Some consequences of Schanuel's conjecture

Abstract

In this paper we prove Shapiro's 1958 Conjecture on exponential polyno-mials, assuming Schanuel's Conjecture.

On the Nature of γ-th Arithmetic Zeta Functions

“…Schanuel's conjecture implies many famous theorems and conjectures about transcendental numbers. For an interesting consequence, see [16]. From now on, we use this conjecture to prove some interesting facts on the exceptional set of the functions ζ γ .…”

Algebraic Numbers of the form αT with α Algebraic and T Transcendental

“…There are many research topics on the deep consequences of the veracity of the Schanuel's conjecture (for some of them, we refer the reader to [10,11] and references therein). In the opposite direction of Theorem 1, here, we still prove the following conditional result: Theorem 2.…”

Semi-abelian analogues of Schanuel Conjecture and applications

“…In [14] C. Cheng et al proved that Schanuel conjecture implies the algebraic independence of the values of the iterated exponential and the values of the iterated logarithm, answering a question of M. Waldschmidt. In [26] the second, third and fourth authors have investigated Waldschmidt's question in the context of abelian varieties: more precisely, they first introduce after G.Vallée [29] an Abelian analogue of Schanuel conjecture (see Conjecture 3.4) and a weak version of it, as a consequence of the Generalized Period conjecture applied to 1-motives without toric part.…”

“…Then, our main result is that the Relative Semi-abelian conjecture 0.3 (also cited in the text as conjecture 3.7) implies the algebraic independence of the values of iterated semiabelian exponential and the values of iterated generalized semi-abelian logarithms. To state explicitly our main Theorem we use the notion of algebraic independence of fields, which for algebraically closed fields coincides with that of linear disjointness, see for example [23], [14] and [26, Lemma 1]. Let F be a field and F 1 , F 2 be two extensions of F contained in a larger field L, then: Definition 0.4.…”