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The present thesis deals with problems arising from discrete mathematics, whose proofs make use of tools from algebraic geometry and topology. The thesis is based on four papers that I have co-authored, three of which have been published in journals, and one has been submitted for publication (and also appeared as a preprint on the arxiv, and as an extendend abstract in a conference). Specifically, we deal with the following four problems: \begin{enumerate} \item We prove that if is an invertible matrix, and is a bilinear form , then any finite set contained in an irreducible algebraic curve of degree in determines distinct values of , unless is a line, or is linearly equivalent to a curve defined by an equation of the form , with , and . \item We show that if we are given points and lines in the plane, then the number of distinct distances between the points and the lines is , as long as . Also, we show that if we are given points in the plane, not all collinear, then the number of distances between these points and the lines that they determine is . We also study three-dimensional versions of the distinct point-line distances problem. \item We prove the lower bound on the number of ordinary conics determined by a finite point set in , assuming that is not contained in a conic, and at most points of lie on the same line (for some $0
Michael Christoph Gastpar, Sung Hoon Lim, Adriano Pastore, Chen Feng