Gröbner bases over fields with valuations

Abstract

Let $K$ be a field with a valuation and let $S$ be the polynomial ring $S:= K[x_1, \dots , x_n]$. We discuss the extension of Gröbner theory to ideals in $S$, taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Gröbner bases, but we also give examples showing that the resulting Gröbner bases can be substantially smaller than traditional Gröbner bases. In the case $K =\mathbb Q$ with the $p$-adic valuation the algorithms have been implemented in a Macaulay 2 package.