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September 10, 2025

Pythagoras numbers for ternary forms

Key Points

Pythagoras number equals d + 1 for ternary forms of specific degrees, offering a fresh perspective on mathematical boundaries.
We confirm that for certain degrees, the pythagorean number lies between d + 1 and d + 2, establishing critical thresholds.
The research employs connections to optimality conditions, enhancing the understanding of sums of squares in ternary forms.
Our findings advance the study of syzygies in the context of Gorenstein algebras, showing their relevance to structure in algebra.

Abstract

We study the Pythagoras numbers py ⁡ (3, 2 d) py (3, 2d) of real ternary forms, defined for each degree 2 d 2d as the minimal number r r such that every degree 2 d 2d ternary form which is a sum of squares can be written as the sum of at most r r squares of degree d d forms. Scheiderer Math. Z. 286 (2017), pp. 559–570 showed that d + 1 ≤ py ⁡ (3, 2 d) ≤ d + 2 d+1 py (3, 2d) d+2. We show that p y (3, 2 d) = d + 1 py (3, 2d) = d+1 for 2 d = 8, 10, 12 2d = 8, 10, 12. Our approach is to connect the problem with optimality conditions for a Euclidean distance problem. When paired with Diesel’s characterization of height 3 Gorenstein algebras, this allows us to control the syzygies of the forms involved in the decomposition into sums of squares.

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