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Applications d’un corps fini dans lui-même. (Self-applications of a finite field). (French) Zbl 0601.12027
Let \({\mathbb{F}}\) be a finite field, and let \(f: {\mathbb{F}}\to {\mathbb{F}}\) be a polynomial map. The author investigates the structure of repeatedly applying f to elements of \({\mathbb{F}}\). In particular he is interested in cyclic behaviour, i.e. for a given element \(x\in {\mathbb{F}}\) which \(n\in {\mathbb{N}}\) we have with \(f^ k(x)=f^{k+n}(x)\), for some \(k\in {\mathbb{N}}\). These numbers may be of importance by analyzing the Pollard-rho method. The author only arrives at partial results.
The author does not proof the results he obtaines. He refers to his thesis at the university of Rennes. Some of his results will appear shortly in English, with proofs.
Reviewer: F.J.van der Linden

11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
11A41 Primes
12-04 Software, source code, etc. for problems pertaining to field theory