Some factorization results for bivariate polynomials

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 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 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 K(x).