Fundamental algebraic sets and locally unit-additive rings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909849911033856 |
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| author | Epstein, Neil |
| author_facet | Epstein, Neil |
| contents | The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be fundamental, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of local fundamentality is introduced and its behavior developed. On the algebraic side, the notions of locally, geometrically, and generically unit-additive rings are introduced, thus complementing unit-additivity as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental algebraic sets and locally unit-additive rings Epstein, Neil Algebraic Geometry Commutative Algebra The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be fundamental, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of local fundamentality is introduced and its behavior developed. On the algebraic side, the notions of locally, geometrically, and generically unit-additive rings are introduced, thus complementing unit-additivity as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout. |
| title | Fundamental algebraic sets and locally unit-additive rings |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2502.20544 |