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The Four Exponentials Problem
and the Schanuel Conjecture
Michel Waldschmidt1,1*
1* Sorbonne Université and Université de Paris, CNRS IMJ-PRG,

F-75005 Paris France.

Corresponding author(s). E-mail(s):michel.waldschmidt@imj-prg.fr;
Abstract∗

Let ( ) t be a real number such that 2t and 3t are integers; does it
follow that t is a nonnegative integer? A positive answer would follow from the solution of the four exponentials Problem, a very special
case of the so–called weak Schanuel’s Conjecture, namely the conjecture of algebraic independence of logarithms of algebraic numbers. These
questions are open both in the complex case and in the p–adic case.
One of the main motivations for investigating the p–adic situation is
Leopoldt’s Conjecture, which will be sort of a leitmotiv in this survey.
2020 Mathematics Subject Classification : 11J81

Keywords: Transcendental numbers, four exponentials Problem, six
exponentials Theorem, Schanuel’s Conjecture

May 26, 2022
Festschrift in honor of Springer’s Editorial Director Dr. Catriona Byrne“Mathematics Going Forward”

1 Introduction
This Festschrift in honor of Springer’s Editorial Director Dr. Catriona Byrne
is a good opportunity for me to thank Catriona for her support in publishing∗

Note to the Editor: the mathematical symbols in the abstract should not be boldface. Pleasealso remove the semi colon at the end of my e-mail address above.

1

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Four Exponentials Problem and Schanuel’s Conjecture

a good part of my works with Springer Verlag. I had the pleasure of working
with her also as an editor of this publisher. I am thankful also to the editors of
this special volume of the Lectures Notes in Mathematics for their invitation
to share my favorite open problems and the ones dear to my heart, with somebackground and context.
(1 ) I have worked on several conjectures. The one on which I spent much
more time than the others is the so–called four exponentials Problem [Lang(1966), Chap. II § 1 p. 11], which is also the first of the eight problems in
Schneider’s book on transcendental numbers [Schneider (1957)]. This
question is a very special case, arguably one of the easiest unsolved cases sofar, of Schanuel’s Conjecture [Lang (1966), Chap. III, Historical note p. 30].
Over the years, I tried to prove Schanuel’s Conjecture; since very few results
are known, in the process of trying to solve it, I added some hypotheses
which might help, and, quite often, after some time, I came back trying tosolve the four exponentials Problem. Without success so far! It is hard to
predict whether the special case of the four exponentials Problem will besolved before the very general case of Schanuel’s Conjecture. As an example,
the following simple looking statement is open:
Let t be a real number such that 2t and 3t are integers. Prove that t is a nonnegative
integer.

While the four exponentials Problem is still open, a weaker statement,
the six exponentials Theorem (Theorem 3), is known to be true; a special
case is the following:
Let t be a real number and p1 , p2 , p3 be three distinct primes. Assume that the threenumbers pt1 , pt2 and pt3 are integers. Then t is a nonnegative integer.

For a complete proof of this result using interpolation determinants, see[Waldschmidt (2021)].
The present paper deals only with Schanuel’s Conjecture and some of its
consequences, including the four exponentials Problem and the problem ofalgebraic independence of logarithms of algebraic numbers (of which Leopoldt’s
Conjecture is a special case in the p–adic case). Further conjectures (including Grothendieck’s Conjecture on Abelian periods, André’s Conjecture on
motives, the Conjecture of Kontsevich–Zagier on periods,...) would deserve to
be discussed also – see for instance [Waldschmidt (2012)].

2 Leopoldt’s Conjecture on the p–adic rank of
the group of units of an algebraic number field
When I started to do research in 1969 in Bordeaux, my thesis advisor, Jean
Fresnel, suggested me to study Leopoldt’s Conjecture [Leopoldt (1962)]. At
that time, Fresnel was interested in p–adic L–functions [Amice and Fresnel(1972)] and Leopoldt’s Conjecture was comparatively recent. The goal is toprove that the p–adic rank of the group of units of an algebraic number fieldis the same as the usual rank given by Dirichlet’s unit Theorem. It amounts
1

Note to the Editor: I do not understand why this section is not right justified.

