aws lecture 4swc.math.arizona.edu/aws/2008/08waldschmidtslides4.pdf · 2012-03-02 · close...

AWS Lecture 4 Tucson, Monday, March 17, 2008

Auxiliary functions in

transcendence proofs

Michel Waldschmidt

http://www.math.jussieu.fr/∼miw/

Michel Waldschmidt

Abstract

Transcendence proofs most often involve an auxiliary function.Such functions can take several forms. Historically, the firstones were Pade approximations, in Hermite’s proof of thetranscendence of e (1873). Next came functions whose existenceis proved by means of the Dirichlet’s box principle, with thework of Thue (early 1900) and Siegel (in the 1920’s). Anothertool was provided by interpolation formulae, mainly Newtoninterpolation (involving Hermite’s formulae again) in the studyby G. Polya (1914) and A.O. Gel’fond (1929) of integer valuedentire functions. Along these lines, recent developments are dueto T. Rivoal (to appear), who renewed the forgotten rationalinterpolation formulae of R. Lagrange (1935). In 1991M. Laurent introduced interpolation determinants, and twoyears later J.B. Bost used Arakhelov theory to prove slopeinequalities, which dispens of the choice of bases.

Michel Waldschmidt http://www.math.jussieu.fr/∼miw/

Existence of transcendental numbers

Theorem [Liouville, 1844] Let α be a real algebraicnumber. There exists κ > 0 such that, for any rationalnumber p/q distinct from α with q ≥ 2,∣∣∣∣α− p

q

∣∣∣∣ ≥ 1

qκ·

Corollary Let ξ be a real number. Assume that for anyκ > 0 there exists a rational number p/q with q ≥ 2 suchthat

0 <

∣∣∣∣ξ − p

q

∣∣∣∣ < 1

qκ·

Then ξ is transcendental.


Proof of Liouville’s inequality

α is algebraic means that there exists a non–zeropolynomial f ∈ Z[X] such that f(α) = 0.Let d be the degree of f . Since p/q is distinct from α wehave f(p/q) 6= 0.Hence qdf(p/q) is a non–zero rational integer

|f(p/q)| ≥ 1

qd·

On the other hand

|f(p/q)| ≤ c(α)

∣∣∣∣α− p

q

∣∣∣∣ .Therefore ∣∣∣∣α− p

q

∣∣∣∣ ≥ c(α)

qd·

Auxiliary function : f .Michel Waldschmidt http://www.math.jussieu.fr/∼miw/

Irrationality proofs

Early methods : Involve continued fractions.Lambert (1767) : irrationality of πClose relation with Pade ApproximationHermite, (1849) :Tout ce que je puis, c’est de refaire ce qu’a deja faitLambert, seulement d’une autre maniere.All I can do is to repeat what Lambert did, just in anotherway.Reference : C. BrezinskiThe long history of continued fractions and Pade approximants.Pade approximation and its applications, Amsterdam 1980, pp.1–27, Lecture Notes in Math., 888, Springer, Berlin-New York,1981.History of continued fractions and Pade approximants. SpringerSeries in Computational Mathematics, 12. Springer-Verlag,Berlin, 1991.


Fourier’s proof (1815) for the irrationality of e

Truncate the Taylor expansion at the origin : the auxiliaryfunction is a polynomial (or rather the remainder : thedifference between ez and the initial polynomial).

First examples of transcendental numbers : Liouville 1844,continued fractions, fast converging series.


Idea of Ch. Hermite

Ch. Hermite (1822 - 1901).Approximate the exponential function ez

by rational fractions A(z)/B(z).Means : Taylor developments matchfor the first terms.

Auxiliary function :

B(z)ez − A(z)

with a zero at the origin of high multiplicity.


Simultaneous approximation to the exponential

function

Irrationality results follow from rational approximationsA/B ∈ Q(x) to the exponential function ex.

One of Hermite’s ideas is to consider simultaneous rationalapproximations to the exponential function, in analogy withDiophantine approximation.

Let B0, B1, . . . , Bm be polynomials in Z[x]. For 1 ≤ k ≤ mdefine

Rk(x) = B0(x)ekx −Bk(x).

Set bj = Bj(1), 0 ≤ j ≤ m and

R = a0 + a1R1(1) + · · ·+ amRm(1).

If 0 < |R| < 1, then a0 + a1e+ · · ·+ amem 6= 0.


