Every Ternary Quintic is a Sum of Ten Fifth Powers

TL;DR

This paper improves the known upper bound for the Waring rank of ternary quintic forms from twelve to ten, advancing understanding of polynomial decompositions in algebraic geometry.

Contribution

It establishes that every ternary quintic polynomial can be expressed as a sum of ten fifth powers, reducing the previous upper bound from twelve.

Findings

Lowered the maximum Waring rank for ternary quintics to ten

Confirmed all ternary quintic forms can be decomposed into ten fifth powers

Advances the algebraic understanding of polynomial decompositions

Abstract

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree $d$ (with coefficients in an algebraically closed field of characteristic zero) is known only for $d\le 4$. The best upper bound that is known for $d=5$ is twelve, and in this work we lower it to ten.