C.6.2.3 The algorithm of Hosten and Sturmfels
The algorithm of Hosten and Sturmfels (see [HoSt95]) allows to
compute without any auxiliary variables, provided that contains a vector
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
is again computed via the
LLL-algorithm. The saturation step is performed in the following way:
First note that induces a positive grading w.r.t. which the ideal
corresponding to our lattice basis is homogeneous. We use the following
lemma:
Let be a homogeneous ideal w.r.t. the weighted reverse
lexicographical ordering with weight vector and variable order
. Let denote a Groebner basis of w.r.t. to
this ordering. Then a Groebner basis of
is obtained by
dividing each element of by the highest possible power of .
From this fact, we can succesively compute
in the -th step we take as the cheapest variable and apply the
lemma with instead of .
This procedure involves Groebner basis computations. Actually, this
number can be reduced to at most (see [HoSh98]), and the single
computations -- except from the first one -- show to be easy and fast in
practice.
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