Date: May 14, 2013 4:38 AM
Author: eric.fennessey@gmail.com
Subject: Can a small degree polynomial exist in this Ideal?

I am feeling dim and my brain is failing me on this one.

Let n,m >=2 and consider the subset, I, of polynomials in Z[X,Y] given by,

I = {P.(X^n - 1) + Q.(Y^m - 1): P,Q in Z[X,Y]},

I.e. the ideal generated by X^n -1, Y^m -1 in Z[X,Y].

Let S be the subset of Z[X,Y] consisting of polynomials where each non-zero term of the form n.X^i.Y^j has i<n and j<m.

My question is: Is I intersected with S the zero polynomial? Or, can some strange cancellation occur which wipes out all terms containing X^n's and Y^m's and higher, but leave something non-zero behind?

My intuition says it ought to be the former, but I seem to have ended up chasing my tail trying to show it. Can someone enlighted me as to how to show it, or give me a cunning counterexample?

Regards,
Eric