Number Theory Seminar

Date: February 1, 2017

Time: 1:45PM - 2:45PM

Location: BLOC 220

Speaker: Andrew Bridy, Texas A&M University

Title: Dynamically distinguishing polynomials

Abstract: For p prime, consider the directed graphs induced by the polynomials x^2,x^2+1,...,x^2+p-1, viewed as mappings F_p -> F_p. Experiments suggest that these graphs are pairwise non-isomorphic for all p not in {2,17}. It is unclear how to show that this holds for all large primes. Turning the question around, we aim to construct large sets of polynomials of the form {x^k+c_1,...,x^k+c_m} so that their reductions mod p induce m pairwise non-isomorphic directed graphs for almost all primes p. We show that m can be arbitrarily large for every degree k, and in fact most m-tuples of integers (c_1,...,c_m) work. The proof uses the Galois theory of periodic points largely developed by Morton. This is joint work with Derek Garton.

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