Title: Equal sums of two cubes of quadratic forms
Abstract: Number theorists since Euler have been interested in finding parameterizations of equal sums of two cubes of rational numbers. We present a complete solution to the equation f_1^3 + f_2^3 = f_3^3 + f_4^3 over C[x,y] in quadratic forms f_i, as well as finding when there is a change of variables leading to forms in Q[x,y]. There are two “flips” of this equation: f_1^3 + (-f_3)^3 = (-f_2)^3 + f_4^3 and f_1^3 + (-f_4)^3 = (-f_2)^3 + f_3^3. One novelty of this solution is showing that in two of these three cases, there is a third representation of the sum as f_5^3 + f_6^3, where these are also binary quadratic forms. This work began as an attempt to understand an example of Ramanujan.
(The work is based on the paper: “Equal sums of two cubes of quadratic forms”, which appeared in an issue in honor of Bruce Berndt's retirement: Int. J. Number Theory 17 (2021), 761-786, MR4254775.)