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On an estimate for the orders of zeros of Mahler type functions. (English) Zbl 0719.11041
Using results analogous to Yu. V. Nesterenko’s [Mat. Sb., Nov. Ser. 128(170), 545-568 (1985; Zbl 0603.10033); translation in Math. USSR, Sb. 56, 545-567 (1987)] the author proves the
Theorem. Let \(f_ 1(z),...,f_ m(z)\in C[[z]]\) be formel power series with coefficients in a field C of characteristic zero and satisfy \[ f_ i(z^ d)=\frac{A_ i(z,f_ 1(z),...,f_ m(z))}{A_ 0(z,f_ 1(z),...,f_ m(z))}\quad (1\leq i\leq m) \] where \(d\geq 2\) is a rational integer and \(A_ i(z,x_ 1,...,x_ m)\in C[z,x_ 1,...,x_ m](0\leq i\leq m)\) are polynomials with \(\deg_ zA_ i\leq s\) and \(tot \deg_ xA_ i\leq t.\) Suppose that \(t^ m<d\) and \(Q(z,x_ 1,...,x_ m)\in C[z,x_ 1,...,x_ m]\) is a nonzero polynomial with \(\deg_ z Q\leq M\), \(tot \deg_ x Q\leq N\) where \(M\geq N\geq 1\). If \(Q(z,f_ 1(z),...,f_ m(z))\neq 0\), then \[ ord Q(z,f_ 1(z),...,f_ m(z))\leq C_ 0 MN^ mN^{m^ 2\log t/(\log d-m \log t)} \] where \(c_ 0\) is a constant, which is explicitly given.

11J85 Algebraic independence; Gel’fond’s method
11J91 Transcendence theory of other special functions
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