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C.6.2.4 The algorithm of Di Biase and Urbanke

Like the algorithm of Hosten and Sturmfels, the algorithm of Di Biase and Urbanke (see [DBUr95]) performs up to n∕2 Groebner basis computations. It needs no auxiliary variables, but a supplementary precondition; namely, the existence of a vector without zero components in the kernel of A.

The main idea comes from the following observation:

Let B be an integer matrix, u1,,ur a lattice basis of the integer kernel of B. Assume that all components of u1 are positive. Then

       u+    u−
IB =< x i − x i |i = 1,...,r >,
i.e., the ideal on the right is already saturated w.r.t. all variables.

The algorithm starts by finding a lattice basis v1,,vr of the kernel of A such that v1 has no zero component. Let {i1,,il} be the set of indices i with v1,i < 0. Multiplying the components i1,,il of v1,,vr and the columns i1,,il of A by 1 yields a matrix B and a lattice basis u1,,ur of the kernel of B that fulfill the assumption of the observation above. We are then able to compute a generating set of IA by applying the following “variable flip” successively to i = i1,,il:

Let > be an elimination ordering for xi. Let Ai be the matrix obtained by multiplying the i-th column of A with 1. Let

  rj a    b
{xi xj − xj|j ∈ J}
be a Groebner basis of IAi w.r.t. > (where xi is neither involved in xaj nor in xbj). Then
{xaj − xrjxbj|j ∈ J}
       i
is a generating set for IA.


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