A Note on Complex p-Adic Exponential Fields

arXiv:1811.04683v1 [math.LO] 12 Nov 2018 A NOTE ON COMPLEX p-ADIC EXPONENTIAL FIELDS ALI BLEYBEL Abstract. In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of p-adic fields in order to get some results towards algebraic independence of p-adic exponentials for almost all primes p. 1. Introduction Let Qp be the field of p-adic numbers, for p a prime number. Given an algebraic closure Qalg p of Qp , it comes naturally equipped with a norm |·|p , uniquely extending the usual norm on Qp . Recall that the standard normalization for |·|p is |p|p = p−1 . Denote by Cp the completion of Qalg p with respect to the norm |·|p . Then Cp is also algebraically closed. It is called a complex p-adic field. The p-adic exponential map ∞ X xn , expp : Ep → C× p , x 7→ n! n=0 1 where Ep is the set Ep = {x ∈ Cp : |x|p < p− p−1 } (the domain of convergence of the defining power series of the exponential) shares several properties with the complex exponential map exp (such as expp (x + y) = expp (x) expp (y), (expp (x))′ = expp (x) where ()′ denotes the usual derivative). There are important open problems regarding the exponential map over a non-archimedean valued field. One of these concerns the algebraic independence of the values of the exponential map at different arguments. Such issues are encapsulated in the following well-known conjecture (p-adic Schanuel’s conjecture) (p-SC) Let x̄ := (x1 , . . . , xn ) ∈ Cnp be an n-tuple of complex p-adic numbers satisfying the requirement |x̄|p := max {|xi |p } < p−1/p−1 . 1≤i≤n Assume that x1 , . . . , xn are Q-linearly independent, then tdQ (x1 , . . . , xn , expp (x1 ), . . . , expp (xn )) ≥ n, where tdQ denotes the transcendence degree of the extension Q(x̄, expp (x̄))/Q. In the following we will denote by G the algebraic group Ga ×Gm , with Ga denoting the additive group of a field (say Cp ) and Gm its multiplicative group. In the above statement we used the abbreviation f (x̄) := (f (x1 ), . . . , f (xn )) for any n-tuple x̄. An equivalent statement to (p-SC) is the following: (p-SC)’ Let x̄ := (x1 , . . . , xn ) ∈ Cnp be an n-tuple of complex p-adic numbers satisfying |x̄|p < p−1/p−1 . Assume that (x̄, expp (x̄)) ∈ V (Cp ), for some subvariety V of Gn defined over Q (i.e. a Q-variety), which is furthermore of dimension < n. Then, x1 , . . . , xn are Q-linearly dependent, i.e. m1 x1 + · · · + mn xn = 0, 1 2 ALI BLEYBEL for some m1 , . . . , mn ∈ Q, not all zero. In this paper we apply the ultraproduct construction and basic model theory in order to obtain some results in the above direction. The main Theorem can be obtained by applying Ax-Schanuel’s Theorem [1] to a non-principal ultraproduct of Cp , and it reads as: 1.1. Theorem. Let V be a Q-variety of dimension n in an affine 2n space. Assume that for infinitely many primes p, V has a Cp -point of the form (āp , expp (āp )), then there exist a finite set S(V ) ⊂ P, and a finite set of rational tuples ᾱi , i ∈ I, (where I is a finite set) such that for all p ∈ P \S(V ), and for all n-tuples x̄p ∈ Epn satisfying (x̄p , expp (x̄p )) ∈ V (Cp ), there is a rational linear dependence that holds for the tuple x̄p of the form αi,1 xp,1 + · · · + αi,n xp,n = 0, for some i ∈ I. An equivalent statement (with a geometrical flavor) is the following: Let V be a Q-variety of dimension n in a 2n-space. If, for infinitely many primes p, V has a Cp point of the form (āp , expp (āp )), then there exist a finite set S ⊂ P and a finite set of hyperplanes Hi ⊂ AnQ , i ∈ I such that for all p ∈ P \S, we have ∀x̄p ∈ Epn , (x̄p , expp (x̄p )) ∈ V (Cp ) −→ (∃i ∈ I)(x̄p ∈ Hi (Cp )). In the above Theorem, the order of quantifiers is essential: for each variety V as above, there is a set S(V ) of exceptional primes (i.e. primes p for which the stated implication might not hold), and such set is only dependent on the variety V . By almost all primes we mean all except a finite set of primes. This is to distinguish from the notion of U-almost all (for a given ultrafilter U) which will be encountered later. A uniform rational linear dependence is a linear dependence of the form m1 x1,p + · · · + mn xn,p = 0, for some fixed rationals m1 , . . . , mn , not all zero. The Theorem implies in particular that, for each family of n-tuples (x̄)p as above there exists a partition of P \ S(V ) into finitely many sets, on each of which the obtained linear dependence is uniform. The method of proof uses Ax’s result [1] on Schanuel’s property for differential exponential fields. For each variety V ⊂ Gn of dimension n as above, the conclusions of Theorem 1.1 hold for all but possibly finitely many primes belonging to some exceptional set S(V ). For a particular variety V ⊂ Gn having dimension ≤ n, and a given prime p ∈ / S(V ), the conclusion of Theorem 1.1 is strictly stronger than what is given by conjecture (p-SC) (or, more precisely, its equivalent (p-SC)’). That is, according to (p-SC), there might exist Q-linearly independent tuples x̄p (in the domain of expp ) for which (x̄p , expp (x̄p )) ∈ V (Cp ) if V is of dimension n, while this is not the case for Theorem 1.1 whenever p ∈ / S(V ). This is due to the statement of Ax’s Theorem, in which the weak inequality in (p-SC) is replaced by a strict one. Let U be a non-principal ultrafilter over the set P of prime numbers. Consider the ultraproduct Y KU := Cp /U. p∈P Then KU is an algebraically closed valued field (whose valuation is induced by p-adic valuations on each Cp ). We may define a partial exponential map on KU , induced by the maps expp . As explained in [10], KU can be embedded in a differential exponential valued field, to which it is possible to apply Ax-Schanuel’s Theorem. Then by an application of Lós’ Theorem on ultraproducts, we obtain the required result. We will consider stronger versions of these results in a forthcoming paper. Theorem 1.1 will be proved in section 3.3, after several preliminary sections, which contain reminders of known results concerning valued fields, ultraproducts of valued fields and other related concepts. A NOTE ON COMPLEX p-ADIC EXPONENTIAL FIELDS 3 Acknowledgments. I am grateful to the anonymous referee for his careful reading of the manuscript, and for many valuable comments and suggestions. 2. Background In this section we introduce background results that will be needed in the rest of the paper. Recall that the field Cp is the completion (with respect to the norm |·|p ) of an algebraic closure of Qp , the field of p-adic numbers. One may consider instead the additive valuation ordp defined on Cp . This valuation is defined through the relation: |z|p = p−ordp (z) . In [1] J. Ax proved the following result, already conjectured by S. Schanuel: 2.1. Theorem. (Ax [1]) Let K be a differential field equipped with a derivation D, and let C be its field of constants. Let y1 , . . . , yn , z1 , . . . , zn ∈ K × be such that Dyi = Dzi /zi . Assume that the yi , i = 1, . . . , n are Q-linearly independent modulo C, then tdC (y1 , . . . , yn , z1 , . . . , zn ) ≥ n + 