Schanuel property for additive power series (Q1760406)

The paper gives an analogue in finite characteristic \(p\) of Ax's theorems on the functional analogue of the Schanuel conjecture. The field of complex numbers is replaced by the completion \(\mathbb F_p[[Fr]]\) of the ring \(\mathbb F_p[Fr] = End(\mathbb G_{a/\mathbb F_p})\) generated by the Frobenius (with composition), and the exponential map by an additive power series \(F\), assumed to be transcendental over \(\mathbb F_p[Fr]\), or algebraic of sufficiently large degree. If \(x_1, ..., x_n\) are power series without constant terms, and linearly independent over \(\mathbb F_p[Fr]\), it is then shown that the \(x_i\)'s and their images \(F(x_i)\) under \(F\) generate over \(\mathbb F_p\) a field of transcendence degree at least \(n+1\). The proof is based on the study of higher jet spaces, which replace the standard differential forms used by Ax.