Polynomials with a common composite

TL;DR

This paper investigates the intersection of polynomial rings generated by two polynomials over any field, providing complete solutions in characteristic zero and various results and algorithms in positive characteristic.

Contribution

It offers a complete characterization of the intersection of polynomial rings for characteristic zero fields and introduces new results and algorithms for positive characteristic cases.

Findings

Complete resolution for characteristic zero fields

Examples and algorithms for positive characteristic

Insights into polynomial ring intersections

Abstract

Let f and g be nonconstant polynomials over an arbitrary field K. In this paper we study the intersection of the polynomial rings K[f] and K[g], and in particular we ask whether this intersection is larger than K. We completely resolve this question when K has characteristic zero, and in positive characteristic we present various results, examples, and algorithms.