Let
K be an arbitrary field of characteristic zero,
Pn:=
K[
x1,…,
xn] be a polynomial algebra, and
, for
n2. Let
σ′Aut
K(
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
σAut
K(
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 SL
2-invariants.
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