On affine tropical F5 algorithms

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Groebner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Groebner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Groebner bases. Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Groebner bases in an affine setting. We also provide a competitor with an adaptation of the F4 strategy to tropical Groebner bases computations. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm. We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over p-adics.