Cubic Polynomials, Linear Shifts, and Ramanujan Cubics

TL;DR

This paper classifies all monic cubic polynomials with complex coefficients and distinct roots as either translations of x^3 or related to Ramanujan cubics through linear transformations, providing new insights into their roots.

Contribution

It establishes a complete classification of cubic polynomials based on their relation to x^3 and Ramanujan cubics, revealing new structural insights.

Findings

Every monic cubic with distinct roots is a translation of y=x^3 or related to Ramanujan cubics.

Provides a new perspective on the roots of cubic polynomials.

Connects cubic polynomials to Ramanujan cubics through linear transformations.

Abstract

We show that every monic polynomial of degree three with complex coefficients and no repeated roots is either a (vertical and horizontal) translation of $y=x^3$ or can be composed with a linear function to obtain a Ramanujan cubic. As a result, we gain some new insights into the roots of cubic polynomials.