Irreducibility testing over local fields

Irreducibility testing over local fields

The purpose of this paper is to describe a method to determine whether a bivariate polynomial with rational coefficients is irreducible when regarded as an element in Q((x))[y], the ring of polynomials with coefficients from the field of Laurent series in x with rational coefficients. This is achieved by computing certain associated Puiseux expansions, and as a result, a polynomial-time complex...

Deterministic Irreducibility Testing of Polynomials over Large Finite Fields

We present a sequential deterministic polynomial-time algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility testing combined with Berlekamp’s algorithm.

Bertini Irreducibility Theorems over Finite Fields

Given a geometrically irreducible subscheme X ⊆ PnFq of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that H ∩X is geometrically irreducible tends to 1 as d→∞. We also prove variants in which X is over an extension of Fq, and in which the immersion X → PnFq is replaced by a more general morphism.

Factoring and Testing Irreducibility of Sparse Polynomials over Small Finite Fields

Irreducibility and Deterministic r-th Root Finding over Finite Fields

Constructing r -th nonresidue over a €nite €eld is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree r e (where r is a prime) over a given €nite €eld Fq of characteristic p (equivalently, constructing the bigger €eld Fqr e ). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some ne...