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 criteria are over any field and have the property of being absolutely irreducible when one modifies their coefficients arbitrarily in the field, keeping only some of them nonzero. Some results on Minkowski sums of convex sets and polyhedra are also presented.