Generalizations and Applications of Border Bases

This doctoral thesis is devoted to generalize border bases to the module setting and to apply them in various ways. First, we generalize the theory of border bases to finitely generated modules over a polynomial ring. We characterize these generalized border bases and show that we can compute them. As an application, we are able to characterize subideal border bases in various new ways and give a new algorithm for their computation. Moreover, we prove Schreyer's Theorem for border bases of submodules of free modules of finite rank over a polynomial ring. In the second part of this thesis, we study the effect of homogenization to border bases of zero-dimensional ideals. This yields the new concept of projective border bases of homogeneous one-dimensional ideals. We show that there is a one-to-one correspondence between projective border bases and zero-dimensional closed subschemes of weighted projective spaces that have no point on the hyperplane at infinity. Applying that correspondence, we can characterize uniform zero-dimensional closed subschemes of weighted projective spaces that have a rational support over the base field in various ways. Finally, we introduce projective border basis schemes as specific subschemes of border basis schemes. We show that these projective border basis schemes parametrize all zero-dimensional closed subschemes of a weighted projective space whose defining ideals possess a projective border basis. Assuming that the base field is algebraically closed, we are able to prove that the set of all closed points of a projective border basis scheme that correspond to a uniform subscheme is a constructive set with respect to the Zariski topology.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Passau

Year

2016

Author dc:creator

• Kriegl, Markus

Contributors dc:contributor

• Kreuzer, Martin
• Winkler, Franz

• (generalized) border bases
• Schreyer's theorem
• weighted projective spaces
• zero-dimensional projective schemes
• uniformity conditions

Statement dc:rights

• CC by: Creative Commons - Namensnennung

Repository record source_url

https://opus4.kobv.de/opus4-uni-passau/frontdoor/index/index/docId/362

OAI identifier oai:identifier

oai:kobv.de-opus4-uni-passau:362