The dynamics of linearized polynomials
- Let F = GF(q). To any polynomial G element of F[x] there is associated a mapping on the set I_F of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of the mapping for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem. Assume that G is not of the form x^(q^l), where l>= 0 (in which event the dynamics is trivial). Then, for every integer n >= 1 and for every integer k >= 0, there exist infinitely many mü element of I_F having preperiod k and primitive period n with respect to the mapping. Previously, Morton, by somewhat different means, had studied the primitive periods of the mapping when G = x^q - ax, a a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.
| Author: | Stephen D. Cohen, Dirk HachenbergerORCiDGND |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-8665 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/1010 |
| Parent Title (English): | Proceedings of the Edinburgh Mathematical Society (Series 2) |
| Type: | Article |
| Language: | English |
| Year of first Publication: | 2000 |
| Publishing Institution: | Universität Augsburg |
| Release Date: | 2008/06/19 |
| Tag: | finite field; polynomial dynamics; linearized polynomial; period |
| GND-Keyword: | Galois-Feld; Polynom; Abbildung <Mathematik> |
| Volume: | 43 |
| Issue: | 1 |
| First Page: | 113 |
| Last Page: | 128 |
| DOI: | https://doi.org/10.1017/S0013091500020733 |
| Institutes: | Mathematisch-Naturwissenschaftlich-Technische Fakultät |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik | |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Diskrete Mathematik, Optimierung und Operations Research | |
| Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
