Nilpotency indices, degrees of iterations of affine triangular automorphisms, and Schubert calculus
type: Item
instance of: scholarly article
title: Nilpotency indices, degrees of iterations of affine triangular automorphisms, and Schubert calculus
zbMATH Open document ID: 1306.14019
author: Shu Kawaguchi
published in: Manuscripta Mathematica
publication date: 2014-08-29T00:00:00Z
review text: For a commutative \(\mathbb{Q}\)-algebra \(R\) let \(R^N\) denote the affine \(N\)-space \(\mathrm{Spec}\;R[X_1 , X_2 , \dots, X_N ]\). Let \(f : R^2 \to R^2\) be a polynomial mapping given by a pair of polynomials \(p, q\). Define \(\mathrm{deg}\;f = \max {\mathrm{deg} p, \mathrm{deg} q}\). In this paper the author considers triangular automorphisms \(f\) of \(R^2 \), i.e. those for which \(p\) does not involve \(X_2\). Let \(f^n\) denote the \(n\)th iterate of \(f\) for any integer \(n\). If further the Jacobian determinant of \(f\) is \(1\), then the main result of the paper says that deg \(f^n \leq d^2 - d + 1\). In particular, \(\mathrm{deg}\;f^n\) is bounded above as \(n\) varies over all integers. The author introduces a weighted nilpotency index, \(\nu(f )\), for a polynomial automorphism with determinant one, \(f : R^N \to R^N \). He gives a bound for \(\mathrm{deg} f^n\) in terms of \(N, d, \nu(f )\) for all integers n. For \(n = -1\) a formula found by S. S. Abhyankar and the reviewer in 1974 for the formal inverse of \(f\) (based on an old result of Lagrange) is used. The estimate for deg \(f^{-1}\) was found by \textit{P. van Rossum} around 2001 [Tackling problems on affine space with locally nilpotent derivations on polynomial rings, Ph.D. thesis, University of Nijmegen (2001)].
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 6336311
zbMATH Keywords: polynomial automorphism
reviewed by: Rajendra Gurjar
describes a project that uses: Macaulay2
MaRDI profile type: Publication
full work available at URL: s00229-014-0658-x
OpenAlex ID: W1995777505
cites work: Q5563439
cites work: Degree Bounds for Inverses of Polynomial Automorphisms
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Inverse degrees and the Jacobian conjecture
cites work: Dynamical compactifications of \(\mathbb{C}^2\)
cites work: Computation of the maximal degree of the inverse of a cubic automorphism of the affine plane with Jacobian 1 via Gröbner bases
cites work: Q4342000
cites work: On the degree of the inverse of an automorphism of the affine space
cites work: On the degree of iterates of automorphisms of the affine plane
cites work: Q4195061
cites work: An algebraic proof of the Lagrange-Bürmann formula
cites work: Inverse degree of an affine space triangular automorphism
cites work: Q3739243
cites work: The Automorphism Group Of
cites work: Polynomial automorphisms and the Jacobian conjecture
cites work: On Bass' inverse degree approach to the Jacobian conjecture and exponential automorphisms.
cites work: Q5818521
DOI: 10.1007/S00229-014-0658-X
description: scientific article; zbMATH DE number 6336311