We continue the study of the Newton polytope ∏m,n of the product of all maximal minors of an m × n-matrix of indeterminates. The vertices of ∏m,n are encoded by coherent matching fields Λ = (Λσ), where σ runs over all m-element subsets of columns, and each Λσ is a bijection σ → [m]. We show that coherent matching fields satisfy some axioms analogous to the basis exchange axiom in the matroid theory. Their analysis implies that maximal minors form a universal Gröbner basis for the ideal generated by them in the polynomial ring. We study also another way of encoding vertices of ∏m,n for m ≤ n by means of “generalized permutations”, which are bijections between (n − m + 1)–element subsets of columns and (n − m + 1)–element submultisets of rows.
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