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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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