In this paperwe give an alternate combinatorial description of the “(l, 0)-Carter partitions” (see [4]). The representation-theoretic significance of these partitions is that they indicate the irreducibility of the corresponding specialized Specht module over the Hecke algebra of the symmetric group (see [7]). Our main theorem is the equivalence of our combinatoric and the one introduced by Jam...

This essay deals with the concept of truth in the context of a version of internal realism. In §1 I define some variants of realism using a set of realistic axioms. In §2 I will argue that for semantical reasons we should be realists of some kind. In §3 I plead for an internalistic setting of realism starting from the thesis that truth is, at least, not a non-epistemic concept. We have to bear ...

The Eisenstein irreducibility critierion is part of the training of every mathematician. I first learned the criterion as an undergraduate and, like many before me, was struck by its power and simplicity. This article will describe the unexpectedly rich history of the discovery of the Eisenstein criterion and in particular the role played by Theodor Schönemann. For a statement of the criterion,...

He noted then that the conjecture is true if n = p 1 2 or if n = p where p is a prime and r a positive integer. Calculations showed the conjecture also held for n 100. Recently, in a study of more general polynomials, the rst author [2] obtained further irreducibility results for f(x); in particular, he established irreducibility in the case that n+ 1 is a squarefree number 3 and in the case th...

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...