Hermite : approximation to the functions

1, eα1x, . . . , eαmx

Let α1, . . . , αm be pairwise distinct complex numbers andn0, . . . , nm be rational integers, all ≥ 0. SetN = n0 + · · ·+ nm.

Hermite constructs explicitly polynomials B0, B1, . . . , Bm

with Bj of degree N − nj such that each of the functions

B0(z)eαkz −Bk(z), (1 ≤ k ≤ m)

has a zero at the origin of multiplicity at least N .


Pade approximants

Henri Eugene Pade (1863 - 1953)Approximation of complexanalytic functions byrational functions.


Pade Approximants of type II

Let f0, . . . , fm be complex functions which are analytic nearthe origin and n0, . . . , nm be rational integers, all ≥ 0. SetN = n0 + · · ·+ nm.There are two dual points of view, giving rise to the twotypes of Pade Approximants.Pade approximants of second type : polynomialsB0, . . . , Bm with Bj having degree ≤ N − nj, such thateach of the functions

Bi(z)fj(z)−Bj(z)fi(z) (0 ≤ i < j ≤ m)

has a zero of multiplicity > N .

Reference : N.I. Feldman and Yu.V. Nesterenko, NumberTheory IV, Transcendental Numbers, Encyclopaedia ofMathematical Sciences, 44 (1998) Chap. 2.


Pade approximants of type I

Let f1, . . . , fm be complex functions which are analytic nearthe origin and let n1, . . . , nm be non-negative integers. SetM = n1 + · · ·+ nm.

Pade approximants of the first type : polynomialsP1, . . . , Pm with Pj of degree ≤ nj such that the function

P1(z)f1(z) + · · ·+ Pm(z)fm(z)

has a zero at the origin of multiplicity at least M +m− 1.

Studied by Ch. Hermite in 1873 and 1893.


Hermite )1893)

If α1, . . . , αm are pairwise distinct complex numbers,n0, . . . , nm non–negative integers, Hermite constructsexplicitely polynomials P1, . . . , Pm with Pj of degree njsuchthat the function

P1(z)eα1z + · · ·+ Pm(z)eαmz

has a zero at the origin of multiplicity at leastn1 + · · ·+ nm +m− 1.

C. Hermite (1917) : further integral formula for theremainder.

Application to transcendence : effective version of theHermite, Lindemann and Weierstraß theorems byK. Mahler (1930).


Transcendental functions

A complex function is called transcendental if it istranscendental over the field C(z), which means that thefunctions z and f(z) are algebraically independent : ifP ∈ C[X, Y ] is a non-zero polynomial, then the functionP(z, f(z)

)is not 0.

Exercise. An entire function (analytic in C) istranscendental if and only if it is not a polynomial.

Example. The transcendental entire function ez takes analgebraic value at an algebraic argument z only for z = 0.


Weierstrass question

Is–it true that a transcendentalentire function f takes usuallytranscendental values at algebraicarguments ?

Answers by Weierstrass (letter to Strauss in 1886), Strauss,Stackel, Faber, van der Poorten, Gramain. . .If S is a countable subset of C and T is a dense subset ofC, there exist transcendental entire functions f mapping Sinto T , as well as all its derivatives.Also there are transcendental entire functions f such thatDkf(α) ∈ Q(α) for all k ≥ 0 and all algebraic α.


Integer valued entire functions

An integer valued entire function is a function f , which isanalytic in C, and maps N into Z.

Example : 2z is an integer valued entire function, not apolynomial.

Question : Are-there integer valued entire function growingslower than 2z without being a polynomial ?

Let f be a transcendental entire function in C. For R > 0set

|f |R = sup|z|=R

|f(z)|.


Integer valued entire functions

G. Polya (1914) :if f is not a polynomialand f(n) ∈ Z for n ∈ Z≥0, then

lim supR→∞

2−R|f |R ≥ 1.

Further works on this topic by G.H. Hardy, G. Polya,D. Sato, E.G. Straus, A. Selberg, Ch. Pisot, F. Carlson,F. Gross,. . .


Arithmetic functions

Polya’s proof starts by expanding the function f into aNewton interpolation series at the points 0, 1, 2, . . . :

f(z) = a0 + a1z + a2z(z − 1) + a3z(z − 1)(z − 2) + · · ·

Since f(n) is an integer for all n ≥ 0, the coefficients an arerational and one can bound the denominators. If f does notgrow fast, one deduces that these coefficients vanish forsufficiently large n.


