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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 α=(α n ) 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)] α 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 r ) where F(z)=(f 1 (z),···,f p (z)) t , f i (z)= i=0 f i,n z n , f i,n =0 (resp. =1) if α n =a i (resp. a i ) 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(z)=M(z)F(Tz), but z is now a p-complex variable (z 1 ,···,z p ), T is a p×p-matrix of nonnegative integer entries t ij and (Tz) i = j=1 p z j t ij ·

If the functions f i are algebraically independent over (z), the aim is to get the numbers f i (ζ) algebraically independent over for a proper algebraic point ζ of p . Such a theorem requires technical assumptions on T, M and ζ, 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:
11J81Transcendence (general theory)
68Q70Algebraic theory of languages and automata