Geometric interpretation of valuated term (pre)orders

Abstract

Valuated term orders are studied for the purposes of Gr\"{o}bner theory over fields with valuation. The points of a usual tropical variety correspond to certain valuated terms preorders. Generalizing both of these, the set of all ``well-behaved'' valuated term preorders is canonically in bijection with the points of a space introduced in our previous work on tropical adic geometry. In this paper we interpret these points geometrically by explicitly characterizing them in terms of classical polyhedral geometry. This characterization gives a bijection with equivalence classes of flags of polyhedra as well as a bijection with a class of prime filters on a lattice of polyhedral sets. The first of these also classifies valuated term orders. The second bijection is of the same flavor as the bijections from [van der Put and Schneider, 1995] in non-archimedean analytic geometry and indicates that the results of that paper may have analogues in tropical adic geometry.