*(English)*Zbl 0656.10032

[For the entire collection see Zbl 0644.00005.]

This is a report on finite automata and applications of transcendence theory, using functional equations of a type first developed by *K. Mahler* [Math. Ann. 101, 342-366 (1929); J. Number Theory 1, 512-521 (1969; Zbl 0184.076)] and by the author and *A. J. van der Poorten* in several papers [particularly in J. Reine Angew. Math. 330, 159-172 (1982; Zbl 0468.10019)].

Let $\alpha ={\left({\alpha}_{n}\right)}_{n}$ be a sequence with entries belonging to a finite alphabet A of p letters ${a}_{i}$ and generated by a finite automaton. According to *A. Cobham* [Math. Syst. Theory 6, 164-192 (1972; Zbl 0253.02029)] $\alpha $ can be defined as a fixed point of a uniform (of length r) substitution on A. This leads to a functional equation like $F\left(z\right)=M\left(z\right)F\left({z}^{r}\right)$ where $F\left(z\right)={({f}_{1}\left(z\right),\xb7\xb7\xb7,{f}_{p}\left(z\right))}^{t}$, ${f}_{i}\left(z\right)={\sum}_{i=0}^{\infty}{f}_{i,n}{z}^{n}$, ${f}_{i,n}=0$ (resp. $=1)$ if ${\alpha}_{n}={a}_{i}$ (resp. $\ne {a}_{i})$ and M(z) is a $p\times p$-matrix whose entries are polynomials. In case of general substitutions, one also has analogous functional equations, namely $F\left(z\right)=M\left(z\right)F\left(Tz\right)$, but z is now a p-complex variable $({z}_{1},\xb7\xb7\xb7,{z}_{p})$, T is a $p\times p$-matrix of nonnegative integer entries ${t}_{ij}$ and ${\left(Tz\right)}_{i}={\prod}_{j=1}^{p}{z}_{j}^{{t}_{ij}}\xb7$

If the functions ${f}_{i}$ are algebraically independent over $\u2102\left(z\right)$, the aim is to get the numbers ${f}_{i}\left(\zeta \right)$ algebraically independent over $\mathbb{Q}$ for a proper algebraic point $\zeta $ of ${\u2102}^{p}$. Such a theorem requires technical assumptions on T, M and $\zeta $, due to the method, which have been discussed by the author (see the reference above for a complete statement).

Transcendence measures obtained by *A. I. Galochkin* [Mat. Zametki 27, 175-183 (1980; Zbl 0426.10036)] and by *Yu. V. Nesterenko* [Astérisque 147/148, 141-149 (1987; Zbl 0615.10043)] on algebraic independence measures arising from functional equations as above are also considered.