On quadratic homogeneous quasi-translations
A polynomial mapping \(F:k^{n}\rightarrow k^{n}\) (\(k\) is a field of characteristic zero) of the form \[ F(x_{1},\dots ,x_{n})=(x_{1}+H_{1}(x_{1},\dots ,x_{n}),\dots ,x_{n}+H_{n}(x_{1},\dots ,x_{n}))=:x+H \] is called: 1. homogeneous of degree \(d\) if \(\;H_{i}\) are homogeneous of degree \(d\), 2. triangular if \(H_{i}\in k[x_{1},\dots ,x_{i-1}],\) \(i\geq 2,\) \( H_{1}\in k,\) 3. linearly triangularizable if there exists invertible linear mapping \(T\in GL_{n}(k)\) such that \(T^{-1}\circ F\circ T\) is triangular. If additionally \(F\) is a polynomial automorphism then it is called: 1. elementary if \(H_{1}=\dots =H_{i-1}=H_{i+1}=\dots =H_{n}=0\) and \(H_{i}\) does not depend on \(x_{i},\) 2. affine if degree of \(F\) is \(1,\) 3. tame if it is a finite composition of elementary ones and affine ones, 4. quasi-translation if \(F^{-1}=x-H.\) The main result of the article is that for \(n\leq 9\) quadratic (\(d=2\)) homogeneous quasi-translations are linearly triangularizable and in consequence tame. It is a particular affirmative answer to the Rusek conjecture that every quadratic polynomial automorphism is tame.
scientific article; zbMATH DE number 5726602