Newton interpolation series

From

f(z) = f(α1) + (z − α1)f1(z), f1(z) = f1(α2) + (z − α2)f2(z), . . .

we deduce

f(z) = a0 + a1(z − α1) + a2(z − α1)(z − α2) + · · ·

with

a0 = f(α1), a1 = f1(α2), . . . , an = fn(αn+1).


An identity due to Ch. Hermite

1

x− z=

1

x− α+z − αx− α

· 1

x− z·

Repeat :

1

x− z=

1

x− α1

+z − α1

x− α1

·(

1

x− α2

+z − α2

x− α2

· 1

x− z

)·


An identity due to Ch. Hermite

Inductively we deduce the next formula due to Hermite :

1

x− z=

n−1∑j=0

(z − α1)(z − α2) · · · (z − αj)(x− α1)(x− α2) · · · (x− αj+1)

+(z − α1)(z − α2) · · · (z − αn)

(x− α1)(x− α2) · · · (x− αn)· 1

x− z·


Newton interpolation expansion

Application. Multiply by (1/2iπ)f(z) and integrate :

f(z) =n−1∑j=0

aj(z − α1) · · · (z − αj) +Rn(z)

with

aj =1

2iπ

∫C

f(x)dx

(x− α1)(x− α2) · · · (x− αj+1)(0 ≤ j ≤ n− 1)

and

Rn(z)= (z − α1)(z − α2) · · · (z − αn)·1

2iπ

∫C

f(x)dx

(x− α1)(x− α2) · · · (x− αn)(x− z)·


Rational interpolation

Rene Lagrange (1935).

1

x− z=

α− β(x− α)(x− β)

+x− βx− α

· z − αz − β

· 1

x− z·

Iterating and integrating yield

f(z) =N−1∑n=0

Bn(z − α1) · · · (z − αn)

(z − β1) · · · (z − βn)+ RN(z).


Hurwitz zeta function

T. Rivoal (2006) : consider Hurwitz zeta function

ζ(s, z) =∞∑k=1

1

(k + z)s·

Expand ζ(2, z) as a series in

z2(z − 1)2 · · · (z − n+ 1)2

(z + 1)2 · · · (z + n)2·

The coefficients of the expansion belong to Q + Qζ(3).This produces a new proof of Apery’s Theorem on theirrationality of ζ(3).In the same way : new proof of the irrationality of log 2 byexpanding

∞∑k=1

(−1)k

k + z·


Mixing C. Hermite and R. Lagrange

T. Rivoal (2006) : new proof of the irrationality of ζ(2) byexpanding

∞∑k=1

(1

k− 1

k + z

)as a Hermite–Lagrange series in(

z(z − 1) · · · (z − n+ 1))2

(z + 1) · · · (z + n)·


Taylor series and interpolation series

Taylor series are the special case of Hermite’s formula witha single point and multiplicities — they give rise to Padeapproximants.

Multiplicities can also be introduced in Rene Lagrangeinterpolation.


Integer valued entire function on Z[i]

A.O. Gel’fond (1929) : growth of entire functions mappingthe Gaussian integers into themselves.Newton interpolation series at the points in Z[i].

An entire function f which is not a polynomial and satisfiesf(a+ ib) ∈ Z[i] for all a+ ib ∈ Z[i] satisfies

lim supR→∞

1

R2log |f |R ≥ γ.

F. Gramain (1981) : γ = π/(2e).This is best possible : D.W. Masser (1980).


Transcendence of eπ

A.O. Gel’fond (1929).

Ifeπ = 23, 140 692 632 779 269 005 729 086 367 . . .

is rational, then the function eπz takes values in Q(i) whenthe argument z is in Z[i].

Expand eπz into an interpolation series at the Gaussianintegers.


Hilbert’s seventh problem

A.O. Gel’fond and Th. Schneider (1934).Solution of Hilbert’s seventh problem :transcendence of αβ

and of (logα1)/(logα2)for algebraic α, β, α2 and α2.

Duality between the methods of Gel’fond and Schneider :Fourier-Borel transform.


Gel’fond and Schneider methods

Assume α, β and αβ = exp(logα) are algebraic with β 6∈ Qand logα 6= 0. Let K = Q(α, β, αβ).

