Settled polynomials over finite fields

By Rafe Jones, Nigel Boston. 2012

55 citations recorded by the source corpus.

We study the factorization into irreducibles of iterates of a qua- dratic polynomial f over a nite eld. We call f settled when the factorization of its nth iterate for large n is dominated by \stable" polynomials, namely those that are irreducible under post-composition by any iterate of f. We prove that stable polynomials may be detected by their action on the critical orbit of f, and that the the critical orbit also gives information about the split- ting of non-stable polynomials under post-composition by iterates of f. We then dene a Markov process based on the critical orbit of f and conjecture that its limiting distribution describes the full factorization of large iterates of f. This conjecture implies that almost all quadratic f dened over a nite eld are settled. We give several types of evidence for our conjecture.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c77bd99f98dc753159ec7fe544bfa722cb88fd70

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Citations in the corpus

Current trends and open problems in arithmetic dynamics (2018, background): “Under the assumptions that q is odd, that f(x) is quadratic, that g(x) is irreducible of even degree, and that every iterate fn(x) is separable, Boston, Goksel, Jones, and Xia [98,126] describe a conjectural multistage Markov model to explain the factorization of successive values of g ( fn(x) ) .”
Galois representations from pre-image trees: an arboreal survey (2014, background): “1 is a finite field, “if" may be replaced by “if and only if," and this stronger result underlies much of the analysis in [3].”