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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.

##### MSC:
 11J82 Measures of irrationality and of transcendence 11J85 Algebraic independence; Gel’fond’s method
##### Keywords:
exponential function; Schanuels’s conjecture
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