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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.

##### MSC:
 11J91 Transcendence theory of other special functions 11J81 Transcendence (general theory)
##### Keywords:
$$E$$-function; Siegel-Shidlovskii theorem; singular points
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