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Automata and transcendence. (English) Zbl 0656.10032
New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 215-228 (1988).
[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 {\it K. Mahler} [Math. Ann. 101, 342-366 (1929); J. Number Theory 1, 512-521 (1969; Zbl 0184.076)] and by the author and {\it A. J. van der Poorten} in several papers [particularly in J. Reine Angew. Math. 330, 159-172 (1982; Zbl 0468.10019)]. Let $\alpha =(\alpha\sb n)\sb n$ be a sequence with entries belonging to a finite alphabet A of p letters $a\sb i$ and generated by a finite automaton. According to {\it 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 $F(z)=M(z)F(z\sp r)$ where $F(z)=(f\sb 1(z),...,f\sb p(z))\sp t$, $f\sb i(z)=\sum\sp{\infty}\sb{i=0}f\sb{i,n}z\sp n$, $f\sb{i,n}=0$ (resp. $=1)$ if $\alpha\sb n=a\sb i$ (resp. $\ne a\sb i)$ 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 $F(z)=M(z)F(Tz)$, but z is now a p-complex variable $(z\sb 1,...,z\sb p)$, T is a $p\times p$-matrix of nonnegative integer entries $t\sb{ij}$ and $(Tz)\sb i=\prod\sp{p}\sb{j=1}z\sb j\sp{t\sb{ij}}.$ If the functions $f\sb i$ are algebraically independent over ${\bbfC}(z)$, the aim is to get the numbers $f\sb i(\zeta)$ algebraically independent over ${\bbfQ}$ for a proper algebraic point $\zeta$ of ${\bbfC}\sp p$. 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 {\it A. I. Galochkin} [Mat. Zametki 27, 175-183 (1980; Zbl 0426.10036)] and by {\it 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.
Reviewer: P.Liardet

11J81Transcendence (general theory)
68Q70Algebraic theory of languages and automata