Polynomial maps and a conjecture of Samuelson
type: Item
instance of: scholarly article
title: Polynomial maps and a conjecture of Samuelson
zbMATH Open document ID: 0767.13009
DOI: 10.1016/0024-3795(92)90327-7
author: S. Singh
published in: Linear Algebra and its Applications
publication date: 1993-05-16T00:00:00Z
review text: It is shown that, for a polynomial map \(F: \mathbb{C}^ n\to \mathbb{C}^ n\), if the upper left-hand principal minors of all orders in the Jacobian matrix \(JF\) are non-zero at all points, then \(F\) has an inverse polynomial map. The authors first note that the non-vanishing of the minors implies that the minors are non-zero elements of \(\mathbb{C}\), as \(\mathbb{C}\) is algebraically closed. Then by induction on \(n\) they show that \(F\) is the product of a diagonal map of constants with \(n\) elementary polynomial maps (\(F\) is elementary of type \(i\) when \(F(x_ 1,\dots,x_ n)=(x_ 1,\dots,F_ 2(x_ 1,\dots,x_ n),\dots,x_ n)\) with \(F_ i(X_ 1,\dots,X_ n)- X_ i\) a polynomial in \(X_ 1,\dots,\hat X_ i,\dots,X_ n)\).
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 32A99
Mathematics Subject Classification ID: 12D99
zbMATH DE Number: 166226
zbMATH Keywords: Jacobian matrix
zbMATH Keywords: inverse polynomial map
reviewed by: David Kirby
MaRDI profile type: Publication
Wikidata QID: Q123317534
full work available at URL: 0024-3795(92)90327-7
OpenAlex ID: W2019835146
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description: scientific article; zbMATH DE number 166226