Pythagoras Numbers for Ternary Forms
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915002439434240 |
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| author | Blekherman, Grigoriy Dunbar, Alex Sinn, Rainer |
| author_facet | Blekherman, Grigoriy Dunbar, Alex Sinn, Rainer |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_17123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pythagoras Numbers for Ternary Forms Blekherman, Grigoriy Dunbar, Alex Sinn, Rainer Algebraic Geometry 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. |
| title | Pythagoras Numbers for Ternary Forms |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.17123 |