A New Proof of the Ultrametric Hermite-Lindermann Theorem
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A New Proof of the Ultrametric Hermite-Lindermann Theorem∗ A. Escassut** Universite´ Clermont Auvergne, UMR CNRS 6620, LMBP, F-63000 Clermont-Ferrand, France Received May 27, 2020; in final form, July 3, 2020; accepted July 3, 2020
Abstract—We propose a new proof of the Hermite-Lindeman Theorem in an ultrametric field by using classical properties of analytic functions. The proof remains valid in zero residue characteristic. DOI: 10.1134/S207004662004007X Key words: p-adic transcendence numbers, p-adic analytic functions.
Definitions and Notations. The Archimedean absolute value of C is denoted by | . |∞ . Let IK be an algebraically closed complete ultrametric field of characteristic 0 and residue characteristic p. We denote by | . | the ultrametric absolute value of IK and we denote by Ω an algebraic closure of Q in IK. Given a ∈ IK and r > 0, we denote by d(a, r − ) the disk {x ∈ IK |x − a| < r}, we denote by ∞ − an (x − a)n converging in d(a, r − ) A(d(a, r )) the IK-Banach algebra of power series f (x) = n=0
and for all s ∈]0, r[, we put |f |(s) = supn∈IN |an |r n . Given a ∈ Ω, we call denominator of a any strictly positive integer n such that na is integral over Z and we denote by den(a) the smallest denominator of a. Next, considering the conjugates a2 , ..., an of a over Q and putting a1 = a, we put |a| = max{|a1 |∞ , ..., |an |∞ }. Next, we put s(a) = Log(max(|a|, den(a)). Let P (X1 , ..., Xn ) = j1 ,...,jn aj1 ,...,jn X1j1 , ..., Xnjn ∈ Ω[X1 , ..., Xn ]. We put |P | = maxj1 ,...,jn |aj1 ,...,jn |∞ and t(P ) = max(Log(|P |), 1 + maxi=1,...,n(degXi (P )). We denote by D0 the disk d(0, 1− ) and in the case when the residue characteristic of IK is p > 0 we −1
put R1 = p p−1 and then we denote by D1 the disk d(0, R1− ). Given a positive real number a, we denote by [a] the biggest integer n such that n ≤ a. Remark. In particular Levi-Civita’s fields have residue characteristic 0 [3]. We aim at proving the Hermite-Lindeman Theorem in an ultrametric field. Our proof is different from the one given by Kurt Mahler in [2] and holds in any ultrametric field, no matter what the residue characteristic. Here, at the beginning, we will follow the method used in the complex context [4]. But it is impossible to get to the conclusion by considering the zeros of an auxiliary analytic function on only two points, as done in the complex context. So, here we assign an auxiliary function FN to vanish at a number of points roughly equal to Log(N ). Theorem. Suppose that IK has residue characteristic p > 0 (resp. 0). Let α ∈ D1 be algebraic, (resp. let α ∈ D0 be algebraic). Then eα is transcendental. Examples: When p > 0, ep , ep
√ n
q
(n ∈ IN) are transcendental.
Let us first recall the following two lemmas: Lemma 1. Let a ∈ Ω be of degree q over Q. Then −2qs(a) ≤ Log(|a|). Lemma 2 is a particular form of Siegel’s Lemma due to M. Mignotte [4]: ∗ **
The text was submitted by the author in English. E-mail: [email protected]
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Lemma 2. Let E be a finite extension of Q of degree q and le
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