TITLE: Criteria for irreducibility of polynomial
QUESTION [23 upvotes]: If $f, g\in \mathbb C[a,b]$ are  polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible? 
Thank you very much,
best

REPLY [11 votes]: David is correct: my REU students have determined the complex polynomials $F(x)$ and $G(y)$ for which $F(x)-G(y)$ has an irreducible factor defining a curve of genus 0 or 1.  By Faltings' theorem (a.k.a. Mordell's conjecture), this lets us write down all $F(x)$ and $G(y)$ with algebraic coefficients for which there is a number field $K$ such that $F(K)$ and $G(K)$ have infinite intersection.  By one of Picard's theorems, it also means we've solved the functional equation $F\circ A = G \circ B$ in complex polynomials $F,G$ and meromorphic functions $A,B$.  This last problem has been studied in the context of finding variants of Nevanlinna's theorem that a nonconstant meromorphic function is uniquely determined by its preimages at each of five points: our equation implies that the preimage under $A$ of the multiset of zeroes of $F(x)$ comprises the same multiset as the preimage under $B$ of the multiset of zeroes of $G(x)$.
We crucially use the genus condition.  If one asks for reducibility of $F(x)-G(y)$ without imposing any hypothesis on the genus, then (as noted previously) one can find all examples when $F$ and $G$ are indecomposable: this was the goal of the paper by Cassou-Nogues and Couveignes, although it should be noted that their list of pairs $(F,G)$ is incomplete, so another thing we did this summer was to find the full list of pairs $(F,G)$ in this situation.  In the decomposable case, much remains to be discovered about reducibility of $F(x)-G(y)$, although progress has been made.  In particular, Mike Fried showed that, if $F$ is indecomposable but $G$ is decomposable, then reducibility of $F(x)-G(y)$ implies that  $G=A \circ B$  for some polynomials $A$ and $B$ such that  $A$  is indecomposable  and  $F(x)-A(y)$  is reducible.  It follows that the pair $(F,A)$ occurs on the (corrected) Cassou-Nogues--Couveignes list, and in particular, we may assume that either $F = A$ or $\deg(F)=\deg(A)\le 31$.  This is shown in [Michael Fried, The field of definition of function fields and a problem in the reducibility of polynomials in two variables], via a novel argument combining Galois theory and representation theory (if you're interested in this, please ask me, since one of my REU students found a very simple proof of Fried's result that I'd love to share).  I note that, while the Cassou-Nogues--Couveignes result depends on the classification of finite simple groups (via the classification of finite groups $G$ that have a cyclic subgroup $C$ which acts transitively in two inequivalent doubly transitive permutation representations of $G$).  However, Fried's result is elementary.
When both $F$ and $G$ are decomposable, in the same paper Fried proved the following result: if $F(x)-G(y)$ is reducible, then we can write  $F = A \circ B$  and  $G = C \circ D$  in such a way that  $A(x)-C(y)$  is reducible  and  the splitting field of  $A(x)-t$  over  $\mathbb{C}(t)$  equals the splitting field of  $C(x)-t$  over  $\mathbb{C}(t)$  (where $t$ is transcendental over $\mathbb{C}$).  The condition on equal splitting fields is extremely restrictive -- for instance, it implies that $\deg(A)=\deg(C)$ (by considering the size of the inertia groups at infinite places), and also that $A$ and $C$ have the same critical values, or more precisely that, for each complex number $\theta$, the least common multiple of the multiplicities of the roots of $A(x)-\theta$ equals the corresponding least common multiples for $C(x)-\theta$.  Fried's proof is remarkably simple; a nice exposition of it is Theorem 8.1 in [Yuri Bilu and Robert Tichy, The Diophantine equation $f(x)=g(y)$, Acta Arith. 95 (2000), 261--288].
Surprising phenomena in the decomposable case can be found in Peter M\"uller's papers [Kronecker conjugacy of polynomials, Trans. Amer. Math. Soc. 350 (1998), 1823--1850] and [An infinite series of Kronecker conjugate polynomials, Proc. Amer. Math. Soc. 125 (1997), 1933--1940].  The most recent work on the decomposable case (to my knowledge) is [Michael Fried and Ivica Gusic, Schinzel's problem: Imprimitive covers and the monodromy method, arXiv:1104.1740].  I have not digested everything in these three papers, so I would be thrilled if someone wanted to summarize their main achievements.