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 Delta such that for any r, the dimension of the spline space in degree d = 2 r 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 k-formality by Brandt and Terao. In this chapter we prove a criteria for k-formal arrangements, using a complex constructed from vector spaces introduced by Brandt and Terao. As an application, we give a simple description of k-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.
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