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...

I Abstract n this paper we prove by entirely elementary means a very effective version of the Hilbert Irreducibility-n Theorem. We then apply our theorem to construct a probabilistic irreducibility test for sparse multivariate poly omials over arbitrary perfect fields. For the usual coefficient fields the test runs in polynomial time in the input K size.

In this article, we give two different sufficient conditions for the irreducibility of a polynomial more than one variable, over an algebraically closed field, that can be written as sum polynomials which depend on mutually disjoint sets variables. These are derived from analyzing Newton polytope such and then applying ‘Irreducibility criterion’ introduced by Gao.

We prove the irreducibility of ceratin polynomials whose coefficients are in arithmetic progression with common difference 2 . We use Sylvester type of results on the greatest prime factor of a product with terms in an arithmetic progression and irreducibility results based on Newton Polygons.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, int...

Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.

In the current paper we prove the irreducibility of Severi varieties on Hirzebruch surfaces in arbitrary characteristic. Our approach is of purely algebro-geometric nature, and it works in any characteristic. As a result, we obtain a deformation-theoretic proof of the irreducibility of moduli spaces Mg in positive characteristic, which does not involve reduction to characteristic zero.

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.

In the paper we introduce the notion of →-irreducibility and we show that the set of all →-irreducible elements of a finite Heyting lattice L forms the skeleton of L. We also discuss a parallel concept of ↔-irreducibility and give a similar characterization. Finally, we present generalizations of these results for some class of infinite Heyting lattices.