A note on one-dimensional varieties over the complex $p$-adic field


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 4, 2018, 1046-1057
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

A note on one-dimensional varieties over the complex
p-adic field

Amran Dalloul

Department of Mathematics, Beirut Arab University, Beirut, Lebanon

Abstract. In this paper, we study the varieties V ⊆ C4
p of dimension one that contain points of

the form (x1, x2, exp(x1), exp(x2)) by using tools from Non-Archimedian Analysis.

2010 Mathematics Subject Classifications: 11E95, 11F85, 11J81

Key Words and Phrases: p-adic analysis, Transcendence Theory

1. Introduction

The algebraic (in)dependence between elements of the form x, exp(x) in the p−adic
domain plays a fundamental role in the p−adic Transcendental Number Theory. Many
results have been made towards this direction. For example, in 1932 K.Mahler, [N],
proved that exp(α) is transcendental over Q for any non-zero algebraic element α ∈ E
(the domain of convergence of the exponential function). In 2008, Yu.V. Nesterenko proved
that if α1, ..., αn ∈ E are algebraic over Q and form a basis of a finite extension of degree
n of Q. Then, there exist at least bn2 c among the elements exp(α1), ...., exp(αn) which are
Q−algebraically independent. This result is usually called half of Lindemann-Weierstrass
Conjecture in the p−adic domain, [N].
In this paper, we use Weierstrass Preparation Theorem to give necessary and sufficient
conditions on a class of polynomials over Z so that each one of them has a root of the
form (x, exp(x)). Similarly, we use Hilbert Theorem on the ring of strictly convergent
power series to give necessary and sufficient conditions on a class of polynomials over Z
so that each one of them has a root of the form (exp(x1), exp(x2)). That enables us to
put necessary conditions on certain varieties V ⊆ C4

p of dimension one over Q in order
to have points of the form (x1, x2, exp(x1), exp(x2)). Also, we give a class of varieties
V ⊆ C4

p of dimension one over Q such that each variety contains a point of the form
(x1, x2, exp(x1), exp(x2)). This point does not contradict Schanuel’s conjecture for two
elements. The conjecture asserts that for a given variety V ⊆ C4

p over Q of dimension one
and a tuple (x1, x2, exp(x1), exp(x2)) ∈ V , then x1, x2 are Q-linearly dependent. Finally,

DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3281

Email address: amrandalloul@hotmail.com (A. Dalloul)

http://www.ejpam.com 1046 c© 2018 EJPAM All rights reserved.



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1047

we give some applications on Weierstrass Preparation Theorem and Hilbert Theorem
concerning the algebraic dependence over Qp and other related topics.
Many results concerning the existence of roots of p−adic exponential polynomials have
been made by Poorten ( see [P] and [PR]) and others. These results imply the existence
of roots of polynomials P [X,Y ] ∈ Q[X,Y ] of the form (x, exp(x)). In our work, we
consider these polynomials directly where the coefficients and degrees of the variables
play a role in the existence of such roots. We prove, as a Corollary, that there exist
polynomials in Q[X,Y ] which do not contain any root of the form (x, exp(x)). This
implies the existence of varieties V ⊆ C4

p over Q of dimension one which do not contain
points of the form (x1, x2, exp(x1), exp(x2)). Using the same technic, we prove that there
exist polynomials over Z with 2n variables which do not contain any root of the form
(x1, .., xn, exp(x1), .., exp(xn)). Furthermore, we use Weierstrass Preparation Theorem to
prove the existence of varieties V ⊆ C4

p over Qp of dimension one that contain points of
the form (x1, x2, exp(x1), exp(x2)).

2. Background

We recall some basic notations and results regarding the field of p−adic numbers and
some elementary Non-Archimedian Analysis that will be needed later. For more details,
see [BGR] and [G].
Let p be a prime number, Qp the completion of Q with respect to the non-archimedian
absolute value |.| and Cp the completion of the algebraic closure of Qp. This field is
Non-Archimedian (with respect to the extended p-adic absolute value |.|), complete and
algebraically closed with the residue class field F̄p (the algebraic closure of the field Fp)
and the value group pQ∪{0}. Moreover, The field Cp is endowed by the exponential map:

exp : E → 1 + E,

x 7−→
∑
n≥0

xn

n!

where E = {x ∈ Cp; |x| < p
−1
p−1 }.

