# zbMATH — the first resource for mathematics

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.

##### MSC:
 11J85 Algebraic independence; Gel’fond’s method 11J91 Transcendence theory of other special functions
Full Text: