Irreducibility criteria for compositions of multivariate polynomials

Irreducibility criteria for compositions of multivariate polynomials over arbitrary fields

“…In the sequel it will be, as in [8], more convenient to make use of a parameter in order to derive a reasonably sharp and simple condition on deg p which will force at least one of the inequalities |R(g m , F 1 )| < 1 and |R(g m , F 2 )| < 1 to hold. So let us fix an arbitrarily chosen real λ ≥ 0 and assume that…”

“…The aim of this paper is to extend the results from [5] and [8] by providing results similar to Theorem B for the case when k is coprime to mn. Unlike Theorem B, the results in this paper will not use Lemma 1.1, whose proof relies on a Newton polygon argument, but instead will benefit from a simultaneous analysis of some resultants associated to the hypothetical factors of f (X, g(X, Y )).…”

“…Here f 1 and f 2 are relatively prime, and in order to adapt Cavachi's method, information on the canonical factorization of a n proves to be helpful in many cases. This idea of writing compositions of polynomials as linear combinations of two relatively prime polynomials was also used in the multivariate case [5], [6], [8], and moreover, similar techniques have been useful to study the more general concept of multiplicative convolutions of polynomials [2]. For instance, the following irreducibility criterion for compositions of bivariate polynomials that uses this idea was proved in [5].…”

“…A Newton polygon method allowed one to generalize in [8] the results from [5], by considering the case when the leading coefficient a m of f is of the form p k q with p, q ∈ K[X], p irreducible over K, q not divisible by p, and k ≥ 1 a suitable positive integer. One such result, which extends Theorem A, is the following.…”