It is well-known in the Non-Archimedian fields that a series
∑

n an is convergent if and only
if lim
n→∞

|an| = 0. Therefore, Let f(X) =
∑∞

n=0 anX
n ∈ Cp[[X]] be a power series. Then,

f(X) is convergent for each x in the closed ball B(0, c) if and only if lim
n→∞

|an|cn = 0. Since

(|an|cn) is convergent, it is bounded. i.e, it has a maximum. Therefore, the norm ‖ . ‖c
on f(X) is defined as follows:

‖ f(X) ‖c:= max{|an|cn}.

We summaries the properties of ‖ . ‖c as follows, [G]:
1) ‖ f(X) ‖c= 0⇔ f(X) ≡ 0,
2) ‖ f(X) + g(X) ‖c≤ max{‖ f(X) ‖c, ‖ g(X) ‖c},
3) ‖ α ‖c= |α|, for any constant α ∈ Cp.



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1048

4) |f(x)| ≤‖ f(X) ‖c, for any x ∈ B(0, c),
where f(X), g(X) are convergent power series on B(0, c) and |.| stands for the p−adic
absolute value on Cp.
Now, we are able to state Weierstrass Preparation Theorem:

Theorem 1. (Weierstrass Preparation Theorem [G]) Let c be a positive real number of
the form pα, α ∈ Q, and let

f(X) = a0 + a1X + ..+ anX
n + .. ∈ Cp[[X]]

be a power series convergent on the closed ball B(0, c). Let N ∈ N be a number defined by
the conditions:
|aN |cN = maxn{|an|cn} and |aN |cN > |an|cn, ∀n > N.
Then, there exist a polynomial g(X) ∈ Cp[X] of degree N , and a power series h(X)
convergent on the closed ball B(0, c) such that
1) f(X) = h(X)g(X). In addition, each root of g(X), if exists, belongs to B(0, c).
2) ‖ h(X)− 1 ‖c< 1. In particular, h(X) has no roots in B(0, c).

We need the following notions and results related to the ring of strictly convergent
power series in order to study the polynomials in Z[X1, X2] that admit roots of the form
(exp(x1), exp(x2)). See [S] and [BGR] for more details.
Let (K, |.|) be a Non-Archimedian, complete and algebraically closed field. Then, a formal
power series f(X1, .., Xn) =

∑
I=(i1,..,in)

aIX
i1
1 ...X

in
n ∈ K[[X1, .., Xn]] is convergent on a

ball B(0, ρ) := {x̄ = (x1, .., xn) ∈ Cnp : max |xi| ≤ ρ} if and only if

|aI |ρ(i1+...+in) → 0 as i1 + ...+ in →∞. We define a norm |.|ρ on f as follows:

|f |ρ := max
I=(i1,..,in)

{|aI |ρ(i1+...+in)}.

This norm is usually called Gauss norm, [S]. Let Tn(ρ) be the set of all formal power series
in K[[X1, .., Xn]] which are convergent on the ball B(0, ρ). Then, Tn(ρ) forms a complete
normed K−algebra embeds K[X1, .., Xn] as a dense K− subalgebra. In particular, for
ρ = 1, K〈X1, .., Xn〉 denotes the ring of all power series which are convergent on the unit
ball. Each element of this ring is usually called strictly convergent power series, [S]. Then,
we have the following:

Lemma 1. ([S] Lemma 4.9, p.9) A strictly convergent power series
f =

∑
I=(i1,..,in)

aIX
i1
1 ...X

in
n ∈ K〈X1, .., Xn〉 is unit in K〈X1, .., Xn〉 if and only if |a(0,..,0)| =

|f | and |a(i1,..,in)| < |f | for all i1 + ..+ in > 0.

This lemma immediately implies that if |a(0,0,..,0)| < |f |, then f is not unit inK〈X1, .., Xn〉.

