Irreducibility Criteria for the Sum of Two Relatively Prime Multivariate Polynomials

Abstract

We provide irreducibility criteria for multivariate polynomials over a field K, of the form $f+p^{k}g$, where $f,g\in K[X_{1},\dots ,X_{r}]$, $\deg _{r}f<\deg _{r}g$, $p\in K[X_{1},\dots ,X_{r-1}]$ is irreducible over $K(X_{1},\dots ,X_{r-2})$, and $k\ge 1$ is an integer prime to $\deg _{r}g$. More precisely, we prove that if f and g regarded as polynomials in $X_{r}$ with coefficients in $K[X_{1},\dots ,X_{r-1}]$ are relatively prime over $K(X_{1},\dots ,X_{r-1})$, k is prime to $\deg _{r}g$, and $\deg _{r-1}p$ is sufficiently large, then the polynomial $f+p^{k}g$ is irreducible over $K(X_{1},\dots ,X_{r-1})$. This complements some previous results where k was assumed to be prime to $\deg _{r}g-\deg _{r}f$.