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Four Exponentials Problem and Schanuel’s Conjecture3

to say that the p–adic regulator, which is defined as the usual regulator byreplacing logarithms with p–adic logarithms, does not vanish. According to
Fresnel, since it amounts to prove that a determinant does not vanish, it should
not be so difficult!
For a subfield of an abelian extension of an imaginary quadratic field, thedecomposition, due to Frobenius, of the Gruppendeterminant of the Galois
group – see for instance [Fresnel (1969), Waldschmidt (1971), Kanemitsu andWaldschmidt (2013)] – shows that the regulator splits into a product of linear forms with algebraic coefficients of logarithms of algebraic numbers. As
a consequence, in this special case, as shown by J. Ax [Ax (1965), Conjecture p. 587], Leopoldt’s Conjecture is a consequence of the p–adic version ofa conjecture of A.O. Gel’fond on the linear independence, over the field Q
of algebraic numbers, of Q–linearly independent logarithms of algebraic numbers. This linear independence result in the complex case has been achievedby the seminal work of A. Baker [Baker (1966)], by means of a far reachingdevelopment of Gel’fond’s method.
The p–adic analog of Baker’s result was proved the year after by A. Brumer[Brumer (1967)], who therefore solved Leopoldt’s Conjecture for these abelian
extensions. As pointed out by Brumer, the translation to the p–adic case oftranscendence methods had been worked out by J-P. Serre [Serre (1967)]. Asa matter of fact, Serre was interested in an extension to several variables (inthe p–adic case) of the six exponentials theorem for an application to `–adic
abelian representations [Serre (1968), Henniart (1982)].
In the general case, Leopoldt’s Conjecture is a special case of the p–adic
version of the conjecture on algebraic independence of logarithms of algebraic numbers. This is how Fresnel suggested me to study the theory oftranscendental numbers.

3 Conjecture on the algebraic independence oflogarithms of algebraic numbers
According to A.O. Gel’fond [Gel’fond (1952), Chap. III § 5 p.177], one may
assume . . . that the most pressing problem in the theory of transcendental numbers is the investigation of the measures of transcendence of finite sets oflogarithms of algebraic numbers. From a qualitative point of view, the statement is the following one [Lang (1966), Chap. III, Historical note p. 31], which,according to Lang, has been conjectured for a long time (by anybody who has
looked at the subject).
Conjecture 1 (Algebraic independence of logarithms of algebraic numbers) Let
λ1 , . . . , λn be Q–linearly independent complex numbers, such that the numbers
αi = eλi (i = 1, . . . , n) are algebraic numbers. Then λ1 , . . . , λn are algebraicallyindependent.

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Four Exponentials Problem and Schanuel’s Conjecture

In [Calegari and Mazur (2009), Conjecture 3.9], this conjecture is called
weak Schanuel.
Under the assumptions of Conjecture 1, the conclusion of Baker’s Theorem[Baker (1966)] is that the numbers 1, λ1 , . . . , λn are Q–linearly independent,
while the conclusion of Conjecture 1 is that, for any nonzero polynomial P (with rational or algebraic coefficients) in n variables, the number
P (λ1 , . . . , λn ) does not vanish.
By abuse of language, we sometimes write λi = log αi (i = 1, . . . , n); theway Conjecture 1 is stated avoids the need to select a branch of the complex logarithm. For instance with λ1 = log 2 and λ2 = log 2 + 2πi, hence
α1 = α2 = 2, Baker’s Theorem yields the linear independence of the numbers
1, log 2, π over Q, while Conjecture 1 claims that log 2 and π are algebraicallyindependent (which is not yet proved).
Conjecture 1 is true for n = 1: a nonzero logarithm of an algebraic number
is transcendental, according to the Theorem of Hermite–Lindemann [Schneider (1957), Chap. II § 4], [Lang (1966), Chap. III Corollary 1], [Waldschmidt(1974), Th. 3.1.1], [Waldschmidt (2000), Th. 1.2]. This is essentially the only
case where Conjecture 1 has been proved. Under the assumptions of Conjecture 1, the conclusion should be that the transcendence degree of the field
Q(λ1 , . . . , λn ) is n. As a matter of fact, it is not yet known that the field generated by all logarithms of all nonzero algebraic numbers has a transcendence
degree over Q at least 2. However, for a conjecture which is equivalent to Conjecture 1, half of the result is proved (see inequalities (2) in § 5 and (3) in§ 6). Hence, depending on the point of view, one may consider that we are half
way on proving Conjecture 1.

