The uniform Schanuel conjecture over the real
numbers
Jonathan Kirby
Boris Zilber
October 2004
Abstract
We prove that Schanuel’s conjecture for the reals is equivalent to
a uniform version of itself.
Schanuel’s conjecture is one of the central conjectures in transcendental
number theory. Some of its consequences in that field are discussed by Lang
in [1]. Macintyre and Wilkie gave another application when they showed in
[3] that if Schanuel’s conjecture is true at least over the real numbers then
the theory of the real field with exponentiation is decidable. The uniform
Schanuel conjecture for the complex numbers and its consequences were a
subject of discussion in [7]. It was shown in that paper that the standard
Schanuel conjecture is equivalent to the uniform Schanuel conjecture if one
assumes a certain Diophantine-type conjecture about intersections of complex algebraic varieties with tori, that is algebraic subgroups of (C∗ )n . Here
we prove that the Schanuel conjecture for the reals is equivalent to the corresponding uniform conjecture without any extra assumptions.
Indeed, the reader can see that the proof follows quickly from o-minimality
considerations. (See [4] for an introduction to the theory of o-minimality.)
The idea that the uniform conjecture for the reals should follow from Schanuel’s
conjecture appeared in [6], a preprint version of [7].
The real form of Schanuel’s conjecture is the statement
(SCR ) Suppose a1 , . . . , an ∈ R such that tdQ Q(a1 , . . . , an , ea1P
, . . . , ean ) < n.
Then there are m1 , . . . , mn ∈ Z, not all zero, such that ni=1 mi ai = 0.
The uniform real version of Schanuel’s conjecture puts a bound on the
coefficients.
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(USCR ) Let V ⊆ R2n be an algebraic variety over Q of dimension < n. Then
there exists a natural number N such that if (a1 , . . . , an , ea1 , . . . , ean ) ∈
V then therePare m1 , . . . , mn ∈ Z, not all zero, with |mi | 6 N for each
i, such that ni=1 mi ai = 0.
It is clear that (USCR ) implies (SCR ), and in this paper we show the
converse also holds.
Definition. Let R be an expansion of the real field hR; +, ·i. With respect
to R, cells in Rn are defined inductively as follows.
1. Singletons are 0-cells in R; open intervals are 1-cells in R.
2. If C is an m-cell in Rn and f : C → R is a continuous function definable
in R then its graph {(x̄, y) ∈ C × R | y = f (x̄)} is an m-cell in Rn+1 .
3. If C is an m-cell in Rn , f, g : C → R are continuous functions definable
in R or are the constant functions ±∞ and f (x̄) < g(x̄) for every x̄ ∈ C
then {(x̄, y) ∈ C × R | f (x̄) < y < g(x̄)} is an (m + 1)-cell in Rn+1 .
If in addition we require all the functions f and g to be analytic, we say the
cells are analytic.
An expansion R of the real field hR; +, ·i is said to be o-minimal when
every definable subset of R is a finite union of points and intervals (that
is, of cells). The cell decomposition theorem for o-minimal structures which
can be found in [4] states that when R is o-minimal, for each n ∈ N, every
definable subset of Rn can be partitioned into finitely many definable cells.
It is shown in [5] that the structure Rexp = hR; +, ·, expi is o-minimal. In
[2] it is shown that this particular structure has analytic cell decomposition,
that is that every definable subset of Rn can be partitioned into finitely many
analytic cells. We use this and the following elementary lemma.
Lemma. If C ⊆ Rn is an m-cell definable in an expansion R of the real field
hR; +, ·i then there is a homeomorphism θ : B → C, definable in R, where
B is a product of m open intervals in Rm (an open box). If C is an analytic
cell then θ may be taken to be an analytic diffeomorphism.
Proof. The proof proceeds by induction on the dimension n of the ambient
space. If n = 1 then C is already a product of intervals, so we may take θ to
be the identity map.
Suppose C is the graph of f : C 0 → R and C 0 is an m-cell in Rn−1 . Then
by the induction hypothesis there is θ0 : B → C 0 where B is a product of m
intervals. Define θ(x̄) = (θ0 (x̄), f (θ0 (x̄))).
