A New Absolute Irreducibility Criterion for Multivariate Polynomials over Finite Fields

One of the key problems in algebraic geometry and its applications in coding theory, cryptography, and other disciplines is to determine whether a variety defined by a set of polynomials is absolutely irreducible; i.e., it remains irreducible in the algebraic closure of the defining field. The famous Eisenstein criterion for irreducibility works only over the defining fields. One important place where this is needed is when one applies the Riemann-Roch theorem. Another important application is in bounding the number of rational points or exponential sums through the Weil conjectures. In this chapter, we consider the hypersurfaces defined by generalized trinomials. We present a new absolute irreducibility criterion for generalized trinomials over finite fields. Our criterion does not require testing for irreducibility in the ground field or in any extension field. We just require multivariate GCD computations and the square-free property. Since almost all polynomials are known to be square-free, our absolute irreducibility criterion proves the absolute irreducibility of almost all generalized trinomials.