LIP RR2002 13

Laboratoire de l’Informatique du Par-
all´ elisme
´
Ecole Normale Sup´erieure de Lyon
Unit´e Mixte de Recherche CNRS-INRIA-ENS LYON no5668
The Theory of Liouville Functions
Pascal Koiran March 2002
Research Report No2002-13
´
Ecole Normale Sup´ erieure de Lyon
46 All´ee d’Italie, 69364 Lyon Cedex 07, France
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The Theory of Liouville Functions
Pascal Koiran
March 2002
Abstract
A Liouville function is an analytic function H:C→Cwith a Taylor
series ∞
n=1 xn/ansuch the an’s form a “very fast growing” sequence of
integers. In this paper we exhibit the complete first-order theory of the
complex field expanded with H.
Keywords: model theory, amalgamation, complex variable,
analytic functions
R´esum´e
Une fonction de Liouville est une fonction analytique H:C→Cavec
une s´erie de Taylor ∞
n=1 xn/antelle que les anforment une suite d’en-
tiers `a croissance “tr`es rapide”. Dans cet article nous exhibons la thorie
du premier ordre de la structure obtenue en enrichissant le corps des
nombres complexes d’une telle fonction.
Mots-cl´es: th´eorie des mod`eles, amalgames, variable complexe,
fonctions analytiques

The Theory of Liouville Functions
Pascal Koiran
14th March 2002
Abstract
A Liouville function is an analytic function H:C→Cwith a Taylor
series ∞
n=1 xn/ansuch the an’s form a “very fast growing” sequence of
integers. In this paper we exhibit the complete first-order theory of the
complex field expanded with H.
Keywords: model theory, amalgamation, complex variable, analytic
functions
1 Introduction
In [8] Wilkie calls “Liouville function” a function H:C→Cwith a Taylor
series of the form
H(x)=
∞
i=1
xi/ai
where the aiare non-zero integers satisfying the condition:
for every l≥1,|ai+1|>|ai|ilfor all sufficiently large i. (1)
A fragment of the first-order theory of the complex field expanded with His de-
scribed in [8]. In this paper we exhibit the complete first-order theory. It turns
out that this theory is the “limit theory of generic polynomials” recently studied
in [4] (this answers a question of Zilber [9]). We recall the axiomatization of the
theory in section 2, where we also present another equivalent axiomatization
which is closer in spirit to [8]. In section 3 we give a short overview of the proof
of our main result and present two of the main tools: continuity of the roots of
polynomial systems and effective quantifier elimination. The last two sections
are devoted to the proof of the main result.
A model of our theory can be constructed by a Hrushovski-style amalga-
mation method [4, 10]. It is therefore natural to ask whether analytic models
exist for other theories constructed by this method. The limit theory of generic
curves [3] and the theory of generic functions with derivatives [10] are two nat-
ural candidates. Additional examples and further discussion can be found in
the surveys [6, 9].
1

