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A refined version of the Siegel-Shidlovskii theorem. (English) Zbl 1133.11044
Let \(f_1(z),\dots, f_n(z)\) be a set of \(E\)-functions satisfying the system of first-order equations \({d\over dz}y= Ay\), where \(y= (y_1,\dots, y_n)^t\), and \(A\) is an \(n\times n\)-matrix with entries in \(\overline{\mathbb{Q}}(z)\). According to the Siegel-Shidlovskii theorem, for any \(\xi\in\overline{\mathbb{Q}}\) with \(\xi T(\xi)\neq 0\) we have \(\text{deg\,tr}_{\overline{\mathbb{Q}}}(f_1(\xi),\dots, f_n(\xi))= \text{deg\,tr}_{\overline{\mathbb{Q}}(z)}(f_1(x),\dots, f_n(z))\), where \(T(z)\) is the common denominator of the entries of \(A\). Yu. V. Nesterenko and A. B. Shidlovskii [Sb. Math. 187, No. 8, 1197–1211 (1996); translation from Mat. Sb. 187, No. 8, 93–108 (1996; Zbl 0990.11051)] proved that for all \(\xi\in\overline{\mathbb{Q}}\setminus S\), where \(S\) is a finite set, any homogeneous polynomial relation \(P(f_1(\xi),\dots, f_n(\xi))= 0\) with \(P\in\overline{\mathbb{Q}}[X_1,\dots, X_n]\) implies a polynomial relation \(Q(z, f_1(z),\dots, f_n(z))\equiv 0\), where \(Q\in\overline{\mathbb{Q}}[z, X_1,\dots, X_n]\), homogeneous in \(X_i\), and \(Q(\xi,X_1,\dots, X_n)= P(X_1,\dots, X_n)\).
In the present paper the author proves for any \(\xi\in\overline{\mathbb{Q}}\) with \(\xi T(\xi)\neq 0\), the conclusion stated above holds using Y. AndrĂ©’s result on differential equations satisfied by \(E\)-functions [Ann. Math. (2) 151, No. 2, 741–756 (2000; Zbl 1037.11050)]. Furthermore, the problem of removing nonzero singularities is also considered.

11J91 Transcendence theory of other special functions
11J81 Transcendence (general theory)
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