Smooth values of polynomials

TL;DR

This paper studies the factorization properties of polynomial compositions over integers, demonstrating that quadratic polynomials can produce values with small prime factors infinitely often.

Contribution

It introduces the concept of auxiliary polynomials to analyze the factorization of polynomial compositions and establishes results on the distribution of prime factors for quadratic polynomial values.

Findings

Existence of auxiliary polynomials for polynomial factorization

Infinitely many polynomial values with small prime factors for quadratic polynomials

New methods for analyzing polynomial value smoothness

Abstract

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in\mathbb{Z}[t]$ and any $\epsilon > 0$, there are infinitely many $n\in\mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\epsilon}$.