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Maximal minors and their leading terms. (English) Zbl 0776.13009
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.

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
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