1. Recall that a derivation over a (commutative) field K is a map D : K → K satisfying additivity (D(x + y) = Dx + Dy) and Leibniz rule (D(xy) = xDy + yDx). The field of constants for D is the set of x ∈ K for which Dx = 0. Using additivity and Leibniz rule, one can see that C is indeed a subfield of K. In [10] this result was restated as follows: k 2.2. Theorem. Let y1 , . . . , yn , z1 , . . . , zn ∈ K × be such that Dyk = Dz zk for k = 1, . . . , n. Pn If tdC C(y1 , . . . , yn , z1 , . . . , zn ) ≤ n, then i=1 mi yi ∈ C for some m1 , . . . , mn ∈ Q not all zero. A corollary of the above is given by (this is essentially Corollary 3 in [10]): 2.3. Proposition. Let (K, exp) be a partial differential exponential field (that is, a field equipped with a partial exponential map exp, satisfying D exp(x) = exp(x)Dx), with a field of constants k. Then, for any n-tuple x̄ := (x1 , . . . , xn ) ∈ K n of elements of K, where x1 , . . . , xn belong to the domain of the exponential map. If (x̄, exp(x̄)) P ∈ V (K) for some algebraic variety V of dimension n with rational coefficients, n V ⊂ Gn , then i=1 mi xi ∈ k, for some m1 , . . . , mn ∈ Q not all zero. 2.4. Language and Logical Setting. Let L = (+, −, ·, ( )−1 , 1, 0) be the language of fields, with the standard interpretation of the symbols involved. We consider the expansion L of L: L = L ∪ {R, Exp}, where R is a unary predicate symbol while Exp is a function symbol (to be interpreted as an exponential map K → K × , with K × being the set of invertible elements of K). Let K be a differentially valued partial exponential field. Denote by val the valuation on K, Γ its value group, and let R be the valuation ring with maximal ideal P. As we will see below, in many cases of interest one can extend the partial exponential on K to a total exponential map (which is not uniquely determined though). Notable exceptions are Laurent power series fields, as well as generalized power series fields (whose definition will be recalled below). It follows that by making the appropriate interpretation of each symbol of L, the field K is then naturally an L-structure (with Exp denoting the (extended) exponential map). Furthermore, the maximal ideal P of R can be defined as follows: x∈P iff x ∈ R & x−1 ∈ / R. Assume now that R is a discrete valuation ring, and let π ∈ R be a uniformizer, i.e. val(π) = 1. Let Lπ be the expansion L ∪ {π}, with π denoting a constant in K. 4 ALI BLEYBEL Using π, the valuation val can be defined using the predicate R in a standard way: val(x) ≥ 0 iff R(x) (x is in the valuation ring), and for all x ∈ K, val(x) = n ∈ Z iff val(x/π n ) ≥ 0 & val(x/π n ) ≤ 0. For any valued field (K, val) (with a possibly non discrete, or even, a non-archimedean value group G where we fix a (generally non-canonical) embedding Z ֒→ G and identify Z with its image in G), one can still use the language Lπ (for some π satisfying val(π) = 1), and in this case any set of the form {x ∈ K | val(x) > e}, e ∈ Q is Lπ -definable. Explicitly, the above set is defined through the formula (below e = n/m): xm πn ϕe (x) : R( n ) & ¬R( m ). π x The complex p-adic field Cp falls in particular in the above case: the value group of the standard valQ uation ordp is Q, and any non-principal ultraproduct p Cp /U (see below) has a non-archimedean value group. From the above remarks, one can see that a formula of the form val(x) = val(y) is an abbreviation of R(x/y) & R(y/x). Note that the expressive power of the language L falls short of defining every ball in K (a set of the form val(x − a) ≥ g for g ∈ G and some a ∈ K), since g might be a non-standard element. An L-structure is a tuple (K, R, Exp), where K is a valued field, R its valuation ring and Exp is an exponential map Exp : K → K × . 