Gr\"{o}bner Basis Theory for Modules over Polynomial Rings over Fields with Valuation

A motivation to study Gr\"{o}bner theory for fields with valuations comes from tropical geometry, for example, they can be used to compute tropicalization of varieties \citep{maclagan2009introduction}. The computational aspect of this theory was first studied in (Chen \& Maclagan, 2013). In this paper, we generalize this Gr\"obner basis theory to free modules over polynomial rings over fields with valuation. As the valuation of coefficients is also taken into account while defining the initial term, we do not necessarily get a monomial order. To overcome this problem we have to resort to other techniques like the use of ecart function where the codomain is the well-ordered set $\mathbb{N}$, and thereby give a method to calculate the Gr\"{o}bner basis for submodules generated by homogeneous elements.