STNB2025 (38th edition)
On some recent progress of the Schinzel hypothesis over polynomial rings
Co-ordinators
Description
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.
Talks
- Review of Dirichlet's theorem on primes in arithmetic progressions (Alberto Fernandez Boix)
- Basics on Hilbertian fields (Alberto Fernandez Boix)
- Classic Hilbertian fields and Hilbertian rings (Alberto Fernandez Boix)
- The Schinzel hypothesis for some polynomial rings (Alberto Fernandez Boix)
References
\begin{thebibliography} \cite{Schinzelhypothesis} A. Schinzel and W. Sierpi ́nski. Sur certaines hypoth`eses concernant les nombres premiers. Acta Arith., 4:185–208; erratum 5 (1958), 259, 1958. \cite{bodin2020schinzel} A. Bodin, P. D`ebes, and S. Najib. The Schinzel hypothesis for polynomials. Trans. Amer. Math. Soc., 373(12):8339–8364, 2020. \cite{Murtyprogressions} M. Ram Murty and N. Thain. Prime numbers in certain arithmetic progressions. Funct. Approx. Comment. Math., 35:249–259, 2006. \cite{FriedJardenFieldArithmeticbook} M. D. Fried and M. Jarden. Field arithmetic, volume 11 of Ergebnisse der Mathematik und ihrer Grenzge- biete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Cham, fourth edition, 2023. \end{thebibliography}
