Probabilistic absolute irreducibility test for polynomials

Absolute Irreducibility of Polynomials via Newton Polytopes

A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein’s criterion. Polynomials from these ...

Irreducibility of Polynomials with Low Absolute Values

Absolute Irreducibility of Bivariate Polynomials via Polytope Method

For any field F, a polynomial f ∈ F [x1, x2, . . . , xk] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F, i.e., irreducible over every algebraic extension of F.We present some results giving new integrally indecomposable classes of polygons. C...

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.

Fast Parallel Absolute Irreducibility Testing

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the cla...