Algebraic independence of certain numbers over a field of finite transcendence type
Let \(a_{1},\ldots,a_{n}\) (resp. \(b_{1},\ldots,b_{m}\)) be linearly independent complex numbers satisfying a suitable measure of linear independence over the rational field; define \(d=mn/(m+n)\). Let \(K\) be a subfield of \(\mathbb{C}\) with transcendence type \(\leq \tau\). Then the transcendence degree \(t\) over \(\mathbb{Q}\) of the field \(L\) generated by the \(mn\) numbers \(\exp(a_{i}b_{j})\) satisfies \(2^{t}\tau\geq d\). A similar result holds for the transcendence degree \(t'\) (resp. \(t''\)) of the field \(L'\) (resp. \(L''\)) generated by the \(mn+m\) numbers \(a_{i}\) and \(\exp(a_{i}b_{j})\) (resp. by the \(mn+m+n\) numbers \(a_{i}\), \(b_{j}\) and \(\exp(a_{i}b_{j})\)), provided that \(d\) is replaced by \(d'=(mn+m)/(m+n)\) (resp. by \(d''=(mn+m+n)/(m+n)\)). The proof rests on a transcendence criterion due to the author. Reviewer's remark. A sharper transcendence criterion, due to \textit{P. Philippon} [Publ. Math., Inst. Hautes Etud. Sci. 64, 5-52 (1986; Zbl 0615.10044); Un critère pour l'indépendance algébrique dans les anneaux diophantiens, C. R. Acad. Sci., Paris, Sér. I (to appear)]\ enables one to replace \(2^{t}\) by \(t+1\).
scientific article; zbMATH DE number 60804
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