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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 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 $F\left(z\right)={\left({f}_{1}\left(z\right),···,{f}_{p}\left(z\right)\right)}^{t}$, ${f}_{i}\left(z\right)={\sum }_{i=0}^{\infty }{f}_{i,n}{z}^{n}$, ${f}_{i,n}=0$ (resp. $=1\right)$ if ${\alpha }_{n}={a}_{i}$ (resp. $\ne {a}_{i}\right)$ and M(z) is a $p×p$-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 $\left({z}_{1},···,{z}_{p}\right)$, T is a $p×p$-matrix of nonnegative integer entries ${t}_{ij}$ and ${\left(Tz\right)}_{i}={\prod }_{j=1}^{p}{z}_{j}^{{t}_{ij}}·$

If the functions ${f}_{i}$ are algebraically independent over $ℂ\left(z\right)$, the aim is to get the numbers ${f}_{i}\left(\zeta \right)$ algebraically independent over $ℚ$ for a proper algebraic point $\zeta$ of ${ℂ}^{p}$. 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.

Reviewer: P.Liardet
##### MSC:
 11J81 Transcendence (general theory) 68Q70 Algebraic theory of languages and automata