Graduate Student Colloquium
Abel Castillo
UIC
Remarks on Effective Versions of Hilbert's Irreducibility Theorem
Abstract: Let $f(X, t_1, t_2, \cdots, t_r)$ be a polynomial with integer coefficients that is irreducible over $\mathbb Q$. It can be shown that there exists a specialization $t_i \mapsto a_i$ with $a_i \in \mathbb Z$ such that $f(X, a_1, a_2, \cdots, a_r)$ is also irreducible over $\mathbb Q$. ``Hilbert's Irreducibility Theorem'' refers to the fact that this is actually the case for ``almost all'' integer specializations.
In this talk we will discuss how techniques from probabilistic Galois theory can give rise to so-called ``effective versions'' of Hilbert's Irreducibility Theorem. We will also discuss how Hilbert's Irreducibility Theorem relates to other questions of number-theoretic interest, such as the Inverse Galois problem and conjectures about prime values of polynomials. As time permits, we will also talk about how Hilbert's Irreducibility Theorem can be interpreted in the language of algebraic varieties, using the notion of ``thin'' sets (in the sense of Serre)
Monday November 11, 2013 at 5:30 PM in SEO 636