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A motivation to study Gr\"{o}bner theory for fields with valuations comesfrom tropical geometry, for example, they can be used to computetropicalization of varieties \citep{maclagan2009introduction}. Thecomputational aspect of this theory was first studied in (Chen \& Maclagan,2013). In this paper, we generalize this Gr\"obner basis theory to free modulesover polynomial rings over fields with valuation. As the valuation ofcoefficients is also taken into account while defining the initial term, we donot necessarily get a monomial order. To overcome this problem we have toresort to other techniques like the use of ecart function where the codomain isthe well-ordered set $\mathbb{N}$, and thereby give a method to calculate theGr\"{o}bner basis for submodules generated by homogeneous elements.

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