# The Schinzel Hypothesis for Polynomials

## TL;DR

This paper extends the Schinzel hypothesis to polynomial rings, proving it for various classes of polynomials over integers and fields, and introduces related conjectures and spectral results.

## Contribution

It develops a version of the Hilbert specialization property over rings and proves the Schinzel hypothesis for broad classes of polynomials.

## Key findings

- Schinzel hypothesis proven for polynomials over integers and fields
- A polynomial Goldbach conjecture is proposed
- Results on spectra of rational functions are established

## Abstract

The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is replaced by a polynomial ring and prove the Schinzel hypothesis for a wide class of them: polynomials in at least one variable over the integers, polynomials in several variables over an arbitrary field, etc. We achieve this goal by developing a version over rings of the Hilbert specialization property. A polynomial Goldbach conjecture is deduced, along with a result on spectra of rational functions.