Sturmfels, Bernd; Zelevinsky, Andrei V. Maximal minors and their leading terms. (English) Zbl 0776.13009 Adv. Math. 98, No. 1, 65-112 (1993). The authors study the Newton polyhedra of the polynomial given by the product of all maximal minors of a \(m \times n\) matrix of indeterminates \(X=(x_{ij})\). It is a polytope in \(\mathbb{R}^{mn}\). The description of this polytope is well known in the following cases: If \(m=n\) it is the Birkhoff polytope of doubly stochastic \(n \times n\) matrices. – If \(m=2\) it is the convex hull in \(\mathbb{R}^{2n}\) of all \(n!\) matrices obtained from \(\begin{pmatrix} n-1 & n-2 & \ldots & 1 & 0 \\ 0& 1 & \ldots & n-2 & n-1 \end{pmatrix}\) by permuting columns. The description of this polytope is really difficult and interesting. The authors give some motivations and applications. Reviewer: M.Morales (Saint-Martin-d’Heres) Cited in 3 ReviewsCited in 28 Documents MSC: 13C40 Linkage, complete intersections and determinantal ideals 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14M12 Determinantal varieties 14N10 Enumerative problems (combinatorial problems) in algebraic geometry Keywords:maximal minors of matrix of indeterminates; Newton polyhedra PDF BibTeX XML Cite \textit{B. Sturmfels} and \textit{A. V. Zelevinsky}, Adv. Math. 98, No. 1, 65--112 (1993; Zbl 0776.13009) Full Text: DOI