Remarks on the Jacobian conjecture
type: Item
instance of: scholarly article
title: Remarks on the Jacobian conjecture
zbMATH Open document ID: 0892.14010
author: Jie-Tai Yu
published in: Journal of Algebra
publication date: 1998-07-19T00:00:00Z
review text: Let \(F=(F_1,\dots,F_n),\;F_i\in\mathbb C[X_1,\dots,X_n] \) and \(\text{Jac}(F)\in\mathbb C^*\). Let \(M_i=m_{i0}(Y)+m_{i1}(Y)T+ \dots +m_{id_i}(Y)T^{d_i}_i \in \mathbb C[Y_1,\dots,Y_n,T] \) be the minimal polynomial of the algebraic dependence of \(F_1,\dots,F_n,X_i\) over \(\mathbb C\), i.e. the non-zero minimal irreducible polynomial such that \(M_i(F_1,\dots,F_n,X_i)=0\). Then the polynomials \(m_{i0}(Y),\dots,m_{id_i}(Y)\) have no common zero in \(\mathbb C^n\). From this theorem the author obtains the flatness of \(\mathbb C[F,X_i]\) over \(\mathbb C[F]\) for \(i=1,\dots,n\), and two known theorems that polynomial birational mappings of \(\mathbb C^n\) with \(\text{Jac}(F)\in\mathbb C^*\) and injective polynomial mappings of \(\mathbb C^n\) into \(\mathbb C^n\) are automorphisms.
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 1013915
zbMATH Keywords: jacobian conjecture
zbMATH Keywords: polynomial birational mappings
zbMATH Keywords: injective polynomial mappings
Wikidata QID: Q123132636
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
full work available at URL: jabr.1996.6816
OpenAlex ID: W1987465467
cites work: Injective Morphisms of Real Algebraic Varieties
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Q4314530
cites work: Flatness of an extension of a commutative ring
cites work: Injective Polynomial Maps are Automorphisms
cites work: Q3947744
cites work: Computing minimal polynomials and the degree of unfaithfulness
cites work: Face polynomials and inversion formula
cites work: On relations between Jacobians and minimal polynomials
DOI: 10.1006/JABR.1996.6816
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description: scientific article; zbMATH DE number 1013915