4 Four exponentials Problem and six
exponentials Theorem
We would like to solve at least some special cases of Conjecture 1. For instance
we would like to prove that there are no algebraic relations like(log α1 )2 = log α2involving nonzero logarithms of algebraic numbers log α1 and log α2 . Very few
results are known even for this very specific case. We will mainly work withhomogeneous relations; among the simplest nonlinear ones is the following:(log α1 )(log α4 ) = (log α2 )(log α3 ),which amounts to consider the vanishing of the determinant
log α1 log α2.
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Four Exponentials Problem and Schanuel’s Conjecture

5Here, log αi denote complex numbers such that αi = elog αi are algebraic numbers. The four exponentials Problem states that such a determinant can vanish
if and only if either the two rows are linearly dependent over the rational number field Q, or the two columns are linearly dependent over Q. Since a 2 × 2
matrix has rank ≤ 1 if and only if it can be written as
x1 y1 x1 y2,
x2 y1 x2 y2
an equivalent form of the four exponentials problem is the following:
Conjecture 2 (Four exponentials Problem) Let x1 , x2 be two complex numbers
which are linearly independent over Q and let y1 , y2 be two complex numbers which
are linearly independent over Q. Then one at least of the four numbers
ex1 y1 , ex1 y2 , ex2 y1 , ex2 y2
is transcendental.

Here is a sketch of proof of Conjecture 2 as a consequence of Conjecture
1 on algebraic independence of logarithms of algebraic numbers [Waldschmidt(2000) Exercise 1.8]. As pointed out by D. Roy [Roy (1995), p. 52], Conjecture 1 is equivalent to the following statement: Let λ1 , . . . , λn be logarithms
of algebraic numbers and let P ∈ Q[X1 , . . . , Xn ] be a nonzero polynomial withalgebraic coefficients such that P (λ1 , . . . , λn ) = 0. Then there is a vector subspace V of Cn , rational over Q, which is contained in the set of zeroes of P
and contains the point (λ1 , . . . , λn ). To completes the proof, one uses the factthat if V a vector subspace of C4 , which is rational over Q and is contained in
the hypersurface z1 z4 = z2 z3 , then there exists (a : b) ∈ P1 (Q) such that V is
included either in the plane{(z1 , z2 , z3 , z4 ) ∈ C4 ; az1 = bz2 , az3 = bz4 }or in the plane{(z1 , z2 , z3 , z4 ) ∈ C4 ; az1 = bz3 , az2 = bz4 }.
This four exponentials problem was proposed explicitly by S. Lang [Lang(1966), Chap. II § 1 p. 11] and K. Ramachandra [Ramachandra (1968) p. 87–88]; it is also the first of the eight problems at the end of Schneider’s book
[Schneider (1957)].
The following statement is weaker than Conjecture 2 but is proved:
Theorem 3 (Six exponentials Theorem) Let x1 , x2 be two complex numbers which
are linearly independent over Q, and let y1 , y2 , y3 be three complex numbers which
are linearly independent over Q. Then one at least of the six numbers
ex1 y1 , ex1 y2 , ex1 y3 , ex2 y1 , ex2 y2 , ex2 y3

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Four Exponentials Problem and Schanuel’s Conjectureis transcendental.

Equivalently, a 2 × 3 matrix with entries logarithms of algebraic numbers,
having its two rows linearly independent over Q and its three columns linearly
independent over Q, has rank 2.
The six exponentials Theorem 3 was proved by [Lang (1966), Chap. II
Th. 1] and [Ramachandra (1968)]. The footnote p. 67 of [Ramachandra(1968)] reads:
After writing this manuscript I came to know from professor C.L. Siegel that thisis a result first due to Schneider and Siegel. The result is unpublished. This result
may also be found in a recent paper by S. Lang, Algebraic values of meromorphic
functions, Topology 5 (4), (1966), pp. 363–370. The results of this paper havesomething in common with Lang’s results.

