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C.6.2.3 The algorithm of Hosten and Sturmfels

The algorithm of Hosten and Sturmfels (see [HoSt95]) allows to compute IA without any auxiliary variables, provided that A contains a vector w with positive coefficients in its row space. This is a real restriction, i.e., the algorithm will not necessarily work in the general case.

A lattice basis v1,,vr is again computed via the LLL-algorithm. The saturation step is performed in the following way: First note that w induces a positive grading w.r.t. which the ideal

      v+   v−
I = < x i − x i |i = 1,...,r >
corresponding to our lattice basis is homogeneous. We use the following lemma:

Let I be a homogeneous ideal w.r.t. the weighted reverse lexicographical ordering with weight vector w and variable order x1 > x2 > > xn. Let G denote a Groebner basis of I w.r.t. to this ordering. Then a Groebner basis of (I : xn) is obtained by dividing each element of G by the highest possible power of xn.

From this fact, we can successively compute

                  ∞         ∞    ∞        ∞
IA = I : (x1 ⋅...⋅xn) = (((I : x1 ) : x2 ) : ...: xn );
in the i-th step we take xi as the cheapest variable and apply the lemma with xi instead of xn.

This procedure involves n Groebner basis computations. Actually, this number can be reduced to at most n∕2 (see [HoSh98]), and the single computations – except from the first one – show to be easy and fast in practice.


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