New Problem

Okay, so I haven’t updated my blog in a long time, which goes to show I am not stress out with work.  I should be!  I have this new assignment and I haven’t able to focus on it.  It’s not that I don’t like the problem – in fact, I think it’s very interesting, and I enjoy learning all the stuff I need to understand and solve it.  It’s just that I really can’t work unless there is some sort of deadline looming in the near future.  It’s hard to motivate myself without having some sort of pressure.

Since wordpress can interpret latex, I’m going to try to describe my problem for future reference.  Let be a rational map defines over a number field , and take both not preperiodic for .  Then I would like to know if there are finitely many which are S-integral for as and varies over the positive integers (here is the th iterate of ).  This is equivalent to asking if there are finitely many which are S-integral to the diagonal .  If I can show all the points which are S-integral to the diagonal all lie on finitely many curves, then using the result in a paper by Tom Tucker, et al., would give the result when has no periodic critical point.

Here’s how I intend to solve the problem for the map .  View as a polynomial map on the surface .  We first need to show that there are finitely many which lie on a curve and are S-integral to the diagonal in .  To do this, we need to know how the diagonal pulls back under .  I think I’ll be able to show that if 0 has period three or more for , then components of pullback intersect in such a way that it will meet any curve and the line in at least 3 places.  This will finish the first part of the problem.  Secondly, I need to show that the divisor satisfies the conditions in Aaron Levin’s paper.  This would mean that the S-integral points on all lie on finitely many curves, and this would imply the results for  (I think we get the full result when the critical point 0 for has period ).