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 750#750 without any auxiliary variables, provided that 190#190 contains a vector 346#346 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 585#585 is again computed via the LLL-algorithm. The saturation step is performed in the following way: First note that 346#346 induces a positive grading w.r.t. which the ideal 749#749 is homogeneous corresponding to our lattice basis. We use the following lemma: Let 251#251 be a homogeneous ideal w.r.t. the weighted reverse lexicographical ordering with weight vector 346#346 and variable order 767#767. Let 189#189 denote a Groebner basis of 251#251 w.r.t. this ordering. Then a Groebner basis of 768#768 is obtained by dividing each element of 189#189 by the highest possible power of 303#303. From this fact, we can succesively compute 769#769 in the 57#57-th step we take 126#126 as the smallest variable and apply the lemma with 126#126 instead of 303#303. This procedure involves 17#17 Groebner basis computations. Actually, this number can be reduced to at most 770#770(see [HoSh98]), and each computation -- except for the first one -- proves to be simple and fast in practice.

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