Lemma 2. (Hilbert Theorem [S], Corollary 5.10, p.14) There is a one to one corre-
spondence between the maximal ideals of K〈X1, .., Xn〉 and the points in the unit ball
B(0, 1) := {x̄ = (x1, .., xn) ∈ Cnp : max{|xi|} ≤ 1}. Under this correspondence, a point
x̄ = (x1, .., xn) ∈ B(0, 1) determines the maximal ideal 〈X1 − x1, .., Xn − xn〉.

Throughout the paper, we use the standard notation (x̄, exp(x̄)) for the 2n- tuple
(x1, ..., xn, exp(x1), ..., exp(xn)) , [K].



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1049

3. The Main Results

It is clear that finding roots of a polynomial with rational coefficients can be reduced
to the case of coefficients in Z. Therefore, without loss of generality, we can take the
polynomials over Z. We only consider the class of polynomials P [X,Y ] ∈ Z[X,Y ] in
which at least one of the degrees of the variable Y is relatively prime to p. Furthermore,
we exclude the case of polynomials that contain the variable X in each term since they
have the trivial root (0, exp(0)).

Theorem 2. The polynomial with rational integer coefficients

P [X,Y ] = c+
m∑
i=1

diY
αi + e1XY

β1,2 +
s∑

k=1

fkX
γk,1Y γk,2 ; γk,1 ≥ 2,

in which (d1α1 + ..+dmαm+e1, p) = 1, has a root of the form (x, exp(x)) ∈ Cp×C∗p; p ≥ 3
if and only if |c+ d1 + ..+ dm| ≤ p−1.

Proof. (Proof of the necessary condition) If (x, exp(x)) is a root of P [X,Y ], then x is
a root of the power series f(X) := P [X, exp(X)] which is convergent on E (since at least
one of the degrees of the variable Y is relatively prime to p, see [PR, Theorem 1] ). Thus,
x ∈ E. So,

c+

m∑
i=1

di exp(αix) = −
(
e1x exp(β1,2x) +

s∑
k=1

fkx
γk,1 exp(γk,2x)

)
.

We have Z ⊆ Zp and | exp(w)| = 1 for every w ∈ E. Using the strong triangle inequality,
it follows that

|c+
m∑
i=1

di exp(αix)| ≤ max
k
{|e1x exp(β1,2x)|, |fkxγk,1 exp(γk,2x)|}

≤ max
k
{|e1||x|| exp(β1,2x)|, |fk||xγk,1 || exp(γk,2x)|)}

≤ max
k
{|x|, |x|γk,1}

< p
−1
p−1 < 1.

We define zi = αix; i = 1, 2, ..,m. Then,

|c+ d1 exp(z1) + ...+ dm exp(zm)| < 1. (1)

Therefore, |c+ d1 + ...+ dm| < 1. This is because,

|c+d1 exp(z1)+...+dm exp(zm)| = |c+d1+...+dm+d1(exp(z1)−1)+..+dm(exp(zm)−1)|.

If |c+ d1 + ...+ dm| = 1, then we find that

|d1(exp(z1)− 1) + ..+ dm(exp(zm)− 1)| ≤ max
1≤i≤m

{|di(exp(zi)− 1)|}



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1050

≤ max
1≤i≤m

{|(exp(zi)− 1)|}

( using the fact |w| = | exp(w)− 1|, ∀w ∈ E) ≤ max
1≤i≤m

{|(zi|}

< p
−1
p−1

< 1.

So, by the isosceles triangle inequality, we find that

|c+ d1 exp(z1) + ...+ dm exp(zm)| =
= max{|c+ d1 + ...+ dm|, |d1(exp(z1)− 1) + ..+ dm(exp(zm)− 1)|} =

= |c+ d1 + ...+ dm| = 1.

