Quadratic and cubic invariants of unipotent affine automorphisms

Let K be an arbitrary field of characteristic zero, Pn:=K[x1,…,xn] be a polynomial algebra, and , for n 2. Let σ′ AutK(Pn) be given by It is proved that the algebra of invariants, , is a polynomial algebra in n−1 variables which is generated by quadratic and cubic (free) generators that are given explicitly. Let σ AutK(Pn) be given by It is well known that the algebra of invariants, , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n−1, and that one can give an explicit transcendence basis in which the elements have degrees 1,2,3,…,n−1. However, it is an old open problem to find explicit generators for Fn. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants is a polynomial algebra over in n−2 variables which is generated by quadratic and cubic (free) generators that are given explicitly. The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL2-invariants.