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Otherwise C is {(x̄, y) ∈ C 0 × R | f (x̄) < y < g(x̄)} for some (m − 1)-cell
C ⊆ Rn−1 , and we have θ0 : B → C 0 of the appropriate form. If f =
6 −∞
and g 6= +∞, define θ : B × (0, 1) → C by
0
θ(x̄, y) = (θ0 (x̄), (1 − y)f (x̄) + yg(x̄)).
If f 6= −∞ and g = +∞, define θ : B × (0, +∞) → C by
θ(x̄, y) = (θ0 (x̄), f (x̄) + y).
If f = −∞ and g 6= +∞, define θ : B × (−∞, 0) → C by
θ(x̄, y) = (θ0 (x̄), g(x̄) + y).
If f = −∞ and g = +∞, define θ : B × R → C by
θ(x̄, y) = (θ0 (x̄), y).
In each case θ is a homeomorphism. If all the functions f and g are analytic
then θ will be an analytic diffeomorphism, as required.
We now prove that (SCR ) implies (USCR ).
Let V ⊆ R2n be an algebraic variety over Q of dimension < n, and let
W = {x̄ ∈ Rn | (x, ex̄ ) ∈ V }. Then W is definable in the structure Rexp , so
by the result mentioned earlier can be partitioned into finitely many analytic
cells. Let C be one of these cells and θ : B → C a definable analytic
diffeomorphism from an open box to C, as given by the lemma.
Let a, b ∈ C and let σ : [0, 1] → B be the path of uniform speed along
the line segment from θ−1 (a) to θ−1 (b). Let γ = θ ◦ σ. Then γ is a definable
analytic path from a to b in C.
By (SCR ), every point
Pn in C and thus every point x ∈ Im γ satisfies an
all zero. There
equation of the form i=1 mi xi = 0 with the mi ∈ Z and notP
are only countably many such equations so one, say h(x) = ni=1 mi xi = 0,
must be satisfied by infinitely many points in Im γ. Then {t ∈ [0, 1] | (h ◦
γ)(t) = 0} is an infinite subset of [0, 1] which is definable in Rexp and so by
o-minimality must contain an open interval. Now h◦γ is an analytic function
[0, 1] → R which is zero on an open interval, hence by uniqueness of analytic
continuation is zero on all of [0, 1]. Thus every point in Im γ, in particular a
and b, satisfy h(x) = 0. Since a and b were arbitrary in C, we deduce that
every point in C satisfies the same equation h. Since every point in W lies
in one of finitely many cells, the uniform bound in (USCR ) follows at once.
Acknowledgement We would like to thank Tamara Servi for reminding
the second author of this question and for bringing it to the attention of the
first.
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References
[1] S. Lang, Introduction to transcendental numbers (Addison Wesley, Reading, MA, 1966).
[2] T. L. Loi, ‘Analytic cell decomposition of sets definable in the structure
Rexp ’, Annales Polonici Mathematici, 59 (1994) 255–266.
[3] A. Macintyre A. J. Wilkie, ‘On the decidability of the real exponential
field’, Kreiseliana (ed P. Odifreddi, A K Peters, Wellesley, MA, 1996).
[4] L. van den Dries, Tame topology and o-minimal structures, LMS lecture
notes 248, (CUP, Cambridge, 1998).
[5] A. J. Wilkie, ‘Model completeness results for expansions of the ordered
field of real numbers by restricted pfaffian functions and the exponential
function’, J. Amer. Math. Soc. 9 (1996) 1051–1094.
[6] B.
Zilber,
‘Intersecting
http://www.maths.ox.ac.uk/~zilber.
varieties
with
tori’,
[7] B. Zilber, ‘Exponential sums equations and the Schanuel conjecture’, J.
London Math. Soc. (2) 65 (2002) 27–44.
Mathematical Institute
University of Oxford
24-29 St Giles
Oxford OX1 3LB
[email protected]
[email protected]
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