2 Axiomatization
In this section only we work within an arbitrary algebraically closed field Kof
characteristic 0. In the remainder of the paper we set K=C.Wedonotwork
in the language of “curved fields” (the language of fields expanded with a binary
predicate) as in [4] but in the language of fields expanded with a unary function
symbol H. We call this language L. Given a tuple xof elements of K,H(x)
denotes the tuple obtained by applying Hcomponentwise. This notation will
be used freely throughout the paper, for Has well as for other unary functions.
2.1 The Limit Theory of Generic Polynomials
A generic polynomial of degree dis of the form gd(x)=d
i=1 αixiwhere the co-
efficients αiare algebraically independent over Q.LetFbe a sentence of L.We
haveshownin[4]thatFis either true for all generic polynomials of sufficiently
high degree, or false for all generic polynomials of sufficiently high degree. The
set Tof sentences which are ultimately true therefore forms a complete theory.
We recall that this theory is defined by the following axioms.
1. The axioms of algebraically closed fields of characteristic 0.
2. H(0) = 0.
3. The universal axioms. Let φ(x1,...,x
n,y 1,...,y n) be a conjunction of
polynomial equations with coefficients in Q. If the subset of K2ndefined
by φis of dimension <n,weaddtheaxiom
∀x1,...,x
n
i
xi=0∧
i=j
xi=xj→¬φ(x, H(x)).(2)
4. The inductive axioms. Let φ(x1,y
1,...,x n,y n, z) be a conjunction of
polynomial equations with rational coefficients. For any fixed value of
the parameter z,φdefines an algebraic subset Vzof K2n.Letξ(z)bea
formula of the language of fields which states that Vzis irreducible, has
dimension nand is not contained in a subspace of the form xi=xjfor
some i=j,oroftheformxi=cfor some element cin the model.
Let be a function which chooses one variable u
i∈{xi,y
i}for ev-
ery i∈{1,...,n}. For each value of the parameter z, the formula
∃u
1,...,u
nφ(x, y,z) defines a constructible set C
z⊆Kn.Aspointed
out in [3], there is a formula ψ(z) of the language of fields which states
that C
zis dense in Kn.Letψ(z) be the disjunction of the 2nformulas
ψ(z). Let θbe the conjunction of ξand ψ. We add the following axiom:
∀z∃x1,...,x
nθ(z)→φ(x, H(x), z).(3)
In fact these inductive axioms are slightly different from those of [4]. Indeed,
there was not requirement that dim Vz=nin that paper. An inspection of the
completeness proof in [4] reveals that the inductive axioms are used only in the
case dim Vz=n. The corresponding theories are therefore identical.
2

We have shown in [4] that a universal axiom is satisfied by a generic poly-
nomial gdof degree das soon as d≥nand that an inductive axiom is satisfied
as soon as d≥n(nr +n+r+ 1). More precisely, we have obtained the following
result.
Theorem 1 Let φ(x1,...,x
n,y 1,...,y n, z)be a conjunction of polynomial
equations with coefficients in Q. Given a tuple zof parameters we denote by Vz
the algebraic subset of C2ndefined by φ.
Fix a tuple zwhich satisfies the associated condition θ(z)from (3). Let rbe
the transcendence degree of Q(z)over Q. For any k≥0,ifd≥n(nr+n+r+1)
there exists x∈Cnsuch that (x, gd(x)) is a generic point of Vz(i.e., a point of
transcendence degree nover Q(z)).
2.2 Axiomatization `a la Wilkie
Given I⊆{1,...,n},sayI={i1,...,i
k}(in increasing order), and a n-
tuple x=(x1,...,x
n) of elements or variables, we denote by xIthe k-tuple
(xi1,...,x
ik).
Consider npolynomials fi(x1,...,x
n,y 1,...,y n,z 1,...,z r) with integer co-
efficients. For each value of the parameter zwe obtain a polynomial map
Fz:Kn×Kn→Kn. Recall that a zero (a, b)ofFzis said to be regular if the
Jacobian matrix of Fzat (a, b)hasrankn,orinotherwordsif
det ∂Fz
∂(xI, yJ)(a, b)=0 (4)
for some I,J ⊆{1,...,n}such that |I|+|J|=n.LetV(Fz) be the set of zeros
of Fz. A regular zero lies on a unique irreducible component of V(Fz)andthis
component is of dimension n. Following Wilkie, we say that (a, b) is a balanced
zero1of Fzif a1,...,a
nare all non-zero and pairwise distinct, and if one can
choose Iand Jso that I∪J={1,...,n}(or equivalently, so that I∩J=∅).
In a geometric language, condition (4) means that the tangent space to
V(Fz)at(a, b) has dimension n. If this tangent space is not included in a
subspace of the form xi=cfor some i∈{,...,n}and some constant c∈K
and if additionally (a, b) is a balanced zero of Fz,wesaythat(a, b)isawell
balanced zero.
In this axiomatization we keep the axioms 1,2 and 3 from section 2.1 but
we replace the inductive axioms by:
4’. Let θ(z) be a formula of the language of fields which expresses that Fz
has a well balanced zero. We add the following axiom.
∀z1,...,z
r∃x1,...,x nθ(z)→F(x, H(x), z)=0.(5)
The following lemma is standard.
1Wilkie uses the terminology “balanced, non-singular zero”. We just write “balanced zero”
for short.
3