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

Sturmfels, Bernd; Zelevinsky, Andrei V.

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)

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