Quadratic and cubic invariants of unipotent affine automorphisms

Let K   be an arbitrary field of characteristic zero, Pn:=K[x1,…,xn]Pn:=K[x1,…,xn] be a polynomial algebra, and Pn,x1:=K[x1−1,x1,…,xn], for n⩾2n⩾2. Let σ′∈AutK(Pn)σ′∈AutK(Pn) be given byx1↦x1−1,x2↦x2+x1,…,xn↦xn+xn−1. It is proved that the algebra of invariants, Fn′:=Pnσ′, is a polynomial algebra in n−1n−1 variables which is generated by [n2] quadratic and [n−12] cubic (free) generators that are given explicitly.Let σ∈AutK(Pn)σ∈AutK(Pn) be given byx1↦x1,x2↦x2+x1,…,xn↦xn+xn−1. It is well known that the algebra of invariants, Fn:=Pnσ, is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n−1n−1, and that one can give an explicit transcendence basis in which the elements have degrees 1,2,3,…,n−11,2,3,…,n−1. However, it is an old open problem to find explicit generators for FnFn. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants Pn,x1σ is a polynomial algebra over K[x1,x1−1] in n−2n−2 variables which is generated by [n−12] quadratic and [n−22] 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 SL2SL2-invariants.