Counting sums of two powers

TL;DR

This paper generalizes classical enumerative results about expressing binary forms as sums of powers, extending known counts for specific degrees and forms.

Contribution

It provides new generalized enumerative formulas for sums of powers representations of binary forms, building on classical results by Clebsch, Zariski, and Vakil.

Findings

General binary sextic forms are expressible as the sum of a cube and a square in 40 ways.

A general pencil of binary 12-ics contains exactly 3762 sums of cubes and squares.

Extends classical enumerative counts to broader settings.

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.