Orienting matroids representable over both GF(3) and GF(5)
The paper provides a simple, constructive proof of the fact that matroids that are representable over both GF(3) and GF(5) are also orientable. The proof relies on a result proved by the author in [J. Comb. Theory, Ser. B. 50, No. 2, 265-287 (1990; Zbl 0657.05017)]. In the course of the proof, a special \(D_0^8\)-mapping for \(\mathcal M\) is constructed, where \(D_0^8=\{0, \pm 1, \pm 2, \pm 4, \pm 8\}\). For such a mapping, there is a naturally arising \(\Sigma\)-mapping for \(\mathcal M\) into \(\Sigma=\{-, 0, +\}\). This \(\Sigma\)-mapping is derived from the \(D_0^8\)-mapping, and it is checked against the necessary conditions for orientability. The paper is concluded with a number of examples.
scientific article; zbMATH DE number 1417659
scientific article