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Pythagoras Numbers for Ternary Forms

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arxiv 2410.17123 v2 pith:FKEG64HT submitted 2024-10-22 math.AG

Grigoriy Blekherman , Alex Dunbar , Rainer Sinn This is my paper · ORCID
classification math.AG
keywords degreeformsternarynumberspythagorassquaresalgebrascharacterization
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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.

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  1. On quadratic persistence and Pythagoras numbers of totally real projective varieties

    math.AG 2025-06 conditional novelty 6.0 of 10

    For totally real varieties of codimension 3 (degree at least 6) or higher codimension with large degree, next-to-maximal quadratic persistence is equivalent to lying as a codimension-1 subvariety in a variety of minim...

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