Indeed, one can infer from [Alaoglu and Erdős (1944) p. 455] that the six
exponentials Theorem 3 and the four exponentials Problem (Conjecture 2)
were also known to C.L. Siegel. When I met A. Selberg at a conference organized by Kai-Man Tsang in Hong Kong in December 1993, he told me that heknew the proof of the six exponentials Theorem 3, but he did not publish it
because it was too easy. He said he tried to solve the four exponentials Problem(Conjecture 2), which was much more interesting, but he did not succeed.
A proof of Theorem 3 is given in [Waldschmidt (1974), Chapter 2 (Schneider’s method)]. In his plenary lecture for the Journées Arithmétiques in
Luminy in 1989, M. Laurent introduced a new idea for transcendence proofs,
by means of interpolation determinants in place of an auxiliary function; theexample he worked out was the six exponentials Theorem [Laurent (1991)].
See also [Waldschmidt (2021)] for the simplest case of rational integers.
The four exponentials Problem (Conjecture 2) has been solved under the
extra assumption that the field generated by the four numbers x1 , x2 , y1 , y2
has transcendence degree ≤ 1 [Brownawell (1974), Corollary 7], [Waldschmidt(1973), Corollary 4]. The proof uses a method of algebraic independence
devised by A.O. Gel’fond. This result has been extended in [Roy and Waldschmidt (1995)], where the determinant X1 X4 − X2 X3 is replaced by any
homogeneous quadratic form; for the proof, Gel’fond’s criterion is replaced bya Diophantine approximation result due to Wirsing.
For an explanation of the fact that the transcendence machinery fails so
far to solve the four exponentials Problem, see Corollary 8.3 of [Roy (2002)].
As mentioned above, the p–adic analog of the six exponentials Theorem has
been proved by J-P. Serre [Serre (1967)]. As shown in [Roy (1993), Corollaryp. 450], a positive solution of the p–adic version of the four exponentials Conjecture implies Leopoldt’s Conjecture for Galois extensions of Q with Galois
group a dihedral group of order 6, 8 or 12 (hence, in particular, it implies
Leopoldt’s Conjecture for number fields which are Galois extensions of Q of
degree ≤ 7).

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5 Rank of matrices
We have seen that the four exponentials Problem can be stated as the nonvanishing of the determinant (1). More generally, Conjecture 1 shows that adeterminant, the entries of which are logarithms of algebraic numbers, canvanish only in trivial cases. A precise statement, with a definition of the meaning of trivial, is the following ([Roy (1995), p. 54] and [Waldschmidt (2000),
Lemma 12.8]).
Definition 1 Let M be a matrix with entries in C and K a subfield of C. Let
e1 , . . . , et be a basis of the K–vector space spanned by the entries of M . Hence M
can be written as
M = M1 e1 + · · · + Mt et ,where the matrices M1 , . . . , Mt have entries in K. Let X1 , . . . , Xt be indeterminates. The rank of the matrix M1 X1 + · · · + Mt Xt , with coefficients in the ringK[X1 , . . . , Xn ] of polynomials in n variables, does not depend on the choice of thebasis e1 , . . . , et and is denoted as rstr,K (M ), which is called the structural rank of
M with respect to K.

For any matrix M with complex coefficients and any field K, the upperbound rk(M ) ≤ rstr,K (M ) is plain. Assume now that the entries of M arelogarithms of algebraic numbers. Conjecture 1 implies rk(M ) = rstr,K (M ).
From the six exponentials Theorem, one deduces that when rstr (M ) ≥ 3, thenrk(M ) ≥ 2. More generally, the lower boundrk(M ) ≥1
rstr,Q (M )2(2)follows from [Waldschmidt (1981)]. The lower bound (2) also holds in the p–
adic case; it proves that the p–adic rank of the group of units of an algebraic
number field is at least half of its usual rank [Waldschmidt (1984)].
The proof of [Waldschmidt (1981)] also yields an answer to the above
mentioned question on `–adic representations [Serre (1968), Henniart (1982)],
while the complex version answers a question of A. Weil [Weil (1956)] on thecharacters of the idèle class group of an algebraic number field [Waldschmidt(1982)].

