Relative negligibility of linear automorphisms
Let \(U\) and \(V\) be finite dimensional vector spaces over a field \(k\), \(\alpha\in \text{GL}(U)\), \(\beta\in \text{GL}(V)\) and \(I\) be the identity transformation on \(V\). Denote by \(\alpha*\beta\) and \(\alpha*I\) the induced linear automorphism on \(U\oplus V\). Then \(\alpha*\beta\) and \(\alpha*I\) can be regarded as \(k\)-automorphisms of the function field \(k(U\oplus V)\). In this paper the authors give necessary and sufficient conditions for \(\alpha*\beta\) and \(\alpha*I\) to be conjugate in \(\Aut_k (k(U\oplus V))\). For this characterization, the authors introduce the concept of the generalized order. Through this invariant, they also settle the question of when two different polynomials are minimal polynomials of the same linear automorphism of a rational function field. After presenting an interesting and informative introduction, the authors begin the work to answer the following two questions: Question 1. What are the conditions on \(f(T), g(T)\in k[T]\) that are necessary and sufficient for \(f\) and \(g\) to be minimal polynomials of the same linear automorphism? Question 2. If \(\alpha\) and \(\beta\) are \(k\)-linear automorphisms having \(f\) and \(g\) as minimal polynomials, respectively, then what conditions on \(f\) and \(g\) are necessary and sufficient for \(\beta\) to be negligible relative to \(\alpha\). For Question 1, the authors answer the question in terms of what they call the generalized order of \(f\). This is a pair \((R(f),\omega(f))\), where \(R(f)\) is the multiplicative group generated by the zeros of \(f\) in some algebraic closure \(\overline k\) of \(k\), and where \(\omega(f)\) is a non-negative integer that depends on the maximum multiplicity among the zeros of \(f\) in \(\overline k\). They show that \(f\) and \(g\) can serve as minimal polynomials for the same linear transformation if and only if they have the same generalized order, i.e., if \((R(g)=R(f)\) and \(\omega(g)= \omega (f)\). The authors say that \(\beta\) is negligible relative to \(\alpha\) if \(\alpha*\beta\) and \(\alpha*I\) are conjugate in \(\Aut_k(K*L)\), where \(K\) and \(L\) are rational extensions of \(k,\alpha\) and \(\beta\) are \(k\)-automorphisms of \(K\) and \(L\), respectively, \(K*L\) is the field of quotients of the tensor product \(K\otimes_kL\), and \(\alpha*\beta\) is the natural extension of \(\alpha \otimes_k \beta\). For Question 2, the authors show that \(\beta\) is negligible relative to \(\alpha\) if and only if the generalized order of \(g\) divides that of \(f\), in the sense that \(R(g)\subseteq R(f)\) and \(\omega (g)\leq\omega(f)\).
scientific article; zbMATH DE number 5354928