Pythagoras Numbers for Ternary Forms

TL;DR

This paper investigates the minimal number of squares needed to represent certain ternary forms as sums of squares, establishing exact values for specific degrees and advancing understanding of Pythagoras numbers in algebraic forms.

Contribution

It determines the exact Pythagoras numbers for ternary forms of degrees 8, 10, and 12, refining previous bounds and utilizing Diesel's Gorenstein algebra characterization.

Findings

py(3,8)=4

py(3,10)=5

py(3,12)=4

Abstract

We study the Pythagoras numbers $py(3,2d)$ of real ternary forms, defined for each degree $2d$ as the minimal number $r$ such that every degree $2d$ ternary form which is a sum of squares can be written as the sum of at most $r$ squares of degree $d$ forms. Scheiderer showed that $d+1\leq py(3,2d)\leq d+2$. We show that $py(3,2d) = d+1$ for $2d = 8,10,12$. The main technical tool is Diesel's characterization of height 3 Gorenstein algebras.