A.O. Gelfond :The two entire functions ez and eβz are algebraicallyindependent, they satisfy differential equations withalgebraic coefficients and they take simultaneously values inK for infinitely many z, viz. z ∈ Z logα.

Th. Schneider :The two entire functions z and αz = ez logα are algebraicallyindependent, they take simultaneously values in K forinfinitely many z, viz. z ∈ Z + Zβ.No use of differential equations (coefficients are not allalgebraic).


Duality between Gel’fond and Schneider

A.O. Gelfond : (d

dz

)t0 (e(s1+s2β)z

)z=t1 logα

Th. Schneider : (zt0αt1z

)z=s1+s2β

Result :(s1 + s2β)t0αt1s1(αβ)t1s2 .


Dirichlet’s box principle

Gel’fond and Schneideruse an auxiliary function,the existence of which followsfrom Dirichlet’s box principle(pigeonhole principle,Thue-Siegel Lemma).


Thue’s work on Diophantine Approximation

A. Thue (∼1910). First improvement of Liouville’sinequality on a lower bound for |α− p/q|.

Idea : in place of evaluating the values at p/q of apolynomial in a single variable (viz. the irreduciblepolynomial of α), consider two approximations p1/q1 andp2/q2 of α and evaluate at the point (p1/q1, p2/q2) apolynomial P in two variables.

This polynomial P ∈ Z[X, Y ] is constructed (or rather isshown to exist) by means of Dirichlet’s box principle. Therequired conditions are that P has zeroes of sufficientlylarge order at (0, 0) and at (p1/q1, p2/q2). The order isweighted (index of P at a point).


Thue’s work on Diophantine Approximation

One of the main difficulties that Thue had to overcome wasto produce a zero estimate (to find a non–zero value ofsome derivative).

For the method to work, one needs to select the secondapproximation p2/q2 depending on the first p1/q1. Hence afirst very sharp approximation p1/q1 is required.The method provides a satisfactory result for all p/q withat most one exception (J.W.S. Cassels, H. Davenport :upper bound for the number of solutions of Diophantineequations).


Diophantine Approximation

E. Bombieri has produced examples where a sufficientlysharp approximation exists for the method to work in aneffective way. Later he produced effective refinements toLiouville’s inequality by extending the argument.

Further improvement by C.L. Siegel in the 1920’s – andapplication of the idea to transcendence questions (periodsof elliptic functions).

K. F. Roth (1955) : introduces many variables – get theessentially sharpest possible exponent in Liouville’sinequality, namely 2 + ε in place of the degree d of θ.

W.M. Schmidt (1970) : higher dimensional generalization ofThue–Siegel–Roth Theorem, Subspace Theorem. Requires arefined zero estimate.


Algebraic values of transcendental functions

Assume f is a transcendental entire function (analytic inC) which takes algebraic values at a sequence of algebraicpoints, say z1, z2, . . . (may include derivatives : repeat thepoints).

For instance f(z) = ez with the points α, 2α, 3α, . . .

We want to get a contradiction (under suitableassumptions).

To say that f is transcendental means that if P is anon–zero polynomial in two variables, then the functionP (z, f(z)) is not the zero function.


Algebraic values of transcendental functions

The idea is to get a contradiction by showing the existenceof a non-zero polynomial P such that the functionF (z) = P (z, f(z)) vanishes at all the zk.

One first show the existence of P such that F vanishes atz1, z2, . . . , zN .

Then, by an extrapolation argument using an induction,one shows that F vanishes also at zN+1, zN+2, . . .


Auxiliary functions

C.L. Siegel (1929) :Hermite’s explicit formulaecan be replaced byDirichlet’s box principle( Thue–Siegel Lemma)which shows the existenceof suitable auxiliary functions.


Auxiliary functions

C.L. Siegel (1929) : auxiliary function for the study ofvalues of E and G functions.

In case of G functions : consider two points, 0 and α, withmultiplicity.

Similar with Hermite-Pade approximants of the first type,but the auxiliary functions are not explicit.

K. Mahler (1930’s) : functions satisfying a functionalequation ; the auxiliary function is constructed by means oflinear algebra.


Transcendence criterion of Schneider–Lang

1949, Th. Schneider, general statement on values ofanalytic functions.Corollaries : Hermite–Lindemann, Gel’fond–Schneider,Six Exponentials1957, variants in his book on transcendental numbers.

