I think that in the proof of your main theorem (Theorem 1, section 3, p.7,8 there is a bug. Equation (6) at page 8 has to be g=g_1 and not g=(1+x_k)g_1 as putting g_1=g_2 into Equation (5) yields g=g_1.
But then deg(g)=deg(g_1) and your inductive argument fails.