Quadratic Factors of $f(X)-g(Y)$
  We classify the pairs of polynomials $f,g$ over a field $K$, such that $f(X)-g(Y)$ has a factor of total degree at most 2. This was done by Y. Bilu for characteristic 0 fields $K$. As his method does not work in positive characteristic, we use a quite different approach. 
Manisha Kulkarni, Peter M\"uller, B. Sury