STNB2025 (38th edition)

On some recent progress of the Schinzel hypothesis over polynomial rings

Coordinación

Alberto Fernandez Boix

Descripción

In \cite[page 188]{Schinzelhypothesis}, it was formulated the so--called \textit{Schinzel (H) hypothesis}, which can be stated as follows. \begin{con} Let $P_1,\ldots,P_s$ be polynomials in $\mathbb{Z}[x]$, all of degree at least one, satisfying the following condition. \[ \text{There is no prime }p\in\mathbb{Z}\text{ dividing all values }\prod_{i=1}^s P_i (m),\ m\in\mathbb{Z}. \] Then, there are infinitely many integers $m\in\mathbb{Z}$ such that $P_1 (m),\ldots,P_s (m)$ are prime numbers. \end{con} The conjecture is, of course, known in the case $s=1$ when $P_1$ is a polynomial of degree one; this is nothing but the classical Dirichlet's theorem on primes in arithmetic progressions. To the best of our knowledge, the case $s>1$ is completely open. The goal of these lectures is to explain how Bodin, Deb\`es and Najib have recently proved \cite{bodin2020schinzel} the Schinzel hypothesis replacing the ring of integers $\mathbb{Z}$ by a polynomial ring $A[x_1,\ldots,x_m]$, where $A$ is, roughly speaking, a ring where the classical Hilbert's irreducibility theorem holds.

Charlas

  1. Review of Dirichlet's theorem on primes in arithmetic progressions (Alberto Fernandez Boix)
  2. Basics on Hilbertian fields (Alberto Fernandez Boix)
  3. Classic Hilbertian fields and Hilbertian rings (Alberto Fernandez Boix)
  4. The Schinzel hypothesis for some polynomial rings (Alberto Fernandez Boix)

Referencias

\cite[page 188]{Schinzelhypothesis} \cite{bodin2020schinzel} \cite{Murtyprogressions} \cite{FriedJardenFieldArithmeticbook}

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