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# Texas A&M Number Theory Seminar

##
Department of Mathematics

Blocker 220

Wednesdays, 1:15–2:15 PM

### Andrew Bridy

Texas A&M University

#### Wednesday, January 31, 2018

#### Blocker 220, 1:15PM

**Title:** *The cycle structure of unicritical polynomials in finite fields
*

**Abstract:**
Let f(x) = x^k+a in Z[x] for k \geq 2. Consider the family of
dynamical systems given by the action of f on F_p as p varies among
primes. The question of how and in what sense this family approximates
a random family of dynamical systems has been studied extensively,
motivated in part by Pollard's "rho" algorithm for integer
factorization. We show that for most choices of a, the cycle structure
in this family is "as random as possible" in an appropriate
sense. As a corollary, we show that most members of these families
have many cycles. This is joint work with Derek Garton.