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Combinatorics of maximal minors. (English) Zbl 0794.13021
In this paper the authors continue [see B. Sturmfels and A. Zelevinsky, Adv. Math. 98, No. 1, 65-112 (1993; Zbl 0776.13009)], the study of the Newton polytope of the product of all maximal minors of an \(m\times n\)-matrix of indeterminates. They prove some of the conjectures made in the paper cited above, and get as a consequence:
The set of all maximal minors of a generic \(m \times n\)-matrix \(X=(x_{ij})\) is a universal Gröbner basis for the ideal generated by them in the polynomial ring \(\mathbb{C} [x_{ij}]\).
This paper is essentially self contained, but the results and methods are mostly combinatorial.

MSC:
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
05B35 Combinatorial aspects of matroids and geometric lattices
15A15 Determinants, permanents, traces, other special matrix functions
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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[1] Sturmfels, B.; Zelevinsky, A., Maximal minors and their leading terms, Advances in Math, 98, 65-112, (1993) · Zbl 0776.13009
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