  We provide upper bounds on the total number of irreducible factors, and in
particular irreducibility criteria for some classes of bivariate polynomials
$f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information
on the degrees of the coefficients of $f$, and on information on the
factorization of the constant term and of the leading coefficient of $f$,
viewed as a polynomial in $y$ with coefficients in $\mathbb{K}[x]$. In
particular, we provide a generalization of the bivariate version of Perron's
irreducibility criterion, and similar results for polynomials in an arbitrary
number of indeterminates. The proofs use non-Archimedean absolute values, that
are suitable for finding information on the location of the roots of $f$ in an
algebraic closure of $\mathbb{K}(x)$.