2.5. The field KU . Let P be the set of prime numbers, and let U be a non-principal ultrafilter on P. Here the predicate R is interpreted as the set C◦p of complex p-adic numbers with non-negative p-adic valuation. Define the field KU as the ultraproduct of the fields Cp : Y KU := Cp /U. p∈P The field KU becomes an L-structure upon interpreting the function and predicate symbols in the standard way, for instance R([(xp )p∈P ]) if and only if the set of p ∈ P for which xp ∈ C◦p is in U. In this case we say that xp ∈ C◦p for U-almost all primes. By application of Loś Theorem on ultraproducts, KU is shown to be an algebraically closed field equipped with the valuation induced by ordp (for p running over P). Equip KU with the valuation val defined as: val([x]) = [(ordp (xp ))p∈P ], where we have used the notation [x] Q := [(xp )p∈P ] ∈ KU . The elements (ordp (xp ))p∈P belong to the Cartesian product of value groups p∈P Q, and [(ordp (xp ))p∈P ] belongs to the ultrapower of Q, i.e. QU . It is immediate to verify that val is indeed a valuation on K× U. For more details about ultraproducts of valued fields (and ultraproducts in general), see, e.g. [14]. Let kU be the residue field, kU = R/P , with R and P the valuation ring and its maximal ideal. It follows from Loś Theorem that kU is an algebraically closed field of characteristic zero, hence KU is an equicharacteristic valued field. 2.6. The exponential map. Let p be a prime number. Fix an extension EXPp of the p-adic exponential expp such that EXPp is an exponential map defined for all elements of Cp , i.e. EXPp : Cp → C× p and ∀x ∈ Cp , |x|p < p−1/p−1 , EXPp (x) = ∀x, y ∈ Cp EXPp (x + y) = expp (x), EXPp (x)EXPp (y). The existence of such an extension is guaranteed by Zorn Lemma (see [13] chap. 5, section 4.4). However, it is not unique. It can be seen that EXPp is a continuous homomorphism from the A NOTE ON COMPLEX p-ADIC EXPONENTIAL FIELDS 5 additive group (Cp , +) to the multiplicative group (C× p , ·). For each prime p, the field Cp equipped with the exponential map EXPp : Cp → C× p is a structure for L. Note that the use of the extension EXPp (rather than just the standard p-adic exponential expp ) seems to be useful from the model-theoretic point of view, in view of the intended application. More precisely, since we are considering an ultraproduct of the Cp ’s, the map E([x]) := [(expp (xp ))p ] (see below) is defined on an open disc around the origin of radius 1 − ǫ, with ǫ > 0 an infinitesimal, whereas the domain of expp is the open disc of radius rp := p−1/(p−1) as already observed. Using instead the maps EXPp , allows us to have a uniform definition of the domain of the exponential map. 2.7. Ordered abelian groups. Let (G, +, ≤) be an ordered abelian group under the law +, where ≤ denotes the order relation on G. Let G>0 be the semi-group of positive elements of G (i.e. elements greater than 0). Let ∆ be the set of archimedean classes of G>0 (see, e.g. [6]). The archimedean class of an element g ∈ G will be denoted by [g]. If ∆ is not a singleton, we say that G is non-archimedean. The set ∆ comes equipped with the inherited order  defined as: δ1 = [g1 ]  δ2 = [g2 ] iff (|g2 | ≤ |g1 |), for any δ1 , δ2 ∈ ∆. Obviously, we may define the induced relations ≺ and ≻ in a similar way. Let [0] = ∞. The order  can then be extended to ∆ ∪ {∞} by setting δ  ∞ for all δ ∈ ∆ ∪ {∞}. Denote by v1 the map (called natural valuation) v1 : G → ∆ ∪ {∞} defined as v1 (g) = [g]. 