6 Strong six exponentials Theorem and strong
four exponentials Problem
There is room between the four exponentials Problem and the six exponentials
Theorem for a result involving five numbers [Waldschmidt (1988)]:
Theorem 4 (Five exponentials Theorem) Let γ be a nonzero algebraic number,
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Four Exponentials Problem and Schanuel’s Conjecture

two complex numbers which are linearly independent over Q. Then one at least of thefive numbers
ex1 y1 , ex1 y2 , ex2 y1 , ex2 y2 , eγx1 /x2
is transcendental.

This result is weaker than the four exponentials Problem (because of theassumption γ 6= 0) but does not imply the six exponentials Theorem. A result
which includes both the five and the six exponentials theorems (Theorems
4 and 3) is the next one [Waldschmidt (1990), Corollary 2.3], [Waldschmidt(1988), Corollary 2.1], [Waldschmidt (2000), § 11.3.3, example 2, p. 386].
Let x1 , x2 be two complex numbers which are linearly independent over Q, let
y1 , y2 , y3 be three complex numbers which are linearly independent over Q and
let βij (i = 1, 2, j = 1, 2, 3) be six algebraic numbers. Then one at least of the six
numbers
ex1 y1 −β11 , ex1 y2 −β12 , ex1 y3 −β13 , ex2 y1 −β21 , ex2 y2 −β22 , ex2 y3 −β23
is transcendental.

A first generalization of this result has been achieved by D. Roy [Roy(1990)], [Roy (1992b), Corollary 2 p. 38], [Waldschmidt (2000), Corollary
11.16], who considers matrices with entries which are linear combinations, withalgebraic coefficients, of 1 and of logarithms of algebraic numbers. Denote by
L the Q–vector space spanned by 1 and all logarithms of all nonzero algebraic
numbers. A typical element of L is of the form
β0 + β1 log α1 + · · · + βn log αnwith algebraic numbers αi and βj .
Theorem 5 (D. Roy, strong six exponentials Theorem) Let x1 , x2 be two complex
numbers which are linearly independent over Q and let y1 , y2 , y3 be three complex
numbers which are linearly independent over Q. Then one at least of the six numbers
x1 y1 , x1 y2 , x1 y3 , x2 y1 , x2 y2 , x2 y3
does not belong to L.

The strong four exponentials Problem is the same statement as Theorem
5 with only two numbers y1 , y2 instead of three.
Several consequences of the strong four exponentials Problem are stated in[Waldschmidt (2005a)]2 and corollaries of the strong six exponentials Theorem
are derived in [Waldschmidt (2005b)].
2
Erratum: The right assumption in corollary 2.12 p. 346 of Waldschmidt (2005a) is that thethree numbers 1, Λ11 and Λ21 are linearly independent over the field of algebraic numbers.

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A much more general statement than Theorem 5 is an extension by D. Roy[Roy (1992a)] of the lower bound (2) to matrices having entries in L: for sucha matrix,
1(3)
rk(M ) ≥ rstr,Q (M )2see [Waldschmidt (2000), Th. 1.17 and Corollary 12.18].
As pointed out by D. Roy ([Roy (1992a)], [Roy (1995), Conjecture 1.1] and[Waldschmidt (2000), Lemma 12.14]), Conjecture 1 on the algebraic independence of logarithms of algebraic numbers is equivalent to the statement that
if the entries of M are in L, then the rank rk(M ) of the matrix M is always
equal to its structural rank rstr,Q (M ) with respect to Q. From this point ofview, we can consider (3) as proving half of Conjecture 1.
As always, the situation is the same in the p–adic case, both for results andfor conjectures. Applications to Leopoldt’s Conjecture on the p–adic rank of theunits of a number field have been derived in the following references: [Emsalem,
Kisilevsky, and Wales (1984), Jaulent (1985), Emsalem (1987), Laurent (1989),
Laurent (1990), Roy (1993)]. See also [Calegari and Mazur (2009), § 3 Remark
p. 127] and [Maksoud (2022)].

