Pythagoras Numbers for Ternary Forms
Grigoriy Blekherman, Alex Dunbar, and Rainer Sinn

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 of real ternary forms, defined for each degree as the minimal number such that every degree ternary form which is a sum of squares can be written as the sum of at most squares of degree forms. Scheiderer showed that . We show that for . The main technical tool is Diesel's characterization of height 3 Gorenstein algebras.
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Taxonomy
TopicsHistory and Theory of Mathematics · Mathematics and Applications · Mathematics Education and Teaching Techniques
