An Analytic Approach to the Problem of Matroid Representibility: Summer REU 2015
نویسندگان
چکیده
A central problem in matroid theory—and by extension coding theory and finite geometry— is the problem of constructing matrices with a maximal number of columns such that every square submatrix formed by a subset of the columns is nonsingular. In this project, we apply the tools of analysis to this problem and demonstrate the value of such an approach. First, we introduce various quantities that can be defined for an arbitrary matroid, and show that certain conditions on these quantities imply that a matroid is not representable over Fq. In particular, for a matroid of rank r, we examine the proportion of size-(r − k) subsets that are dependent, and give bounds, in terms of the cardinality of the matroid and q a prime power, for this proportion, below which the matroid is not representable over Fq. Using an observation about the reduced form of a uniform matrix, we explore connections between the quantities we define. Further, we demonstrate that the defined quantities can be used to prove that random matrices have high proportions of subsets of independent columns. In addition, we construct a uniform matroid using a method analogous to the construction of Pascal’s Triangle, generalized for finite fields of non-prime order. We extend our thanks to Dr Steve Damelin and Dr Ming Yu for their guidance in this project. The following people also contributed to the discussions regarding the Pascal triangle work and we are thankful to them for their input and to fruitful discussions: Karl Winsor, Sam Tenka and Bar Roytman. This research was funded by NSF grants: 0901145, 1160720, 1104696. REU webpage: http://www.ima.umn.edu/∼ damelin/REU.html
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