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Cross ratios for finite field geometries

Siliņš, Pēteris (2024) Cross ratios for finite field geometries. Bachelor's Thesis, Mathematics.


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In this thesis we study matroids whose elements are the points of a projective plane over some finite field. We start by introducing matroids, giving multiple ways of defining a matroid, and explaining what it means for a matroid to be representable. We also touch upon the cross ratio, an essential invariant in projective geometry, and explore the connection to matroid representability. To formally establish this connection, partial fields are introduced. Partial fields are an algebraic structure which was originally studied to classify certain kinds of matroids that can be represented by a matrix whose subdeterminants are constrained to some multiplicative group. The final concept we introduce is the universal partial field. If a partial field is the universal partial field of a matroid, then every other representation of the matroid can be obtained from this universal partial field. We explain how to compute the universal partial field, which is where cross ratios become relevant again. In the results section we define a matroid represented over the finite field of order p. We then show that its universal partial field is exactly equal to this field for infinitely many primes p and for p < 1000.

Item Type: Thesis (Bachelor's Thesis)
Supervisor name: Lorscheid, O. and Top, J.
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 19 Feb 2024 11:13
Last Modified: 19 Feb 2024 11:13

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