Finiteness of fibers in matrix completion via Plücker coordinates
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Let Ω⊆{1,...,m}×{1,...,n}. We consider fibers of coordinate projections π_Ω : M_k(r,m × n) → k^#Ω from the algebraic variety of m × n matrices of rank at most r over an infinite field k. For #Ω = M_k(r,m × n) we describe a class of Ω's for which there exist non-empty Zariski open sets U_Ω⊂M_k(r,m × n) such that π_Ω^-1(π_Ω(X)) ∩U_Ω is a finite set ∀ X ∈U_Ω. For this we interpret matrix completion from a point of view of hyperplane sections on the Grassmannian Gr(r,m). Crucial is a description by Sturmfels & Zelevinsky of classes of local coordinates on Gr(r,m) induced by vertices of the Newton polytope of the product of maximal minors of an m × (m-r) matrix of variables.
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