Irreducibility of multivariate polynomials

Irreducibility of Hecke Polynomials

In this note, we show that if the characteristic polynomial of some Hecke operator Tn acting on the space of weight k cusp forms for the group SL2(Z ) is irreducible, then the same holds for Tp, where p runs through a density one set of primes. This proves that if Maeda’s conjecture is true for some Tn, then it is true for Tp for almost all primes p.

On the irreducibility of Hecke polynomials

Let Tn,k(X) be the characteristic polynomial of the nth Hecke operator acting on the space of cusp forms of weight k for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials Tn,k(X). Using this criterion with some machine computation, we show that if there exists n ≥ 2 such that Tn,k(X) is irreducible and has the full symmetric gr...

Irreducibility testing of lacunary 0, 1-polynomials

A reciprocal polynomial g(x) ∈ Z[x] is such that g(0) 6= 0 and if g(α) = 0 then g(1/α) = 0. The non-reciprocal part of a monic polynomial f(x) ∈ Z[x] is f(x) divided by the product of its irreducible monic reciprocal factors (to their multiplicity). This paper presents an algorithm for testing the irreducibility of the nonreciprocal part of a 0, 1-polynomial (a polynomial having each coefficien...

Irreducibility of Polynomials with Low Absolute Values

Factors and Irreducibility of Generalized Stern Polynomials

We investigate an infinite class of polynomial sequences at(n; z) with integer parameter t 1, which reduce to the well-known Stern (diatomic) sequence when z = 1 and are (0, 1)-polynomials when t 2. These sequences are related to the theory of hyperbinary expansions. The main purpose of this paper is to obtain various irreducibility and factorization results, most of which involve cyclotomic po...