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 be a sequence with entries belonging to a finite alphabet A of p letters and generated by a finite automaton. According to A. Cobham [Math. Syst. Theory 6, 164-192 (1972; Zbl 0253.02029)] can be defined as a fixed point of a uniform (of length r) substitution on A. This leads to a functional equation like where , , (resp. if (resp. and M(z) is a -matrix whose entries are polynomials. In case of general substitutions, one also has analogous functional equations, namely , but z is now a p-complex variable , T is a -matrix of nonnegative integer entries and
If the functions are algebraically independent over , the aim is to get the numbers algebraically independent over for a proper algebraic point of . Such a theorem requires technical assumptions on T, M and , 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.