Degree bounds of minimal polynomials and polynomial automorphisms
Let \(k\) be a field, \(h_ 1,\dots, h_ n\in k[x_ 1,\dots, x_{n-1}]=: k^{[n-1]}\) contain \(n-1\) algebraically independent polynomials. The author has shown earlier, that there exists, up to a non-zero constant factor, a unique irreducible polynomial \(m\) in \(n\) variables such that \(m(h_ 1, \dots,h_ n)=0\), called the minimal polynomial of \(h_ 1,\dots,h_ n\). Here it is proved that \[ \deg m (k^{[n -1]}:\;k(h_ 1,\dots, h_ n))\leq (\max_ j (\deg h_ j))^{n-1}. \] As a consequence a straightforward elementary proof of the degree bound for \(k\)-algebra automorphisms \(F: k^{[n]}\to k^{[n]}\) is obtained, which says that \(\deg (F^{-1})\leq (\deg F)^{n-1}\), \(\deg F\) being the maximum of the total degrees of \(F(x_ i)\), \(i=1,\dots, n\).
scientific article; zbMATH DE number 561566