arXiv:0903.1442v2 [math.LO] 24 May 2009
Henson and Rubel’s Theorem for Zilber’s
Ahuva C. Shkop
May 25, 2009
In 1984, Henson and Rubel () proved the following theorem: If
p(x1,...,xn) is an exponential polynomial with coefficients in C with
no zeroes in C, then p(x1,...,xn) = eg(x1,...,xn)for some exponential
polynomial g(x1,...,xn) over C. In this paper, I will prove the analog
of this theorem for Zilber’s Pseudoexponentiation directly from the
axioms. Furthermore, this proof relies only on the existential closed-
ness axiom without any reference to Schanuel’s conjecture.
In , Zilber constructed an exponential field, Zilber’s Pseudoexponentiation,
of size continuum that satisfies many special properties. Schanuel’s conjec-
ture is true in this field and every definable set is countable or co-countable
(quasiminimality). It is still unknown whether Pseudoexponentiation is iso-
morphic to complex exponentiation.
In , Henson and Rubel prove that the only exponential polynomials
with no zeros are of the form exp(g) where g is some exponential polynomial.
Although this seems to be a question in exponential algebra, this proof uses
The goal of this paper is to prove the following theorem:
Theorem 1. Let p(x1,...,xn) be an exponential polynomial with coefficients
in Zilber’s Pseudoexponentiation K. If p ?= exp(g(x1,...,xn)) for any expo-
nential polynomial g(x1,...,xn), then p has a root in K.
D’Aquino, Macintyre, and Terzo have also explored this problem and offer
an alternate proof of this theorem in . We will use purely algebraic tech-
niques and give a proof directly from the axioms. This proof uses only basic
exponential algebra and is entirely independent of Schanuel’s conjecture.
We will begin with the following definitions.
Definition 2. In this paper, a (total) E-ring is a Q-algebra R with no zero
divisors, together with a homomorphism exp : ?R,+? → ?R∗,·?.
A partial E-ring is a Q-algebra R with no zero divisors, together with a Q-
linear subspace A(R) of R and a homomorphism exp : ?A(R),+? → ?R∗,·?.
A(R) is then the domain of exp.
An E-field is an E-ring which is a field.
We say S is a partial E-ring extension of R if R and S are partial E-ring,
R ⊂ S, and for all r ∈ A(R), expS(r) = expR(r).
We now set some conventions. Let K be any algebraically closed field and
α ∈ N. Throughout this paper, a variety is a (possibly reducible) Zariski
closed subset of Gα(K) := Kα× (K∗)αor some projection of Gα(K). We
will use the notation ¯ y for a finite tuple y1,...,ym, and we will write exp(¯ y)
instead of exp(y1),...,exp(yn). Similarly for a subset S of an E-ring, exp(S) is
the exponential image of S. We write tdA(¯b) to be the transcendence degree
of the field generated by (A,¯b) over the field generated by A.
To prove Theorem 1, we recall that Zilber’s field, which we will call K,
satisfies the following axiom:
Axiom 3. If a variety V ⊆ Gα(K) is irreducible, rotund, and free, then there
are infinitely many ¯ x such that (¯ x,exp(¯ x)) ∈ V .
The definitions of a rotund variety and a free variety will be given later in
the paper. The outline of the proof is as follows:
1. Given an exponential polynomial p(¯ x), we construct a variety Vpsatis-
∀¯ a(∃¯b(¯ a,¯b,exp(¯ a),exp(¯b)) ∈ Vp ⇐⇒ p(¯ a) = 0).
2. We reduce to the case where Vpis irreducible and free.
3. We prove that if p(¯ x) ?= exp(g(¯ x)), then Vpis rotund.
irreducible but not free. Notice that if p = exp(g) − k for some k ∈ K,
there are infinitely many choices for log(k) and thus infinitely many zeros.
So p must be a polynomial. (We excluded this case on page 3.) The only
irreducible polynomials with finitely many solutions are lines.
 P. D’Aquino, A. Macintyre, G. Terzo Schanuel Nullstellensatz for Zilber
Fields. Preprint, 2008
 C.W. Henson and L.A. Rubel Some applications of Nevanlinna theory
to mathematical logic: identities of exponential functions. Trans. Amer.
Math. Soc., 282 (1984), no. 1, 1-32.
 Angus Macintyre Schanuel’s conjecture and free exponential rings. Annals
of Pure and Applied Logic, 51 (1991), no. 3, 241-246.
 David Marker Remarks on Zilber’s Pseudoexponentiation Journal of Sym-
bolic Logic, 71 (2006), no. 3, 791-798.
 Igor Shafarevich Basic Algebraic Geometry Springer-Verlag Telos, 1995
 Lou van den Dries Exponential rings, exponential polynomials and expo-
nential functions. Pacific Journal of Math, 113 (1984), no. 1, 51-66.
 Boris Zilber Pseudo-exponentiation on algebraically closed fields of char-
acteristic zero., Annals of Pure and Applied Logic, 132 (2005) no. 1,