We provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and \(h^*\)-real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are \(h^*\)-real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

我们提供了大小为n且秩为k且基数最少的连通拟阵的 Ehrhart 多项式的公式，也称为最小拟阵。我们证明了它们的多胞体是 Ehrhart 正的并且是\(h^*\) -real-rooted（因此是单峰的）。我们证明了电路超平面弛豫的操作与最小拟阵和拟阵多胞体细分有关，并且还保留了 Ehrhart 的积极性。我们陈述两个猜想：实际上所有拟阵都是\(h^*\) -实根的，并且固定秩和基数的连通拟阵的 Ehrhart 多项式的系数受相应的最小拟阵的限制，并且对应的均匀拟阵。