Singular Manual: Hosten and Sturmfels

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 $v_1, \ldots, v_r$ 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

$\begin{displaymath}I=<x^{v_i^+}-x^{v_i^-}\vert i=1,\ldots,r> \end{displaymath}$

is homogeneous corresponding to our lattice basis. We use the following lemma:

Let be a homogeneous ideal w.r.t. the weighted reverse lexicographical ordering with weight vector and variable order $x_1 > x_2 > \ldots > x_n$ . Let denote a Groebner basis of w.r.t. this ordering. Then a Groebner basis of $(I:x_n^\infty)$ is obtained by dividing each element of by the highest possible power of .

From this fact, we can succesively compute

$\begin{displaymath}I_A= I:(x_1\cdot\ldots\cdot x_n)^\infty =(((I:x_1^\infty):x_2^\infty):\ldots :x_n^\infty); \end{displaymath}$

in the

-th step we take

as the smallest 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 each computation -- except for the first one -- proves to be simple and fast in practice.

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