Polynomial Methods for Grid Covering Problems
L.K.M. Verlinde (TU Delft - Electrical Engineering, Mathematics and Computer Science)
C.E. Groenland – Mentor (TU Delft - Discrete Mathematics and Optimization)
A. Bishnoi – Mentor (TU Delft - Discrete Mathematics and Optimization)
Dion Gijswijt – Mentor (TU Delft - Discrete Mathematics and Optimization)
JMAM Van Neerven – Graduation committee member (TU Delft - Analysis)
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Abstract
Consider an arbitrary finite grid in some field. How many hyperplanes are required so that every point is contained in at least k hyperplanes, except for one point that is not allowed to be contained in any hyperplane? To solve this so-called hyperplane grid covering problem, the polynomial method has proven to be extremely useful in finding bounds on the minimum number of hyperplanes required. This has given rise to a second problem: the polynomial grid covering problem. This problem considers the minimum degree of a polynomial such that every grid point is a root with multiplicity k, except for one point where the polynomial does not vanish. We study these two related problems for multiple grids: the hypercube, the vector space over the binary field and grids in the Cartesian plane. Since polynomial covers in the latter have not been studied before, we provide algorithms and techniques to study these covers. We also explore the link between grid coverings and two results from algebraic geometry: the Footprint Bound and the Cayley-Bacharach Theorems.