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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.

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
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