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.

In this paper we obtain new quantitative forms of Hilbert’s Irreducibility Theorem. In particular, we show that if f(X,T1, . . . , Ts) is an irreducible polynomial with integer coefficients, having Galois group G over the function field Q(T1, . . . , Ts), and K is any subgroup of G, then there are at most Of,ε(H s−1+|G/K|+ε) specialisations t ∈ Zs with |t| ≤ H such that the resulting polynomial...

The notion of subdirect irreducibility in the context of languages without equality, as presented by Elgueta, is extended in order to obtain subdirect representation theorems for abstract and reduced classes of structure systems. Structure systems serve as models of firstorder theories but, rather than having universal algebras as their algebraic reducts, they have algebraic systems in the sens...

We explore several concepts for analyzing the intuitive notion of computational irreducibility and we propose a robust formal definition, first in the field of cellular automata and then in the general field of any computable function f from N to N. We prove that, through a robust definition of what means "to be unable to compute the n th step without having to follow the same path than simulat...

Basic properties in Perron-Frobenius theory are strict positivity, primitivity, and irreducibility. Whereas for nonnegative matrices, these properties are equivalent to elementary graph properties which can be checked in polynomial time, we show that for Kraus maps the noncommutative generalization of stochastic matrices checking strict positivity (whether the map sends the cone to its interior...