Matroids over \(F_ p\) which are rational excluded minors
The characteristic set of a matroid M (the set of integers p such that M is representable over some field of characteristic p) is called a prime- field characteristic set if, for each p in the set, M is in fact representable over the field of p elements. It was shown by Brylawski and Reid that for any prime p there is a matroid with the following two properties: (i) its prime-field characteristic set is \(\{\) \(p\}\) ; (ii) it is an excluded minor for representability over the rationals. Reid conjectured that any such matroid must have \(2p+o(p)\) points. The basic construction in this paper disproves the conjecture by giving a matroid with 3p-5 points having the above two properties. The matroid constructed is moreover 'sequentially unique', in that its points can be ordered \(a_ 1,...,a_ n\) in such a way that for any j, any two matrices representing the submatroid induced by \(a_ 1,...,a_ j\) are projectively equivalent. This construction is then generalized; for any p and any ordered partition of p-1 into, say, k parts, there is a sequentially unique matroid on \(2(p+k-1)\) points with the above two properties. The matroids corresponding to different partitions are non-isomorphic, and so for any p there are at least \(2^{p-1}\) matroids with those two properties. Finally, another set of matroids, dual to the above in a projective sense, is presented, doubling the number of non-isomorphic matroids with the stated properties.
scientific article; zbMATH DE number 3887712