A practical method is described for deciding whether or not a nite-dimensional module for a group over a nite eld is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalisation of the Parker-Norton`Meataxe' algorithm, but it does not depend for its eeciency on the eld being small. The principal tools involved are the calculation of the nullspace and t...

This means that for all n ≥ 2, there exists no first-order sentence Fn in the language Ln = (+,−,×,=, 0, 1, In) satisfying the following property: for any real algebraic variety V of R, if we interpret the n-ary predicate In by membership to V then R |= Fn if and only if V is irreducible. Our proof will imply that irreducibility remains undefinable even if we add the exponential function to Ln....

We construct quasimultiples of affine planes, inversive planes, and unitals from geometrical configurations in various finite geometries. All designs have k = λ. These designs are simple (no repeated blocks), and all appear to be irreducible in the sense that a quasimultiple of type P will have no subdesign isomorphic to P. This is verified for small orders by computer computations.

We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the “generic” irreducibility of the numerator of the zeta functions in a family of curves with large monodromy.

An approach to the computations of upper bounds on the size of large complete arcs is presented. In particular, we obtain geometrical properties of irreducible envelopes associated to a second largest complete arc provided that the order of the underlying field is large enough.