Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

Abstract

Abstract. We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field, that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form with f, f 1 , . . . , f n , g univariate polynomials over an arbitrary field.

Irreducibility criteria for compositions of multivariate polynomials over arbitrary fields

“…. , m}, depending on j, for which g(X, α j ) = γ i , which in view of (9) shows that we must have |g(X, α j…”

“…Brauer and Hopf [13], followed by the works of Wegner [48], Dorwart and Ore [25], Seres [44]- [46] and Győry [29]- [33]. More recent irreducibility criteria for compositions of polynomials can be found in papers by Győry, Hajdu and Tijdeman [34], Ayad [1], and also in [4], [9] and [11]. One of the most influential results in the study of multivariate polynomials and their specializations is the famous Hilbert Irreducibility Theorem [35].…”

On some classes of irreducible polynomials

“…The polynomials in Corollary 14 have already been considered in the literature. In [5] the authors consider the polynomial F (x, y) = h(x) n i=1 (y − f i (x)) + g(x) and they are interested on the irreducibility over the rational function field K(x) with certain conditions on the degree of f i (x). We can drop all those conditions in two different ways:…”

“…Let F (x, y) be the polynomial defined as above, with distinct n polynomials f i (x) and such that with Remark. Observe that b) is also considered in [5].…”

“…Our criterion also allows us to address other problems on irreducibility. In particular, we are able to simplify and generalize some results of [5].…”