I'm currently trying to prove following statement.
Let f,g be homogeneous polynomials of degree n,m respectively, in $k[X,Y,Z]$. Here k is algebraically closed field, and n $\le$ m. Also f does not divide g. Then there exists a homogeneous polynomial h with degree m-n such that fh+g is reducible.
I proved this in the case m=2 or n=1. Mainly I used Bezout's theorem for proof. However, this doesn't work well for higher degrees. I personally believe that this statement should be true. In the sense that irreducible element is prime element in UFD, if fh+g is always irreducible, then we will get infinitely many prime elements by using method.
Please give me any type of advice to help proving this statement.