Irreducibility criteria for compositions of multivariate polynomials (Q1714958)

In this paper the author obtains several results about irreducibility of compositions of multivariable polynomials over a field \(K\). In particular, suppose \[ f(X,Y)=a_0+a_1Y+\dots+a_mY^m, \] \[ g(X,Y)=b_0+b_1Y+\dots+a_nY^n, \] where \(a_0,\dots,a_m,b_0,\dots,b_n\) are polynomials in \(K[X]\), \(m,n \geq 1\), \(a_0a_mb_n \neq 0\). Let \(d\) be the largest index \(i <m\) for which \(a_i \neq 0\). Then, the polynomial \(f(X, g(X,Y))\) is irreducible over \(K(X)\) provided that \(a_m=p^k q\), \(p,q \in K[X]\), \(p\) irreducible over \(K\), \(qa_db_n\) not divisible by \(p\), \(\gcd(k,n(m-d))=1\) and \[ k \deg p > \max\{(m-1)\deg q, (n-1) \deg q +mn \deg b_n\}+\max_{0 \leq i \leq d} \deg a_i. \] The proofs are based on some evaluation of resultants.