7 Schanuel’s Conjecture
Conjecture 1 on the algebraic independence of logarithms of algebraic numbers
is a special case of Schanuel’s Conjecture, which was proposed by Stephen
Schanuel during a course given by Serge Lang at Columbia in the 1960’s [Lang(1966), Chap. III, Historical Note, p. 30–31].
Conjecture 6 (Schanuel’s Conjecture) Let x1 , . . . , xn be Q–linearly independent
complex numbers. Then at least n of the 2n numbers
x1 , . . . , xn , ex1 , . . . , exn
are algebraically independent over Q.

The conclusion is that the transcendence degree over Q of the field
Q(x1 , . . . , xn , ex1 , . . . , exn ) is at least n. This result is known when x1 , . . . , xn
are algebraic numbers: this is the Lindemann–Weierstrass Theorem. Conjecture 1 is the special case of Conjecture 6 where the n numbers ex1 , . . . , exn are
assumed to be algebraic.

8 Roy’s conjecture
In his plenary talk at the Journées Arithmétiques in Rome in 1999 [Roy(2001a), Roy (2001b)], D. Roy proposed a new conjecture of his own andproved the remarkable and surprising result that it is equivalent to Schanuel’s
Conjecture 6.

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Denote by D the derivation
D=

∂
∂
+ X1
∂X0
∂X1

on the field C(X0 , X1 ).
Conjecture 7 (Conjecture of D. Roy) Let ` be a positive integer, y1 , . . . , y` Qlinearly independent complex numbers, α1 , . . . , α` nonzero complex numbers and
s0 , s1 , t0 , t1 , u positive real numbers satisfying1
max{1, t0 , 2t1 } < min{s0 , 2s1 } < u and max{s0 , s1 +t1 } < u < (1+t0 +t1 ).
2
Assume that, for any sufficiently large positive integer N , there exists a nonzero
polynomial PN ∈ Z[X0 , X1 ] with partial degree ≤ N t0 in X0 , partial degree ≤ N t1
in X1 and height H(PN ) ≤ eN , which satisfies
`
`

Y
 X
m
αj j ≤ exp(−N u )D k PN
mj yj ,
j=1

j=1

for any integers k, m1 , . . . , m` in N with k ≤ N s0 and max{m1 , . . . , m` } ≤ N s1 .
Then, we have the following lower bound for the transcendence degree:
trdegQ Q(y1 , . . . , y` , α1 , . . . , α` ) ≥ `.

See also [Waldschmidt (2000), Conjecture 15.36]. Hence Schanuel’s Conjecture is equivalent to a purely algebraic statement, which bears some similarity
with the available criteria of algebraic independence.
The proof of the equivalence between Schanuel’s Conjecture 6 and Roy’s
Conjecture 7 involves a new interpolation formula for holomorphic functionsof two complex variables [Roy (2001a), Roy (2001b)]. Refined interpolation
formulae are proved in [Roy (2002), Nguyen and Roy (2016)].
Several significant steps in the direction of Conjecture 7 were performed
by D. Roy, first for the multiplicative group [Roy (2008)], next for the additive group [Roy (2010)] and then for the product of the additive group by
the multiplicative group [Roy (2013)]. A refinement of Conjecture 7, again
equivalent to Schanuel’s Conjecture, is devised by Nguyen Ngoc Ai Van in[Nguyen (2009)]. The statement which is proved in [Nguyen and Roy (2016)]
is similar to Conjecture 7 and is not restricted to the one parameter subgroupt 7→ (t, exp(t)).
In [Ghidelli (2015)], Luca Ghidelli refines the results of [Roy (2013)] and[Nguyen and Roy (2016)], replacing the total degree with multidegrees; his tool
[Ghidelli (2019)] is an extension of Roy’s multiplicity lemma for the resultant,using the theory of multiprojective elimination initiated by P. Philippon anddeveloped by G. Rémond.
This original point of view of D. Roy suggests a promising approach forproving Schanuel’s Conjecture: so far it is the only available strategy towards
a proof of it.
Transcendence theory is going forward.

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Acknowledgments. Thanks to Damien Roy for his comments on a preliminary draft of this paper and to Claude Levesque for his support.

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