On maximum-sized \(k\)-regular matroids (Q1961762)

The class of \(k\)-regular matroids is a generalization of the classes of regular and near-regular matroids. Let \(\alpha_1, \alpha_2, \dots , \alpha_k\) be \(k\) algebraically independent transcendentals over the rationals \(\mathbb Q\). A matroid is \(k\)-regular if it can be represented by a matrix over \({\mathbb Q} (\alpha_1, \alpha_2, \dots , \alpha_k)\) of which all subdeterminants are products of positive and negative powers of differences of pairs of elements in \(\{ 0, 1, \alpha_1, \alpha_2, \dots , \alpha_k \}\). A simple rank-\(r\) matroid is maximum sized in a clsss if it has the maximum number of points amongst all simple rank-\(r\) matroids in the class. The author determines the maximum size for a rank-\(r\) \(k\)-regular matroid for all \(r\) and \(k\). Moreover, the maximum-sized matroids are classified and described as well. It is concluded, with one exception, that there is a single maximum-sized rank-\(r\) \(k\)-regular matroid. It turns out that a maximum-sized matroid is obtained by freely adding \(k\) independent points to a flat of \(M (K_{r+k+1})\) which is isomorphic to \(M (K_{k+2})\), then contracting each of these points and simplifying the matroid. The matroid obtained this way is isomorphic to the simplification of \(\overline{T}_{M(K_{k+2})} (M(K_{r+k+1}))\).