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  <doc>
    <id>470</id>
    <completedYear>2017</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>doctoralthesis</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2017-04-11</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>2017-04-06</thesisDateAccepted>
    <title language="eng">Border Basis Schemes</title>
    <abstract language="eng">The basic idea of border basis theory is to describe a zero-dimensional ring P/I by an order ideal of terms  whose residue classes form a K-vector space basis of P/I. The O-border basis scheme is a scheme that parametrizes all zero-dimensional ideals that have an O-border basis. In general, the O-border basis scheme is not an affine space. Subsequently, in [Huib09] it is proved that if an order ideal with "d" elements is defined in a two-dimensional polynomial ring and it is of some special shapes, then the O-border basis scheme is isomorphic to the affine space of dimension 2d. This thesis is dedicated to find a more general condition for an O-border basis scheme to be isomorphic to an affine space of dimension "nd" that is independent of the shape of the order ideal with "d" elements and "n" is the dimension of the polynomial ring that the order ideal is defined in.&#13;
We accomplish this in 6 Chapters. In Chapters 2 and 3 we develop the concepts and properties of border basis schemes. In Chapter 4 we transfer the smoothness criterion (see [Huib05]) for the point (0,...,0) in a Hilbert scheme of points  to the monomial point of the border basis scheme by employing the tools from border basis theory. In Chapter 5 we explain trace and Jacobi identity syzygies of the defining equations of a O-border basis scheme and characterize them by the arrow grading. In Chapter 6 we give a criterion for the isomorphism between 2d dimensional affine space and O-border basis scheme by using the results from Chapters 3 and Chapter 4. The techniques from other chapters are applied in Chapter 6.1 to segment border basis schemes and in Chapter 6.2 to O-border basis schemes for which O is of the sawtooth form.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus4-4702</identifier>
    <author>Bilge Sipal</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Border Bases, Border Basis Scheme, Monomial point, Cotangent Space, Hilbert Schemes</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Polynomring</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Basis (Mathematik)</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Kommutative Algebra, Randbasen, Randbasen Schema</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <thesisGrantor>Universität Passau</thesisGrantor>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/470/Sipal_diss.pdf</file>
  </doc>
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