This contradicts (1). Therefore, |c+ d1 + ...+ dm| < 1, so |c+ d1 + ...+ dm| ≤ p−1. This
is because, c+ d1 + ...+ dm ∈ Z, and the value group of Z is pZ ∪ {0}.i.e, for each q ∈ Z∗,
|q| = ps, for some s ∈ Z.
Proof of the sufficient condition. Consider the polynomial

P [X,Y ] = c+
m∑
i=1

diY
αi + e1XY

β1,2 +
s∑

k=1

fkX
γk,1Y γk,2 ∈ Z[X,Y ],

with the condition |c + d1 + ... + dm| ≤ p−1. We have to prove that P [X,Y ] has a
root of the form (x, exp(x)), x ∈ E. This is equivalent to prove that the power series
f(X) := P [X, exp(X)] has a root x ∈ E. Suppose that the power series f(X) takes the
form f(X) = a0 + a1X + ..+ anX

n + ..
In our case, we have

a0 = c+ d1 + ..+ dm,

a1 = d1α1 + ..+ dmαm + e1,

an =
d1α

n
1 + ..+ dmα

n
m

n!
+ e1

βn−11,2

(n− 1)!
;n < min

1≤j≤s
{γj,1},

an =
d1α

n
1 + ..+ dmα

n
m

n!
+ e1

βn−11,2

(n− 1)!
+ f1

γ
n−γ1,1
1,2

(n− γ1,1)!
+ ..+ fs

γ
n−γs,1
s,2

(n− γs,1)!
;n ≥ min

1≤i≤s
{γj,1}.

Let α be any rational number satisfying −1 < α < −1
p−1 . In fact, we have chosen α ∈ Q

to guarantee that pα ∈ |Cp|. Then f(X) is convergent on the closed ball B(0, pα). The
general assumption of the theorem guarantees that ord(a1) = 0. Therefore, |a1| = 1. Also,
by definition of α, we find that p−1 < pα. Thus,

|a0| ≤ p−1 < |a1|p1.α ≤ max
n≥1
{|an|pnα}.

Therefore, the number N , defined in Weierstrass Preparation theorem, is strictly larger
than zero.i.e, N > 0. Weierstrass Preparation theorem guarantees that f(X) can be writ-
ten in the form f(X) = h(X)g(X); h(X) is a power series convergent and non-vanishing



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1051

on B(0, pα) and g(X) is a polynomial with p-adic complex coefficients of degree N > 0.
Since Cp is algebraically closes field, it follows that g(X) has a root x. This root belongs
to B(0, pα).i.e., x ∈ E. Therefore, f(x) = h(x).0 = 0. Thus, P (x, exp(x)) = 0.

Remark 1. In the proof of the necessary condition, we did not use the assumption (d1α1+
..+ dmαm + e1, p) = 1. This implies that any polynomial of the form

P [X,Y ] = c+
m∑
i=1

diY
αi +

s∑
k=1

fkX
ξk,1Y ξk,2 ; ξk,1 ≥ 1,

with (c+ d1 + ..+ dm, p) = 1 and at least one of the degrees of the variable Y is relatively
prime to p does not have any root of the form (x, exp(x)).

Example 1. We can use Remark 1 to prove that the polynomial P [X,Y ] = X2 + Y 2 has
no roots of the form (x, exp(x)) ∈ Cp × C∗p; p ≥ 3.

Example 2. Consider the polynomial

P [X,Y ] = p− 1 + (p+ 1)Y p +X +X3Y p−1 +X7Y 15.

Then, the domain of f(X) = P [X, exp(X)] is E, (d1α1 + ..+ dmαm + e1, p) = (p(p+ 1) +
1, p) = 1 and |c+d1 + ..+dm| = |2p| = p−1. According to Theorem 2, we find that P [X,Y ]
has a root of the form (x, exp(x)) ∈ Cp × C∗p; p ≥ 3. This example shows that there exists
a non trivial tuple of the form (x, exp(x)) satisfies an algebraic dependence relation with
rational integer coefficients relatively prime to p.

Also we can use Hilbert Theorem to get a result concerning the roots of the form
(exp(x1), exp(x2)) to the polynomials with rational integer coefficients and two variables .

