Gröbner Fans of x-homogeneous Ideals in R[[t]][x]
Thomas Markwig, Yue Ying Ren · arXiv (Cornell University) · 2015
We generalise the notion of Gröbner fan to ideals in R[[t]][x_1,...,x_n] for certain classes of coefficient rings R and give a constructive proof that the Gröbner 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 Gröbner 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 Gröbner fan as studied in this paper by some affine hyperplane, as shown in a forthcoming paper.