A classical result due to Segre states that on a real cubic surface in P3 R there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2m-3 in PmR.

Okonek, Christian; Telemann, Andrei (2014). *Intrinsic signs and lower bounds in real algebraic geometry.* Journal fur die Reine und Angewandte Mathematik, 688:219-241.

## Abstract

A classical result due to Segre states that on a real cubic surface in P3 R there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2m-3 in PmR.

## Citations

## Altmetrics

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2014 |

Deposited On: | 15 Oct 2014 08:25 |

Last Modified: | 05 Apr 2016 18:24 |

Publisher: | de Gruyter |

ISSN: | 0075-4102 |

Publisher DOI: | https://doi.org/10.1515/crelle-2012-0055 |

## Download

Full text not available from this repository.View at publisher

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.