Theorem 3. The polynomial

P [X1, X2] = aI0 + aI1X
i1,1
1 X

i1,2
2 + ...+ aImX

im,1

1 X
im,2

2 ∈ Z[X1, X2],

in which at least one of the elements aI1i1,1 + ....+ aImim,1, aI1i1,2 + ....+ aImim,2 and all
the degrees of X1 and X2 are relatively prime to p has a root of the form (exp(x1), exp(x2))
if and only if |aI0 + ....+ aIm | ≤ p−1.

Proof. (Proof of the necessary condition). If P has a root of the form (exp(x1), exp(x2))
for some elements x1, x2 ∈ E, then

P (exp(x1), exp(x2)) = aI0 + aI1(exp(x1))
i1,1(exp(x2))

i1,2 + ...+ aIm(exp(x1))
im,1(exp(x2))

im,2 = 0.

Let zj := ij,1x1 + ij,2x2, ∀j = 1, 2, ..,m. Then zj ∈ E. Using the universal property of the
exponential function, we obtain

aI0 + aI1 exp(z1) + ...+ aIm exp(zm) = 0.



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1052

Thus,

|aI0 + aI1 exp(z1) + ...+ aIm exp(zm)| = 0 < 1.

By a similar argument to the necessary proof of Theorem 2, we find that
|aI0 + ....+ aIm | ≤ p−1.
Proof of the sufficient condition. Consider the polynomial

P [X1, X2] = aI0 + aI1X
i1,1
1 X

i1,2
2 + ...+ aImX

im,1

1 X
im,2

2 ∈ Z[X1, X2],

in which at least one of the elements aI1i1,1 + ....+ aImim,1, aI1i1,2 + ....+ aImim,2 and all
the degrees of X1 and X2 are relatively prime to p.
Let f ∈ Cp[[X1, X2]] be an element defined by the relation

f(X1, X2) = P [exp(X1), exp(X2)].

Then, P [X1, X2] has a root of the form (exp(x1), exp(x2)) if and only if (x1, x2) is a root
of f. It is clear that f is convergent on the ball B(0, ρ) := {(x1, x2) : max |xi| ≤ ρ, i = 1, 2}
for every ρ < p

−1
p−1 (since all the degrees of the variables X1 and X2 are relatively prime to

p). Let α be a rational number satisfying the relation −1 < α < −1
p−1 . Then, f(X1, X2) is

convergent on the ball B(0, pα). We define new variables: Z1 := pαX1, Z2 := pαX2. Also,
we define a new power series g(Z1, Z2) by the relation

g(Z1, Z2) := f(p−αZ1, p
−αZ2).

It’s clear that g(Z1, Z2) is convergent on the unit ball B(0, 1). Furthermore, f(X1, X2)
has a root in the ball B(0, pα) if and only if g(Z1, Z2) has a root in the unit ball. Since
g(Z1, Z2) is convergent on the unit ball, it follows that g(Z1, Z2) ∈ Cp〈Z1, Z2〉. Suppose
that g(Z1, Z2) takes the form g = (g0, g1, ..., gq, ...),where gi is homogeneous polynomial of
degree i. Then, in our case, we have

g0 = g(0, 0) = aI0 + ....+ aIm ,

g1 = (aI1i1,1 + ....+ aImim,1)p
−αZ1 + (aI1i1,2 + ....+ aImim,2)p

−αZ2.

Suppose that α takes the form α = −m
n . Then, we have

|p−α|n = |pm| = p−m ⇒ |p−α| = p
−m
n = pα.

We assume that (aI1i1,1 + ....+aImim,1, p) = 1 (the other case can be done similarly). This
implies that |aI1i1,1 + ....+ aImim,1| = 1.
Now, since −1 < α < −1

p−1 , it follows that p−1 < pα. Hence, we obtain the inequalities

|g0| = |aI0 + ....+ aIm | ≤ p−1 < pα = |(aI1i1,1 + ....+ aImim,1)p
−α| ≤

≤ max
J
{|bJ |} = |g|,



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1053

where {bJ} are the coefficients of the power series g. Thus,

|g(0, 0)| < |g|.

