This dissertation uses methods from homological algebra and computational commutative
algebra to study four problems. We use Hilbert function computations and
classical homology theory and combinatorics to answer questions with a more applied
mathematics content: splines approximation, hyperplane arrangements, configuration
spaces and coding theory.
In Chapter II we study a problem in approximation theory. Alfeld and Schumaker
give a formula for the dimension of the space of piecewise polynomial functions
(splines) of degree d and smoothness r. Schenck and Stiller conjectured that this formula
holds for all d 2r + 1. In this chapter we show that there exists a simplicial
complex such that for any r, the dimension of the spline space in degree d = 2r is
not given by this formula.
Chapter III is dedicated to formal hyperplane arrangements. This notion was
introduced by Falk and Randell and generalized to formality by Brandt and Terao.
In this chapter we prove a criteria for formal arrangements, using a complex constructed
from vector spaces introduced by Brandt and Terao. As an application,
we give a simple description of formality of graphic arrangements in terms of the
homology of the flag complex of the graph.
Chapter IV approaches the problem of studying configuration of smooth rational
curves in P2. Since an irreducible conic in P2 is a P1 (so a line) it is natural to ask if classical results about line arrangements in P2, such as addition-deletion type
theorem, Yoshinaga criterion or Terao's conjecture verify for such configurations. In
this chapter we answer these questions. The addition-deletion theorem that we find
takes in consideration the fine local geometry of singularities. The results of this
chapter are joint work with H. Schenck.
In Chapter V we study a problem in algebraic coding theory. Gold, Little and
Schenck find a lower bound for the minimal distance of a complete intersection evaluation
codes. Since complete intersections are Gorenstein, we show a similar bound for
the minimal distance depending on the socle degree of the reduced zero-dimensional
Gorenstein scheme. The results of this chapter are a work in progress.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/5789 |
| Date | 17 September 2007 |
| Creators | Tohaneanu, Stefan Ovidiu |
| Contributors | Schenck, Henry K. |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | 391293 bytes, electronic, application/pdf, born digital |
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