Quadratic and cubic invariants of unipotent affine automorphisms (Q2378599)

Let \(K\) be a characteristic zero field, and \(P_{n}=K\left[ x_{1},\dots ,x_{n}\right] .\) Let \(\sigma\in\)Aut\(_{K}\left( P_{n+1}\right) \) be the automorphism \(\sigma\left( x\right) =J_{n+1}\left( 1\right) x,\) where \(x=\left( x_{1},\dots,x_{n+1}\right) ^{t}\) and \(J_{n+1}\left( 1\right) \) is the lower triangular Jordan matrix. Additionally, let \(\sigma^{\prime}\in \)Aut\(_{K}\left( P_{n}\right) \) be given by \(\sigma^{\prime}\left( x\right) =J_{n}\left( 1\right) x+e_{1}\), where \(e_{1}=\left( 1,0,\dots,0\right) ^{t}.\) This paper is a study of the algebras of invariants \(F_{n+1} :=P_{n+1}^{\sigma}\) and \(F_{n}^{\prime}:=P_{n}^{\sigma^{\prime}}.\) It is well-known that \(F_{n}\) is finitely generated and has transcendence degree \(n-1\), but explicit generators are not known. It is shown that \(F_{n}^{\prime}=K\left[ y_{2},\dots,y_{n}\right] ,\) where \(y_{i+1} =\sum_{j=1}^{i}\phi_{j-i}x_{j+1}+i\left( \sigma^{\prime}\right) ^{-1}\left( \phi_{-i-1}\right) ,\) \(\phi_{0}:=1\) and \(\phi_{-i}:=x_{1}\left( x_{1}-1\right) \cdots\left( x_{1}-i+1\right) /i!\) for \(1\leq i\leq n-1.\) Explicit bases are given for the quadratic and cubic invariants, and it is shown how these invariants can be used to give a generating set for \(F_{n}^{\prime}.\) The results above can be adapted to the case of the automorphism \(\sigma\) of \(P_{n+1}.\) A transcendence basis for \(F_{n+1}\) is constructed, consisting of \(x_{1}\) along with \(n\) other elements, \(\left[ n/2\right] \) of which have degree \(2\) and \(\left[ \left( n-1\right) /2\right] \) have degree \(3\) (\(\left[ \_\_\right] \) is the floor function).