Using Lemma 1, it implies that g is not unit in the ring Cp〈Z1, Z2〉. Therefore, there exits
a maximal ideal % in Cp〈Z1, Z2〉 such that g ∈ %. Using Lemma 2 and the fact that Cp is
algebraically closed field, it follows that there exist the elements z1, z2 ∈ B(0, 1) such that

% = 〈Z1 − z1, Z2 − z2〉.

Therefore, g can be written in the form g = r1(Z1 − z1) + r2(Z2 − z2), for some r1, r2 ∈
Cp〈Z1, Z2〉. Thus, it is clear that

g(z1, z2) = 0.

Hence, g has a root in the unit ball. Therefore, f has a root in the ball B(0, pα). Thus,
the original polynomial P [X1, X2] has a root of the form (exp(x1), exp(x2)).

Corollary 1. Let V ⊆ C4
p be a variety over Q of dimension one defined by a system of

polynomials with rational integer coefficients of the form

P1[X1, X3] = c(1) +

m∑
i=1

d
(1)
i X

α
(1)
i

3 +

r∑
l=1

f
(1)
k X

ξ
(1)
k,1

1 X
ξ
(1)
k,2

3 ; ξ
(1)
k,1 ≥ 1

P2[X2, X4] = c(2) +

m∑
i=1

d
(2)
i X

α
(2)
i

4 +

r∑
l=1

f
(2)
k X

ξ
(2)
k,1

2 X
ξ
(2)
k,2

4 ; ξ
(2)
k,1 ≥ 1

P3[X3, X4] = aI0 + aI1X
i1,1
3 X

i1,2
4 + ...+ aImX

im,1

3 X
im,2

4 ,

such that there exists a degree of each of the variables X3 and X4 in P1 and P2 respectively
which is relatively prime to p and all the degrees of the variables X3 and X4 in P3 are also
relatively prime to p. If V contains a point of the form (x1, x2, exp(x1), exp(x2)), then the

quantities c(1) +
∑m

i=1 d
(1)
i , c(2) +

∑m
i=1 d

(2)
i and aI0 + ...+ aIm are all divisible by p.

Proof. If V contains a point of the form (x1, x2, exp(x1), exp(x2)), then we have

P1(x1, exp(x1)) = P2(x2, exp(x2)) = P3(exp(x1), exp(x2)) = 0.

Using Theorems 2 and 3, we find that the quantities c(1) +
∑m

i=1 d
(1)
i , c(2) +

∑m
i=1 d

(2)
i and

aI0 + ...+ aIm are all divisible by p.

Remark 2. From the previous corollary, we can deduce that if we have a variety V ⊆ C4
p

defined as in the previous Corollary in which one of the quantities c(1) +
∑m

i=1 d
(1)
i , c(2) +∑m

i=1 d
(2)
i or summation of coefficients of P3 is relatively prime to p, then V has no point

of the form (x1, x2, exp(x1), exp(x2)).

We can also give sufficient conditions on a class of varieties such that each variety
admits a point of the form (x1, x2, exp(x1), exp(x2)) as follows.



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1054

Corollary 2. Let p be an odd prime and let c, d,m ∈ Z,m ≥ 1 with the conditions
(d + 1, p) = (m, p) = 1, p|(c + d). Then the variety V ⊆ C4

p of dimension one defined by
the system of polynomials

P1[X1, X3] = c+ dXm
3 +mX1

P2[X2, X4] = c+ dX4 +X2

P3[X3, X4] = X4 −Xm
3 ,

has a point of the form (x1, x2, exp(x1), exp(x2)).

Proof. In fact, Theorem 2 guarantees that P1 has a root of the form (x, exp(x)). By a
simple calculation, we find that (mx, exp(mx)) is a root of P2 which admits roots of the
form (x, exp(x)) according to Theorem 2.
It’s clear that (exp(x), exp(mx)) is a root of P3 which admits roots of the form (exp(x1), exp(x2))
according to Theorem 3. Hence (x,mx, exp(x), exp(mx)) ∈ V.