∼1964’s, S. Lang, simpler statements,• one for functions satisfying differential equations –contains the Theorem of Hermite–Lindemann and thesolution of Hilbert’s seventh problem by Gel’fond’s method,• one for other functions– contains the solution of Hilbert’sseventh problem by Schneider’s method as well as the SixExponentials Theorem.


Order of growth of an entire functions

Definition : An entire function f has finite order of growthif

|f |r := sup|z|=r|f(z)|

satisfies|f |r ≤ eCr

%

.


Transcendence criterion of Schneider–Lang

Theorem. Let f1, f2 be two algebraically independent entirefunctions of finite order of growth. Let K be a number field.Assume the derivatives f ′1 and f ′2 of f1 and f2 arepolynomials with coefficients in K in f1 and f2. Then theset of w ∈ C such that f1(w) and f2(w) are in K is finite.

Assumption : differential equations

f ′1 = A1(f1, f2), f ′2 = A2(f1, f2)

with A1 and A2 in K[X1, X2].

Conclusion :

S ={w ∈ C ; f1(w) ∈ K , f2(w) ∈ K

}is finite.


Corollaries

Examples.• Hermite–Lindemann’s Theorem on the transcendence ofeβ for algebraic β 6= 0.Take f1(z) = z, f2(z) = ez, the differential equations are

f ′1 = 1, f ′2 = f2,

and the two functions take values in Q(β, eβ) at w = sβ,s ∈ Z.• Gel’fond–Schneider’s Theorem on the transcendence of αβ

for algebraic α 6= 0, 1 and β 6∈ Q.Take f1(z) = ez, f2(z) = eβz, the differential equations are

f ′1 = f1, f ′2 = βf2,

and the two functions take values in Q(α, β, αβ) atw = s logα, s ∈ Z.


Remarks

• Explicit upper bounds for the number of exceptional w,in terms of the growth order %i of fi (i = 1, 2) and thedegree [K : Q] :

CardS ≤ (%1 + %2)[K : Q].

• Extends to meromorphic functions (need to avoid poles).

• More general differential equations are allowed – forinstance elliptic functions.


Higher dimension

Extensions to several variables : Th. Schneider, S. Lang,E. Bombieri (conjecture of M. Nagata). Generalization ofthe finiteness condition to higher dimension : subsets ofalgebraic hypersurfaces.

Replace the number of elements of a finite set by thesmallest degree of an algebraic hypersurface containing theset.

Schwarz’ Lemma in several variables : Schneider forCartesian products, Bombieri–Lang using Lelong’s theoryof functions in several variables, Bombieri usingL2–estimates of L. Hormander.


Idea of the proof

We argue by contradiction : assume f1 and f2 takesimultaneously their values in K for many w ∈ C. We wantto show that there exists a non–zero polynomialP ∈ K[X1, X2] such that the function P (f1, f2) is the zerofunction.

The first step is to show that there exists a non–zeropolynomial P ∈ K[X1, X2] such that F = P (f1, f2) has azero of high multiplicity at each w :(

d

dz

)tF (w) = 0 for 0 ≤ t < T.


Linear algebra vs Thue–Siegel Lemma

(d

dz

)tF (w) = 0 for 0 ≤ t < T

is a finite set of homogeneous linear equations withcoefficients in K. As soon as the number T of equations isless than the number of unknown, namely the coefficients ofP , there is a non–trivial solution.

Thue–Siegel Lemma : estimate for the coefficients of P(rational integers). Needs only to have sufficiently manyunknowns (say twice the number of equations).


Induction

Our goal is to prove that F = 0. We already know(d

dz

)tF (w) = 0 for 0 ≤ t < T.

By induction on T ′ ≥ T we shall prove(d

dz

)tF (w) = 0 for 0 ≤ t < T ′.

At the end of the induction we deduce F = 0, which is thecontradiction with the algebraic independence of f1 and f2.


Extrapolation

If F has a zero of multiplicity ≥ T ′ at each w, then F hasmany zeroes, hence it is small in a disk containing thesepoints (Schwarz Lemma), and also its derivatives (Cauchy’sinequalitites) have small absolute values.

From the assumptions it follows that (d/dz)T′F (w) is an

algebraic number in K with a small absolute value. Fromthe product formula (or the size inequality, or other variantsof Liouville’s inequality) one deduces (d/dz)T

′F (w) = 0.


