Endomorphisms of polynomial algebras with small co-invariants

Let ϕ be an endomorphism of the polynomial algebra C[x1,…,xn] and let co(ϕ) be the dimension of the smallest sub-coalgebra C(ϕ) of the Hopf algebra containing ϕ(x1),…,ϕ(xn). We show that ϕ is a tame automorphism if detJ(ϕ)∈C∗ and one of the following conditions is satisfied: (i) co(ϕ)≤n+2; (ii) ϕ is quadratic and co(ϕ)≤n+5.