Remark 3. Schanuel’s conjecture in the case of two variables asserts that if V ⊆ C4
p is

a variety of dimension one over Q and has a point of the form (x1, x2, exp(x1), exp(x2)),
then the point must take the form (x,mx, exp(x), exp(mx)), for some m ∈ Q.

4. Further Applications of Weierstrass Preparation Theorem and
Hilbert Theorem

We can use Weierstrass Preparation Theorem to get a result concerning the algebraic
dependence over Qp as follows.

Theorem 4. Let P1, P2 ∈ Z[X,Y ] be polynomials defined as in the beginning of the pre-
vious section of the form

P1[X,Y ] = c(1) +
m∑
i=1

d
(1)
i Y

α
(1)
i

i + e(1)XY β
(1)
1,2 +

s∑
k=1

f
(1)
k Xγ

(1)
k,1Y γ

(1)
k,2 ; γ

(1)
k,1 ≥ 2,

P2[X,Y ] = c(2) +
m∑
i=1

d
(2)
i Y

α
(2)
i

i + e(2)XY β
(2)
1,2 +

s∑
k=1

f
(2)
k Xγ

(2)
k,1Y γ

(2)
k,2 ; γ

(2)
k,1 ≥ 2,

in which (d
(1)
1 α

(1)
1 +...+d

(1)
m α

(1)
m , p) = (d

(2)
1 α

(2)
1 +...+d

(2)
m α

(2)
m , p) = 1. If the quantities c(1)+∑m

i=1 d
(1)
i , c(2) +

∑m
i=1 d

(2)
i are divisible by p and (x1, exp(x1)), (x2, exp(x2)) are roots of

P1, P2 receptively, then there exists a variety V ⊆ C4
p over Qp of dimension ≤ 1 containing

the point (x1, x2, exp(x1), exp(x2)).

Proof. Since Q ⊆ Qp and P1(x1, exp(x1)) = 0, it follows that x1 and exp(x1) are
Qp−algebraically dependent. The same holds true for x2 and exp(x2). It remains to show
that x1 and x2 are Qp−algebraically dependent. For this, it suffices to show that x1 and
x2 are algebraic over Qp. We briefly review the proof of Theorem 2. We have considered



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1055

the power series f [X] := P [X, exp(X)] ∈ Q[[X]] which is convergent on the closed ball
B(0, pα), α ∈ (−1, −1p−1) ∩ Q. Weierstrass Preparation Theorem can be applied over any
finite extension K of Qp (For more details, see [G]). Also, the coefficients of f(X) (which
are rationals) can be considered as elements in any finite extension of Qp. Hence, we can
take K to be Qp. Then, f(X) can be factored in the form f(X) = g(X)h(X), where
g(X) ∈ Qp[X] and h(X) ∈ Qp[[X]] is non-vanishing and converging on the ball B(0, pα).
The roots of f(X) are exactly the roots of the polynomial g. That is, each root of f(X) is
algebraic over Qp. From this argument, we deduce that x1 and x2 are algebraic numbers
over Qp. This clearly implies that x1 and x2 are Qp−algebraically dependent. Thus,

tdQpQp(x1, x2, exp(x1), exp(x2) ≤ 1.

Hence, there exists a variety V ⊆ C4
p over Qp of dimension ≤ 1 containing the point

(x1, x2, exp(x1), exp(x2)).

Finally, we generalize Theorem 2 to the case of polynomials P [X1, ..., Xn, Y1, ..., Yn] ∈
Q[X1, ..., Xn, Y1, ..., Yn].
As in the two variables case, we reduce the problem to find the roots of polynomials with
rational integer coefficients and exclude the polynomials that have at least one of the vari-
ables X1, .., Xn in each term since it implies that the trivial point(0, .., 0, exp(0), .., exp(0))
is a root of these polynomials. Also, we only consider the polynomials in which all the
degrees of the variables Y1, ..., Yn are relatively prime to p. Then, we prove

Theorem 5. The polynomial with rational integer coefficients

P [X1, ..., Xn, Y1, ..., Yn] = c+
m∑
i=1

diY
αi,1

1 ...Y
αi,n
n +

n∑
j=1

ejXjY
βj,1
1 ....Y

βj,n
n +

s∑
k=1

fkX
γk,1
1 ...X

γk,n
n Y

γk,n+1

1 ...Y
γk,2n
n ;

γk,1 + ...+ γk,n ≥ 2,

in which at least one of the elements
(d1α1,1 + ...+ dmαm,1 + e1), ..., (d1α1,n + ...+ dmαm,n + en) is relatively prime to p; p ≥ 3,
has a root of the form (x̄, exp(x̄)) if and only if

|c+ d1 + ...+ dm| ≤ p−1.

