Additive polynomials and the elementary properties of 𝔽_{p}((t))

by Franz-Viktor Kuhlmann, Saskatoon

𝔽_{p}((t)) denotes the field of formal Laurent series over the field with p elements; it carries the t-adic valuation v_{t}. A well known open problem in model theoretic algebra is to find a complete recursive axiom system for the elementary theory of 𝔽_{p}((t)). This would yield that it is decidable. A similar result was shown by Ax–Kochen and Ershov for the p-adics ℚ_{p}, v_{p}. An adaptation of their axiom system to the case of 𝔽_{p}((t)) is: (∗) “henselian defectless valued field of characteristic p with value group a ℤ-group and residue field 𝔽_{p}”.

Theorem 1 (K., 1989) This axiom system is not complete. While

K = O_{K} + ℘(K) + tK^{p} + … + t^{p−1}K^{p}  (1)

(with O_{K} the valuation ring and ℘(X) = X^{p} − X) holds for K = 𝔽_{p}((t)), there is an extension of 𝔽_{p}((t)) of transcendence degree 1, with v(t) still the least positive element in the value group, which satisfies (∗) but not (1).

As ℘(K), tK^{p}, …, t^{p−1}K^{p} are the images of K under the additive polynomials ℘(X), tX^{p}, …, t^{p−1}X^{p}, the question arises what one can say elementarily about subgroups of (K, +) of the form

f_{1}(K) + … + f_{n}(K)  (2)

for any additive polynomials f_{1} , …, f_{n} (a polynomial f is called additive if f(a + b) = f(a) + f(b) for all a, b). Definition: Let (K, v) be a valued field and S a subset of K. We say that S has the optimal approximation property (OA) if each x ∈ K has an optimal approximation from S w.r.t. the v-metric, that is, there is some (in general non-unique) closest point in S. This property is elementary if S is definable.

Theorem 2 (K., 1998) There is an elementary condition for the coefficients of f_{1} , …, f_{n} such that (2) has (OA) in every maximally valued field (K, v) which contains the coefficients.

Using the local compactness of 𝔽_{p}((t)), we can do better:

Theorem 3 (van den Dries and K., 1999) In 𝔽_{p}((t)), (2) has (OA) for all choices of f_{1} , …, f_{n}.

Quantifying over the coefficients of all additive polynomials, we obtain a recursive elementary axiom scheme (S) which holds in 𝔽_{p}((t)). Question: Is (∗) + (S) complete?