Gonality, apolarity and hypercubics

TL;DR

This paper explores the relationship between Fermat hypercubics and trigonal curves, establishing their apolarity connection, and provides bounds on the Waring number for hypercubics associated with tetragonal curves, advancing understanding in algebraic geometry.

Contribution

It demonstrates the equivalence between Fermat hypercubics and trigonal curves via apolarity, and derives new bounds on the Waring number for hypercubics linked to tetragonal curves.

Findings

Fermat hypercubics are apolar to trigonal curves.

Waring number bounds are established for hypercubics associated with tetragonal curves.

Bounds depend on the genus of the curves.

Abstract

We show that any Fermat hypercubic is apolar to a trigonal curve, and vice versa. We show also that the Waring number of the polar hypercubic associated to a tetragonal curve of genus $g$ is at most $\lceil 3/2g - 7/2\rceil$, and for a large class of them is at most $4/3g - 3$.