp( n

) and max # | P

( ξ

·· · ξ

) | ;0 ≤ i

,. .. ,i

≤ n

, 0 ≤ j ≤ n

$

> exp( − n

) .

The end

Happ y Birthda y, Jing Y u,

Heureux Anniversaire,

And lot of thanks to all thos e who contributed to the organization of this exceedingly interesting conf erence