Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916103298482176 |
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| author | Fernando, José F. |
| author_facet | Fernando, José F. |
| contents | A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring $A$ as sums of squares of elements of $A$. If $A$ is an excellent ring of dimension $\geq3$, it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in $A$. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the $2$-dimensional case and determine (under some mild conditions) which local excellent henselian rings $A$ of embedding dimension $3$ have the property that every positive semidefinite element of $A$ is a sum of squares of elements of $A$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12572 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings Fernando, José F. Algebraic Geometry Primary: 14P99, 11E25, 32S05, Secondary: 13F25, 13F40, 12D15 A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring $A$ as sums of squares of elements of $A$. If $A$ is an excellent ring of dimension $\geq3$, it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in $A$. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the $2$-dimensional case and determine (under some mild conditions) which local excellent henselian rings $A$ of embedding dimension $3$ have the property that every positive semidefinite element of $A$ is a sum of squares of elements of $A$. |
| title | Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings |
| topic | Algebraic Geometry Primary: 14P99, 11E25, 32S05, Secondary: 13F25, 13F40, 12D15 |
| url | https://arxiv.org/abs/2401.12572 |