Counting sums of two powers

2025/07/11 by Patel, Anand
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.08337

Abstract

We generalize two well-known enumerative facts. The first, due to Clebsch, says that a general binary sextic form is expressible as the sum of a cube and a square in 40 different ways. The second, due to Zariski and later Vakil, states that a general pencil of binary 12-ics contains exactly 3762 cubes plus squares, ignoring scaling.

Citations

• Twelve points on the projective line, branched covers, and rational elliptic fibrations
• Localization in equivariant intersection theory and the Bott residue formula

Related

• On Waring's Problem: Two Cubes and Two Minicubes
• Sum of cubes: Old proofs suggest new q-analogues
• On sums of unequal powers of primes and powers of 2
• On the sum of two squares and at most two powers of 2
• An upper bound of the number of distinct powers in binary words