Schwarz Lemma for functions of a single variable

Lemma. Let f be an analytic function in a disc |z| ≤ Rhaving at least N zeroes (counting multiplicities) in a discof radius r with r < R. Recall |f |r = sup|z|=r |f(z)|. Then

|f |r ≤(

2rR

R2 + r2

)N|f |R.

Proof. Let z1, . . . , zN be zeroes of f in |z| ≤ r, countingmultiplicities. Then the function

g(z) = f(z)N∏j=1

(R2 − zzjR(z − zj)

)Nis analytic in |z| ≤ R, hence |g|r ≤ |g|R.


Universal auxiliary functions

Alternative argument for the construction of the auxiliaryfunction : joint work with M. Mignotte (1974).

Given arbitrary analytic functions f1, . . . , fn, construct anon–zero polynomial P ∈ Z[X1, . . . , Xn] such that the thefirst Taylor coefficients at the origin of F = P (f1, . . . , fn)are small.

To solve a system of finitely many linear inequalities, useDirichlet’s box principle – get also an upper bound for thecoefficients of P in Z.

It follows that |f |r is small. Hence f and its first derivativeshave small absolute values in |z| ≤ r.


Using the universal auxiliary function

If all fi(w) are algebraic (maybe including somederivatives), use Liouville’s inequality to produce manyzeroes of F .

Very efficient with a zero estimate : avoids use of Schwarz’Lemma.

Especially useful in several variables.Example : Transcendence of values of exponential functionsin several variables.


Lehmer’s Problem

Let θ be a non–zero algebraic integer of degree d. Mahler’smeasure of θ is

M(θ) =d∏i=1

max(1, |θi|) = exp

(∫ 1

0

log |f(e2iπt|dt),

where θ = θ1 and θ2, · · · , θd are the conjugates of θ and fthe monic irreducible polynomial of θ in Z[X].

Kronecker : M(θ) ≥ 1, and M(θ) = 1 if and only if θ is aroot of unity.

D.H. Lehmer asked whether there is a constant c > 1 suchthat M(θ) < c implies that θ is a root of unity.

M. Mignotte (1977) : Ordinary Vandermonde determinants.


Auxiliary functions and interpolation

determinants

C.L. Stewart (1978) introduces an auxiliary function, usingThue–Siegel’s Lemma.

E. Dobrowolski (1979) : refined estimate, using congruencesmodulo p.

D. Cantor and E.G. Straus (1982) : replaceStewart-Dobrowolski auxiliary function by a determinant.


Dobrowolski’s Theorem

Theorem [E. Dobrowolski (1979)].There is a constant c such that, for θ a non–zero algebraicinteger of degree d,

M(θ) < 1 + c(log log d/ log d)3

implies that θ is a root of unity.

Best unconditional result so far in this direction –improvements only on the numerical value for c.

Dobrowolski’s Lemma. For θ not a root of unity,∏i,j

|θpi − θj| ≥ pd

for any prime p.


Introducing determinants

D. Cantor and E.G. Straus (1979) : GeneralisedVandermonde determinant.

This determinant is big : has many factors of the form∏i,j |θ

pi − θj|k, for many primes p.

Hadamard’s inequality : upper bound for the determinant,in terms of M(θ).

Remark : lower bounds for the determinants also followfrom Schwarz’ inequality for p–adic function.

Extensions of the argument : F. Amoroso and S. David.


Laurent’s interpolation determinants

Underlying idea : a zero estimate shows that some matrixwhose components are values of polynomials has maximalrank.

Select a non-zero maximal minor, bound it from above andfrom below.

M. Laurent (1991) : instead of using the pigeonholeprinciple for proving the existence of solutions tohomogeneous linear systems of equations, consider thematrices of such systems and take determinants.


Slope inequalities in Arakelov theory

J–B. Bost (1994) :matrices and determinants requirechoices of bases.Arakelov’s Theory producesslope inequalities whichavoid the need of bases.

Periodes et isogenies des varietes abeliennes sur les corpsde nombres, (d’apres D. Masser et G. Wustholz).Seminaire Nicolas Bourbaki, Vol. 1994/95.


AWS Lecture 4 Tucson, Monday, March 17, 2008

Auxiliary functions in

transcendence proofs

Michel Waldschmidt

http://www.math.jussieu.fr/∼miw/


aws lecture 4swc.math.arizona.edu/aws/2008/08waldschmidtslides4.pdf · 2012-03-02 · close...

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