Equal sums of two cubes of binary quadratic forms

2020/02/03 by Reznick, Bruce
#11D45 #14M99 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11E76 #Secondary: 11D41

paper · doi:10.48550/arxiv.2002.00888

Abstract

We give a complete description of all solutions to the equation f₁³ + f₂³ = f₃³ + f₄³ for quadratic forms f_j ∈ \mathbb C[x,y] and show how Ramanujan's example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic p ∈ \mathbb C[x,y] can be written as a sum of two cubes. The extreme example is p(x,y) = xy(x⁴-y⁴), which has six such representations.

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