zbMATH — the first resource for mathematics

An arithmetic criterion for the values of the exponential function. (English) Zbl 0981.11025
Schanuel’s conjecture states that if \(y_1,\ldots,y_l\) are complex numbers and linearly independent over the rationals, then \(\text{tr.deg}_{\mathbb{Q}}\mathbb{Q}(y_1,\ldots,y_l,e^{y_1},\ldots,e^{y_l})\geq l\). In this paper, the author shows that Schanuel’s conjecture is equivalent to an arithmetic statement similar to criteria for algebraic independence. The statement involves a set of positive numbers satisfying \[ \max\{1,t_0,2t_1\}<\min\{s_0,2s_1\},\;\max\{s_0,s_1+t_1\}<u<\tfrac 12(1+t_0+t_1) \] (and, in fact, if it holds for any choice of these parameters, it holds for all such choices). Let \(y_1,\ldots,y_l\) be as in Schanuel’s conjecture, and let \(\alpha_1,\ldots,\alpha_l\) be non-zero complex numbers. Suppose that, for each sufficiently large positive integer \(N\), there is a non-zero polynomial \(P_N\) in \(\mathbb{Z}[X_0,X_1]\) with degree at most \(N^{t_0}\) in \(X_0\), degree at most \(N^{t_1}\) in \(X_1\), height at most \(e^N\) and satisfying \[ \left|({\mathcal D}^kP_N)\left(\sum_{j=1}^l m_jy_j,\;\prod_{j=1}^l \alpha_j^{m_j}\right)\right|\leq\exp(-N^u), \] for any integers \(k,m_1,\ldots,m_l\) with \(k\leq N^{s_0}\) and \(\max\{m_1,\ldots,m_l\}\leq N^{s_1}\). Here, \({\mathcal D}={\partial\over\partial X_0}+X_1{\partial\over\partial X_1}\). Then \(\text{tr.deg}_{\mathbb{Q}}\mathbb{Q}(y_1,\ldots,y_l,\alpha_1,\ldots,\alpha_l)\geq l\). The idea of the proof comes from Waldschmidt’s general constructions of auxiliary functions and interpolation lemmas.

11J82 Measures of irrationality and of transcendence
11J85 Algebraic independence; Gel’fond’s method
Full Text: DOI