Grunwald-Wang theorem, an effective version

Grunwald–wang for Global Character Groups

Let F denote C or Q` for a prime ` 6= char(k), equipped with its natural topology, and let Gk = Gal(ks/k) for a choice of separable closure ks/k. For each place v of k choose a separable closure kv,s/kv and an embedding ks ↪→ kv,s over k ↪→ kv, thereby identifying Gkv with a decomposition group at v in Gk. It is surprising (but confirmed by consultation with experts) that the following natural ...

The Theorem of Grunwald -

The theorem of Grunwald-Wang in the setting of valuation theory

Given a field K with finitely many valuations; does there exist an extension of K which at these valuations has a prescribed local behavior? The Grunwald-Wang theorem answers this question in the case of abelian field extensions. Originally developed for algebraic number fields in the context of class field theory, it has turned out that it is valid quite generally, for arbitrary multi-valued f...

Towards an Effective Version of a Theorem of Stafford

Simplifying a matrix of differential operators by elementary row and column operations is quite a fundamental task. If one considers matrices whose entries are ordinary timevarying differential operators with rational function coefficients (or more generally meromorphic functions) then a classical theorem tells you that by elementary row and column operations an analogue to the Smith form can b...

An Effective Version of Kronecker’s Theorem on Simultaneous Diophantine Approximation

Kronecker’s theorem states that if 1, θ1, . . . , θn are real algebraic numbers, linearly independent over Q, and if α ∈ R, then for any > 0 there are q ∈ Z and p ∈ Z such that |qθi − αi − pi| < . Here, a bound on q is given in terms of the dimension n, of the precision , of the degree of the θi’s and of their height. A possible connection to the square-root sum problem is discussed.