Arithmetic properties related to the shuffle-product

1: Properties of the shuffle product in positive characteristic suggest to consider a p−homogeneous form σ : Fp〈〈X1, . . . ,Xk〉〉 −→ Fp〈〈X1, . . . ,Xk〉〉 on the vector space Fp〈〈X1, . . . ,Xk〉〉 of formal power series in k free noncommuting variables. The form σ preserves rational elements in Fp〈〈X1, . . . ,Xk〉〉, algebraic series of Fp[[X]] and induces a bijection on the affine subspace 1+m of formal power series with constant coefficient 1. Conjecturally, this bijection restricts to a bijection of rational elements in 1+m ⊂ Fp〈〈X1, . . . ,Xk〉〉, respectively algebraic elements in 1 +XFp[[X]].