A polynomial with a root mod $p$ for every $p$ has a real root

TL;DR

This paper proves that a polynomial with roots modulo every large prime must also have a real root, linking number theory and algebraic geometry, and applies this to a problem about quadratic forms covering primes.

Contribution

It establishes a new connection between local roots modulo primes and the existence of real roots for polynomials, with implications for quadratic form theory.

Findings

Polynomials with roots mod p for all large p have real roots.

Application to the non-covering of primes by finitely many positive definite binary quadratic forms.

Provides a bridge between local and global properties of polynomials.

Abstract

We prove that if a polynomial has a root mod $p$ for every large prime $p$, then it has a real root. As an application, we show that the primes can't be covered by finitely many positive definite binary quadratic forms.