The density of prime divisors in the arithmetic dynamics of quadratic polynomials |
| |
Authors: | Jones Rafe |
| |
Affiliation: | Department of Mathematics University of Wisconsin-Madison 480 Lincoln Dr. Madison, WI 53706 USA |
| |
Abstract: | Let f [x], and consider the recurrence given by an = f(an –1), with a0 . Denote by P(f, a0) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a0)). The problem of determining D(P(f, a0)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a0)) was knownonly in a few cases. We show that D(P(f, a0)) = 0 regardlessof a0 for four infinite families of f, including f = x2 + k,k {–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a0)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials. |
| |
Keywords: | |
本文献已被 Oxford 等数据库收录! |
|