
C.6.2.3 The algorithm of Hosten and Sturmfels
.............................................


The algorithm of Hosten and Sturmfels (see [HoSt95]) allows to
compute $I_A$ 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 $v_1,\ldots,v_r$ 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
$$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$
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 $x_1
> x_2 > \ldots > x_n$. Let $G$ denote a Groebner basis of $I$ w.r.t. to
this ordering.  Then a Groebner basis of $(I:x_n^\infty)$ is obtained by
dividing each element of $G$ by the highest possible power of $x_n$.

From this fact, we can successively compute
$$ I_A= I:(x_1\cdot\ldots\cdot x_n)^\infty
=(((I:x_1^\infty):x_2^\infty):\ldots :x_n^\infty); $$
in the $i$-th step we take $x_i$ as the cheapest variable and apply the
lemma with $x_i$ instead of $x_n$.

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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