Schanuel property for additive power series

Abstract

Abstract. We prove a version of Schanuel's Conjecture for a field of Laurent power series in positive characteristic replacing C and a non-algebraic additive power series replacing the exponential map.

Ax–schanuel Condition in Arbitrary Characteristic

“…The aim of this line of research is to find general geometric reasons for such Ax-Schanuel statements (with no restrictions for the characteristic of the base field). It is easy to see that [16,Theorem 1.1] would follow from an arbitrary characteristic version of Theorem 1.2 (in the same way as in [2]). We formulate it as a question below.…”

“…One can state similar (to Theorem 1.3) Ax-Schanuel inequalities in the case of positive characteristic. Actually, we have obtained in [16] an Ax-Schanuel statement for a 'sufficiently non-algebraic' additive power series (so A = B = G n a ) in the positive characteristic case. The aim of this line of research is to find general geometric reasons for such Ax-Schanuel statements (with no restrictions for the characteristic of the base field).…”

“…Thus we need to take C = F p to guarantee that S is commutative. This case is analyzed in [16]. Below is our transcendental statement about formal endomorphisms.…”

“…The main and only result of this subsection is a generalization of [16,Proposition 2.5] from the case of a vector group to the case of an arbitrary commutative algebraic group. It is also a desired improvement of Proposition 2.9, which we are able to show only under the A-limit assumption.…”

“…• The first (to my knowledge) positive characteristic version of [1,(SP)] is [16,Theorem 1.1], where the exponential map is replaced with a non-algebraic additive power series.…”