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A Schanuel property for exponentially transcendental powers. (English) Zbl 1223.11093

The usual exponential function \(\exp:\mathbb R\to\mathbb R\) makes the reals into an exponential field, formally a field of characteristic zero equipped with a homomorphism from its additive to multiplicative groups. In any exponential field \(\langle F;+,\cdot,\exp\rangle\), an element \(x\in F\) is exponentially algebraic in \(F\) if and only if there are \(n\in\mathbb N\), \(x= (x_1,\dots, x_n)\in F^n\), and polynomials \(f_1,\dots, f_n\in\mathbb Z[X, e^X]\), where \(e^X= (e^{X_1},\dots, e^{X_n})\), such that \(f_i(x,e^x)= 0\) \((i= 1,\dots, n)\) for \(x= x_1\), and \(\det(\partial f_i/\partial X_j)_{1\leq i,j\leq n}\) is nonzero at \(x\). If \(x\) is not exponentially algebraic in \(F\), then it is exponentially transcendental in \(F\).
The authors prove the following analogue of Schanuel’s conjecture for raising to the power of an exponentially transcendental real number: Let \(\lambda\in\mathbb R\) be exponentially transcendental, and \(y_1,\dots, y_n\in\mathbb R_{>0}\) be multiplicatively independent. Then \[ \text{tr}\deg_{\mathbb Q(\lambda)}\mathbb Q(y_1,\dots, y_n, y^\lambda_1, \dots,y^\lambda_n, \lambda)\geq n. \] From it we obtain the algebraic independence of the numbers \(\lambda,\lambda^\lambda, \lambda^{\lambda^2},\lambda^{\lambda^3},\dots\) for all but countably many \(\lambda\) (although explicit such a \(\lambda\) is not known). The proof of this result is based on a work of the second author [Bull. Lond. Math. Soc. 42, No. 5, 879–890 (2010; Zbl 1203.03050)]. A more general result for several powers in a context which encompasses the complex case is also given.

MSC:

11J91 Transcendence theory of other special functions
03C64 Model theory of ordered structures; o-minimality

Citations:

Zbl 1203.03050
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