Grobner fans of x-homogeneous ideals in R [t] [x]

摘要

We generalise the notion of a Gobner fan to ideals in R [t] [x(1) . . . , x(n)] for certain classes of coefficient rings R and give a constructive proof that the Grobner 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 Grobner fan, implemented in the computer algebra system SINGULAR. The problem is motivated by the wish to compute tropical varieties over the p-adic numbers. (C) 2016 Elsevier Ltd. All rights reserved.