Let K be a field equipped with valuation. Tropical varieties over can defined theory of Gröbner bases taking into account the valuation K. Because use valuation, tropical has proved to provide settings for computations polynomial rings p-adic that are more stable than classical bases. Beforehand, these strategies were only available homogeneous polynomials. In this article, we extend F5 strateg...

Abstract Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any valuation whose group is not closed in divisible hull definable language of rings, using one parameter. Thereby strengthen known results. Moreover, this case, parameter optimal sense cannot obtain parameters. To end, present construction meth...

Building on the work of K. Chiba (J. Algebra 263 (2003), 75– 87), we present sufficient conditions for the completion of a division ring with respect to the metric defined by a discrete valuation function to contain a free field, i.e. the universal field of fractions of a free associative algebra. Several applications to division rings generated by torsion-free nilpotent groups, skew Laurent se...

We present the project of developing computational algebra inside type theory in an integrated way. As a rst step towards this, we present direct constructive proofs of Dickson's lemma and Hilbert's basis theorem, and use this to prove the constructive existence of Grr obner bases. This can be seen as an integrated development of the Buchberger algorithm, and so far we have a concise formalisat...

Exponent matrices appeared in the study of tiled orders over discrete valuation rings. Many properties of such orders are formulated using this notion. We think that such matrices are of interest in them own right, in particular, it is convenient to write finite partially ordered sets (posets) and finite metric spaces as special exponent matrices. Note that when we defined a quiver Q(E) of a re...

Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k[x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q → k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k(x, y). Although the value groups correspondi...