| Title:
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Generalized polar varieties and an efficient real elimination (English) |
| Author:
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Bank, Bernd |
| Author:
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Giusti, Marc |
| Author:
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Heintz, Joos |
| Author:
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Pardo, Luis M. |
| Language:
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English |
| Journal:
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Kybernetika |
| ISSN:
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0023-5954 |
| Volume:
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40 |
| Issue:
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5 |
| Year:
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2004 |
| Pages:
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[519]-550 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
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Let $W$ be a closed algebraic subvariety of the $n$-dimensional projective space over the complex or real numbers and suppose that $W$ is non-empty and equidimensional. In this paper we generalize the classic notion of polar variety of $W$ associated with a given linear subvariety of the ambient space of $W$. As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that $W$ is affine) conic. We show that for a generic choice of their parameters the generalized polar varieties of $W$ are empty or equidimensional and, if $W$ is smooth, that their ideals of definition are Cohen-Macaulay. In the case that the variety $W$ is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of $W$ by explicit equations. Finally, we use this description in order to design a new, highly efficient elimination procedure for the following algorithmic task: In case, that the variety $W$ is $\mathbb{Q}$-definable and affine, having a complete intersection ideal of definition, and that the real trace of $W$ is non-empty and smooth, find for each connected component of the real trace of $W$ a representative point. (English) |
| Keyword:
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Geometry of polar varieties and its generalizations |
| Keyword:
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geometric degree |
| Keyword:
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real polynomial equation solving |
| Keyword:
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elimination procedure |
| Keyword:
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arithmetic circuit |
| Keyword:
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arithmetic network |
| Keyword:
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complexity |
| MSC:
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14B05 |
| MSC:
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14N05 |
| MSC:
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14P05 |
| MSC:
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68Q25 |
| MSC:
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68W30 |
| idZBL:
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Zbl 1249.14019 |
| idMR:
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MR2120995 |
| . |
| Date available:
|
2009-09-24T20:03:37Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135615 |
| . |
| Reference:
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