On the irreducibility of the generalized Laguerre polynomials

Abstract

Let m ≥ 1 and a m be integers. Let α be a rational number which is not a negative integer such that α = u v with gcd(u, v) = 1, v > 0. Let φ(x) belonging to Z[x] be a monic polynomial which is irreducible modulo all the primes less than or equal to vm + u. Let a i (x) with 0 ≤ i ≤ m − 1 belonging to Z[x] be polynomials having degree less than deg φ(x). Assume that the content of (a m a 0 (x)) is not divisible by any prime less than or equal to vm + u. In this paper, we prove that the polynomialsb j a j (x)φ(x) j ) are irreducible over the rationals for all but finitely many m, where bis irreducible over rationals for each α ∈ {0, 1, 2, 3, 4} unless (m, α) ∈ {(1, 0), (2, 2), (4, 4), (6, 4)}. For proving our results, we use the notion of φ-Newton polygon and some results from analytic number theory. We illustrate our results through examples.

Harmonic divisors and rationality of zeros of Jacobi polynomials

“…It is still an unsolved question whether the Legendre polynomials are irreducible over the rationals, see [23], [24], [30], [40] and [41]. H. Ille has shown that P (0;0) n (x) has no quadratic factors which implies that P (0;0) n p b=c 6 = 0 for all n; b; c 2 N (even for the case b = 1; 3): In passing we note that recent research is devoted to the study of irreducibility of the Laguerre polynomials L n (x) initiated by I. Schur, see [20], [22], [36], and for a family of Jacobi polynomials see [12]. For general questions about irreducibility of polynomial with rational coe¢ cients we refer to [28], [31] and [38].…”

On the Galois group of generalized Laguerre polynomials

“…[10], [11], and Hajir-Wong [12]. In particular, we have the following theorem of Filaseta and Lam [4] on the irreducibility of GLP.…”

“…conditions ii) and iii) of Lemma 3.1 hold. By Filaseta-Lam [4], there is an effectively computable constant N (α) such that f (x) is irreducible for n ≥ N (α). Thus, all the conditions of Lemma 3.1 hold, and the proof of the theorem is complete.…”

“…4. The proofs of the Filaseta-Lam Theorem in [4] and of Theorem 1.1 are both effective. Acknowledgements.…”

On the Factorization of Polynomials Over Discrete Valuation Domains

“…There are many recent results that provide irreducibility conditions for various classes of polynomials by using techniques coming from valuation theory (see for instance [23], [24], [2], [3], [4], [8], [5] and [9]), or Newton polygon method (see for instance [12], [13], [14], [15], [16], [17], [18], [1], [6], [7], [22] and [25]). …”