# Björn Carlsten: Cycles of certain maps between polynomials over finite fields

**Time: **
Fri 2021-04-09 10.00 - 11.00

**Lecturer: **
Björn Carlsten

**Location: **
Meeting ID: 697 4204 2109

### Abstract

In this essay we investigate cycles of a certain map between polynomials in one variable over finite fields using methods from experimental mathematics. We show that a certain cycle of length 4 exists for all fields of prime characteristic, and that all possible cycles must be of even length. Based on previous results on cycles of length 2, we describe a theory of critical equations that serves as the basis for an automated search program to find more examples of such cycles. The results from that automated search are categorized into a taxonomic hierarchy of cycles of length 2. Finally, we explore the necessary conditions that must hold for cycles of length 2 to occur and propose hypotheses regarding these conditions.

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