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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 thatthe Gr\"obner fan is a rational polyhedral fan. For this we introduce thenotion of initially reduced standard bases and show how these can be computedin finite time. We deduce algorithms for computing the Gr\"obner fan,implemented in the computer algebra system Singular. The problem is motivatedby the wish to compute tropical varieties over the p-adic numbers, which arethe intersection of a subfan of a Gr\"obner fan as studied in this paper bysome affine hyperplane, as shown in a forthcoming paper.

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