Proof. Proof of the necessary condition. If (x̄, exp(x̄)) is a root of the polynomial
P [X1, ..., Xn, Y1, ..., Yn], then x̄ is a root of the power series
f(X1, ..., Xn) := P [X1, ..., Xn, exp(X1), .., exp(Xn)] which is convergent on the disk

{x̄ : max |xi| < p
−1
p−1 }. Thus, x̄ ∈ En. So,

c+
m∑
i=1

di exp(αi,1x1)... exp(αi,nxn) =



A. Dalloul / Eur. J. Pure Appl. Math, 11 (4) (2018), 1046-1057 1056

= −
( n∑
j=1

ejxj exp(βj,1x1).... exp(βj,nxn)+

s∑
k=1

fkx
γk,1
1 ...x

γk,n
n exp(γk,n+1x1)... exp(γk,2nxn)

)
.

Using the fact Z ⊆ Zp, | exp(w)| = 1,∀w ∈ E and the strong triangle inequality, we find
that

|c+
m∑
i=1

di exp(αi,1x1)... exp(αi,nxn)| < 1.

Let zi = αi,1x1 + ...+αi,nxn; i = 1, 2, ..,m. Using the universal property of the exponential
function, we find that

|c+ d1 exp(z1) + ...+ dm exp(zm)| < 1.

By a similar fashion to the two variables case, we find that |c+ d1 + ...+ dm| ≤ p−1.
Proof of the sufficient condition. Consider the polynomial

P [X1, ..., Xn, Y1, ..., Yn] =c+
m∑
i=1

diY
αi,1

1 ...Y
αi,n
n +

n∑
j=1

ejXjY
βj,1
1 ....Y

βj,n
n +

s∑
k=1

fkX
γk,1
1 ...X

γk,n
n Y

γk,n+1

1 ...Y
γk,2n
n ; γk,1 + ...+ γk,n ≥ 2,

with the conditions:
1) At least one of the elements (d1α1,1 + ...+ dmαm,1 + e1), ..., (d1α1,n + ...+ dmαm,n + en)
is relatively prime to p,
2) |c+ d1 + ...+ dm| ≤ p−1.
Consider the ring of the formal power series Cp[[X1, ..., Xn]]. Let f ∈ Cp[[X1, ..., Xn]] be
an element defined by the relation

f(X1, ..., Xn) = P [X1, ..., Xn, exp(X1), .., exp(Xn)].

Then, P [X1, ..., Xn, Y1, ..., Yn] has a root of the form (x̄, exp(x̄)) if and only if (x1, .., xn)

is a root of f. It is clear that f is convergent on the ball B(0, ρ) for every ρ < p
−1
p−1 .

Applying the same argument in the proof of Theorem 3, we find that P has a root of the
form (x̄, exp(x̄)).

Remark 4. As in the two variables case, the polynomial over Z

P [X1, ..., Xn, Y1, ..., Yn] = c+

m∑
i=1

diY
αi,1

1 ...Y
αi,n
n

+

s∑
k=1

fkX
ξk,1
1 ...X

ξk,n
n Y

ξk,n+1

1 ...Y
ξk,2n
n

; ξk,1 + ...+ ξk,n ≥ 1,

with (c+ d1 + ...+ dm, p) = 1 has no roots of the form (x̄, exp(x̄)).



REFERENCES 1057

Acknowledgements

I would like to thank the referees for their constructive comments. Also, I would like
to thank Ali Bleybel for proposing this subject, as well as his constant help and support
throughout the preparation of this paper.

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