Irreducibility criteria for compositions of multivariate polynomials over arbitrary fields

We provide irreducibility criteria for compositions of multivariate polynomials over a field $K$, of the form $f(X_{1},\dots,X_{r-1},g(X_{1},\dots,X_{r}))$, with both $f$ and $g$ in $K[X_{1},\dots,X_{r}]$, for the case that $f$, viewed as a polynomial in $X_{r}$, has leading coefficient divisible by the $k^{\rm th}$ power of an irreducible polynomial $p(X_{1},\dots,X_{r-1})$ of sufficiently large degree with respect to $X_{r-1}$, with $k$ coprime to $\deg _{r}f$ and $\deg _{r}g$.