2.7.1. Hahn Embedding Theorem. A central result in the theory of linearly ordered abelian groups is the following: Let G be a linearly ordered abelian group. Then there exists an embedding of ordered groups i : G ֒→ H(∆) ⊂ R∆ where ∆ is the set of archimedean classes of G, and H(∆) (the Hahn group with respect to ∆) is given by H(∆) := {a = (aγ )γ∈∆ : aγ ∈ R and Supp(a) is well ordered}. Here H(∆) is equipped with the lexicographic order, and Supp(a) (for a ∈ R∆ ) is defined as Supp(a) := {γ ∈ ∆ : aγ 6= 0}. P Any element g of G can be written as g = φ∈∆ gφ 1φ where gφ ∈ R and 1φ , φ ∈ ∆ the element of Γ that corresponds through the embedding i to (aψ )ψ∈∆ ∈ R∆ , with aφ = 1 aψ = 0, for ψ 6= φ. We have v1 (g) = min(Supp(g)) ∈ ∆ ∪ {∞}. Q Let Γ be the value group of KU , Γ := ( p∈P Q/U, +). Observe that we have a canonical embedding Q ֒→ Γ, r 7→ [(rp )p∈P ] (with rp = r for all p ∈ P). An element of Γ is called standard if it is in the image of Q by this embedding. Let γ := [(gp )p∈P ] be an element of Γ such that for any ε > 0, there exists p0 ∈ P for which ∀p ∈ P, p > p0 ⇒ |gp | < ε. Then clearly, γ is an infinitesimal element, since it is smaller (in absolute value) than any element of Q>0 . Similarly, an element [(gp )p∈P ] of Γ is infinite iff it satisfies ∀A ∈ Q>0 , ∃p0 ∈ P(p > p0 → |gp | > A). Note that the above definitions are not first-order, since we have no way of quantifying over standard positive rationals in the language. We have: 2.8. Proposition. The group Γ is an ordered abelian group. Furthermore, the set ∆ of archimedean classes of Γ is an infinite, unbounded, densely linearly ordered set having uncountable cofinality. Proof. The first assertion follows using standard properties of ultraproducts, e.g. [2]. To see that ∆ is unbounded we equip Γ with the multiplicative operation (compatible with the order) induced by standard multiplication on Q, endowing Γ with an ordered field structure. Hence ∆ acquires a group structure through v1 (α)+v1 (β) = v1 (αβ), for all α, β ∈ Γ>0 , where v1 : Γ → ∆∪{∞}, g 7→ [g] 6 ALI BLEYBEL as defined above. It follows that ∆ is the value group for the natural valuation v1 . Now the required conclusion follows since Γ is nonarchimedean. Let us now show that ∆ has uncountable cofinality. Assume there exists a countable sequence (δn )n∈N , δn ∈ Γ such that ∀α ∈ Γ, ∃n0 ∈ N, ∀n > n0 , [α]  [δn ]. (†) Writing δn = [(δnk )k∈P ], δnk ∈ Q+ , one can check that the double sequence (δnk )n≥0,k∈P is strictly increasing (beyond some n0 , k0 ). Now let α := [(αk )k ] be defined by αk = δkk . Then it can be checked that (†) does not apply for α, contradiction. Since Γ has cardinality 2ℵ0 , the cofinality of ∆ is at most 2ℵ0 . Finally, to see that ∆ is densely ordered, assume the contrary. It suffices then to observe that the induced order on the set Γ≥0 is of type > ω, on which there exists no possible cancellative semi-group structure compatible with the ordering. This contradiction proves the result.  