Gr\"obner Fans of x-homogeneous Ideals in R[[t]][x]

TL;DR

This paper extends the concept of Gr"obner fans to certain ideals over power series rings with coefficients in R, providing algorithms and implementations for their computation, with applications to tropical geometry over p-adic numbers.

Contribution

It introduces the notion of initially reduced standard bases and proves the rational polyhedral structure of the Gr"obner fan in this context.

Findings

Established that the Gr"obner fan is a rational polyhedral fan.

Developed algorithms for computing the Gr"obner fan.

Implemented the algorithms in the Singular computer algebra system.

Abstract

We generalise the notion of Gr\"obner fan to ideals in R[[t]][x_1,...,x_n] for certain classes of coefficient rings R and give a constructive proof that the Gr\"obner fan is a rational polyhedral fan. For this we introduce the notion of initially reduced standard bases and show how these can be computed in finite time. We deduce algorithms for computing the Gr\"obner fan, implemented in the computer algebra system Singular. The problem is motivated by the wish to compute tropical varieties over the p-adic numbers, which are the intersection of a subfan of a Gr\"obner fan as studied in this paper by some affine hyperplane, as shown in a forthcoming paper.