Algebra Seminar - Andrew Herring (Math)

Speaker: Andrew Herring (Western)

"Dynamics of Rational Quadratic Polynomials: Uniform Boundedness, Hilbert Irreducibility, and Finite Exceptional Sets"

Arithmetic dynamics asks number theoretic questions about dynamical systems (for example, a polynomial over Q which is iteratively composed with itself). One of the "holy grails" of the discipline is the Dynamical Uniform Boundedness Conjecture (DUBC) which purports a uniform bound on the number of points with finite orbit. The simplest case of the DUBC concerns quadratic polynomials g(z) over Q, and Flynn, Poonen, and Schaefer conjecture that no periodic point of g(z) can have period greater than 3. Poonen went on to show that if he, Flynn, and Schaefer are correct, then the largest possible g(z)-orbit has size 9, and the first full case of the DUBC is proved. The "nth dynatomic polynomial," ΦnΦn, has as its zeros the points of period n for g(z), so studying G_n, the Galois group of ΦnΦn, may shed light on Flynn, Poonen, and Schaer's conjecture. By considering Phi_n over Q(t), we have an expectation for each G_n, and Hilbert's Irreducibility Theorem says that our expectation is almost always correct. For periods n=1,2,3, there are infinitely many g(z) with smaller than expected G_n, but for n=4,5,6,7,9 it's known that there are only finitely many such exceptions, and it's conjectured that n=1,2,3 are the only values exhibiting infinitely many exceptions. We will discuss how giving lower bounds on genus of fixed fields of maximal subgroups of the expected G_n can be used to prove that there are only finitely many exceptions.