2.8.1. Kaplansky embedding theorem. Let Γ be an ordered abelian group and k a commutative field. Let k((tΓ )) be the field of generalized power series with a well-ordered set of exponents in Γ and coefficients in k. Denote by v the t-adic valuation of k((tΓ )). We are now able to apply the following: 2.9. Theorem. (Kaplansky [8]) Let (K, val) be a valued field of zero equi-characteristic, with value group Γ and algebraically closed residue field k. Then (K, val) is analytically isomorphic to a subfield of (k((tΓ )), v), i.e. there exists a value preserving embedding of fields K ֒→ k((tΓ )). The original statement in [8] is more general, allowing non-algebraically closed residue fields at the expense of introducing factor sets into the definition of the multiplicative operation of monomials in the power series field. By Theorem 7 of [8], this turns out not to be necessary in the special case of an algebraically closed residue field. 3. A differential exponential valued field Now we consider again the field KU . Observe that the valuation ring of KU is given by R = {x := [(xp )p∈P ] : val(x) ≥ 0}, where the order relation (on the value group of KU ) has already been explained in the previous section. We can easily show that the residue class field kU (kU = R/P where P is the maximal ideal of R) is given by Y kU = Falg p /U, p∈P Falg p where is the algebraic closure of the finite field Fp . By Lefshetz principle (see, e.g. Theorem 2.4.3 [14]) we have kU ≃ C, since both are algebraically closed fields of cardinality 2ℵ0 having characteristic zero. Applying Kaplansky’s result mentioned above, there exists an embedding of valued fields KU ֒→ LU := kU ((tΓ )). For each non-principal ultrafilter U over P, we fix an embedding ιU ιU : KU ֒→ LU , and we will denote by v the canonical valuation on LU . The p-adic exponential map on each Cp can be used to introduce a total exponential map on KU . More precisely, one may show (using Loś Theorem) that the map Exp : [(xp )p ] 7→ [(EXPp (xp ))p ] is indeed an exponential map, KU → K× U (satisfying Exp(x + y) = Exp(x) · Exp(y)). A NOTE ON COMPLEX p-ADIC EXPONENTIAL FIELDS 7 3.0.1. An exponential differential field. In order to be able to apply Ax’s Theorem, we need to define an embedding of KU into a (partial) exponential differentiable field, along the lines of [9] and [10] (see also [11] for a general survey). As will be seen, this embedding need not be an embedding of differential fields, neither this is assumed. First we define a right-shift map σ : ∆ → ∆, φ 7→ σ(φ) such that σ(φ) ≻ φ and σ is orderpreserving. Let δ be the archimedean class of some infinitesimal element of Γ. We set: σ : ∆ → ∆, φ 7→ δ · φ. Here by δ · φ we mean the archimedean class of any product of two elements in δ and φ respectively. It can be seen that this is independent of the choices, and that, indeed σ(φ) ≻ φ, ∀φ ∈ ∆. Then, as in ”Case 1” of Example (6) in [10], one may define a derivation D : LU → LU with field since D is a series derivation of constants kU , and which satisfies furthermore: Dx = D(exp(x)) exp x (see [7], Corollary (3.9)). Let us denote by L◦U the ring of bounded elements of LU , and by L◦◦ U its maximal ideal, i.e. the ◦◦ Γ>0 ideal of infinitesimal elements. Note that LU = kU ((t )) (the set of generalized power series with strictly positive support). Let DU be the set defined as:   1 for U−almost all p ∈ P . DU := x = [(xp )p∈P ] ∈ KU : ordp (xp ) > p−1 × Consider the map E : DU → K× U ⊂ LU defined by [(xp )]p∈P 7→ E([(xp )]p∈P ) := [(expp (xp ))p∈P ]. Using Lós Theorem we see that E is a partial exponential map on LU (i.e. E(x + y) = E(x)E(y)). P n In what follows we note that, using Neumann Lemma (see [12]), the series n∈N xn! is summable >0 for all x ∈ k((tG )), for any field k and ordered abelian group G, and exp(x) ∈ k((tG )). In P × xn particular the map exp : L◦◦ U → LU , x 7→ exp(x) := n∈N n! is well defined. P n 3.1. Theorem. The map E coincides with the map x 7→ exp(x) = n∈N xn! on DU , i.e. E(x) is given by the Taylor formula for the standard exponential map. In other words, the embedding ιU : KU ֒→ LU commutes with the exponential, i.e. ιU (E(x)) = exp(ιU (x)) for all x ∈ DU . 1 Proof. Let α ∈ DU . Then α = [(αp )p ], and ordp (αp ) > p−1 for U-almost all p. Consider the fields F0 = kU (α) and F = kU (α, E(α)) and let Γ0 be the divisible hull of the group val(F × ). By ( [2] Theorem 3.4.3) Γ0 is an ordered abelian group having finite dimension as a linear space over Q. This will be shown directly below. From the embedding F ֒→ LU we get an embedding of valued fields F ֒→ kU ((tΓ0 )) such that the following diagram commutes // LU F :: ✈✈ ✈ ✈ ✈ ✈✈  ✈✈ kU ((tΓ0 )) obtained by identifying F with its image in LU . PN n Let sN (α) be the partial sum sN (α) := n=0 αn! . For every p, the sequence (sN (αp ))N is a Cauchy sequence in Cp , hence in particular it is pseudo-Cauchy, and expp (αp ) is a (pseudo-)limit. Thus we obtain using Lós Theorem that for all positive integers N1 < N2 < N3 KU |= val(sN3 (α) − sN2 (α)) > val(sN2 (α) − sN1 (α)), 8 ALI BLEYBEL (where we used the abbreviation val defined in section 2.4) and (sN (α))N is pseudo-Cauchy in KU . In particular, (sN (α))N is pseudo-Cauchy in F0 . Also, we have KU |= val(E(α) − sN (α)) = val(sM (α) − sN (α)), (by Lós Theorem) for all sufficiently large N , and all M > N . It follows by [8] (Theorems 2 and 3) that the extension F0 ֒→ F0 (E(α)) = F is an immediate valued field extension, and Γ0 = Q · γ where γ = val(α), hence Γ0 has finite dimension as a linear space over Q, as claimed. It follows in particular that the field kU ((tΓ0 )) is Hausdorff and complete with respect to the topology induced by the valuation v|kU ((tΓ0 )) . For any x ∈ DU , one has  N +1  xp Cp |= ordp (expp (xp ) − sN (xp )) = ordp (N + 1)! for all N , for U-almost all p. Hence, in this case we have: αN +1 KU |= val(E(α) − sN (α)) = val (N + 1)!   for all N . From the above observation we see that the sequence val(αN ) is cofinal in Γ0 , hence αN → 0 as N → ∞ (in F ) and the sequence E(α) − sN (α) converges to zero in kU ((tΓ0 )). Also, in kU ((tΓ0 )) we have that sN (α) → exp(α). Consequently, E(α) = exp(α) as required.  Using the above, we reach the following corollary: 3.2. Corollary. For all x ∈ DU , one has: D(E(x)) = E(x)Dx. P xn Proof. This follows from E(x) = n∈N n! = exp(x) and that Dx = subsection 3.0.1. D(exp(x)) exp(x) as observed in  3.3. Proof of Theorem 1.1. In this section we apply the above considerations in order to obtain Theorem 1.1. Let V be a variety of dimension n in the affine 2n-space A2n Q . For each prime p, denote by W (Cp ) the set of tuples āp ∈ Epn for which (āp , expp (āp )) ∈ V (Cp ). Let S ′ ⊂ P denote the set of primes p for which V has a Cp -point of the form (āp , expp (āp )) (hence, in particular, |āp |p < p−1/p−1 ). The following will be assumed throughout: (⋆) There are infinitely many primes p such that V has a Cp -point of the form (āp , expp (āp )). In other words, S ′ is an infinite set. The proof of Theorem 1.1 proceeds by assuming the contrapositive. More precisely, let (†) denote the following statement: (†) There exists no uniform rational linear dependence that holds for āp , for infinitely many p ∈ S ′ . Let U be a non-principal ultrafilter on P, such that S ′ ∈ U. Define x̄ = (x1 , . . . , xn ) = [(x̄p )p ] ∈ KnU ⊂ LnU as follows: If p ∈ S ′ , then x̄p = āp . Otherwise, we let x̄p be an arbitrary n-tuple of complex p-adic numbers which lie in the domain of the corresponding expp (this last assumption, though harmless, is not strictly necessary). Then x̄ is an n-tuple of elements of DU . Applying Lós Theorem to KU it follows that (x̄, Exp(x̄)) ∈ V (KU ), consequently (x̄, E(x̄)) ∈ V (KU ) ⊂ V (LU ). Hence, applying Proposition 2.3 to the (partial) exponential valued field LU , it follows that m1 x1 + · · · + mn xn ∈ kU , for some m1 , . . . , mn ∈ Q (not all zero). Clearing denominators, we can assume that m1 , . . . , mn are integers. A NOTE ON COMPLEX p-ADIC EXPONENTIAL FIELDS 9 Furthermore, since the elements x1 , . . . , xn lie in the maximal ideal L◦◦ U , any Z-linear combination of x1 , . . . , xn will necessarily be in L◦◦ . Writing x = [(x ) ] for i = 1, . . . , n, it follows i p,i p U m1 xp,1 + · · · + mn xp,n = 0, (∗) for U-almost all p ∈ P, i.e., the set of primes p for which (∗) holds belongs to the ultrafilter U. In particular, observing that the intersection of any two sets in U is in U (so it is an infinite set, U being a non-principal ultrafilter), one has, for infinitely many primes p m1 ap,1 + · · · + mn ap,n = 0, (‡) with fixed m1 , . . . , mn (not all zero), contradicting the hypothesis. In particular it follows from the above argument that the set of rational n-tuples (m1 , . . . , mn ) (up to a non-zero multiplicative constant) for which (‡) holds uniformly for infinitely many primes p ∈ S ′ is finite. Let us denote this set by A. In order to show the remaining statement of Theorem 1.1, let us define ᾱ := (ᾱp )p (i.e. ᾱ defines a family of n-tuples of complex p-adic numbers that belong to W (Cp ) for infinitely many p’s) and denote by Sᾱ the set of primes for which (‡) does not hold for the tuple ᾱp for any (m1 , . . . , mn ) ∈ A. By the above reasoning, Sᾱ is a finite set. Assume, for the sake of contradiction, that there is no finite set S for whichSSᾱ ⊂ S for all ᾱ. Choose a countable subset of these ᾱ enumerated as ∞ ᾱ1 , ᾱ2 , . . . , such that i=1 Sᾱi ⊂ P is an infinite set. Without loss of generality the sets Sᾱi can be assumed disjoint. One can then construct an infinite family of tuples (β̄p )p which satisfy (†): / Sᾱi , ∀i. Repeating the above set β̄p = ᾱi,p for p ∈ Sᾱi , and assign arbitrary values to β̄p for p ∈ reasoning, we reach a contradiction. This proves the claim. Combining the above results, we see that Theorem 1.1 is now fully proved. References [1] J. Ax, On Schanuel’s Conjectures, Annals of Mathematics, Second Series, Vol. 93, No. 2 (Mar., 1971), pp. 252-268. [2] A. J. Engler and A. Prestel, Valued Fields, Springer Monographs in Mathematics, 2005. [3] R. Bianconi, Some remarks on Schanuels conjecture, Annals of Pure and Applied Logic, V.108, 2001, 1518. [4] C. C. Chang and H. J. Keisler, Model theory, North-Holland Publishing Co, Amsterdam, 1973. [5] K. Gravett, Ordered Abelian Groups, The Quarterly Journal of Mathematics. Oxford Second Series, V.7, 1956 , 5763. [6] H. Hahn, Über die nichtarchimedischen Grössensystem, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch - Naturwissenschaftliche Klasse (Wien) 116 (1907), no. Abteilung IIa, 601655. 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Robert, A course in p-adic analysis, Graduate Texts in Mathematics, 198, Springer-Verlag, 2000. [14] H. Schoutens, The Use of Ultraproducts in Commutative Algebra, Lecture notes in Mathematics, SpringerVerlag Berlin Heidelberg, 2010. Lebanese University, Faculty of Sciences, Beirut, Lebanon E-mail address: [email protected]