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 752#752 without any auxiliary variables, provided that 191#191 contains a vector 348#348 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 587#587 is again computed via the LLL-algorithm. The saturation step is performed in the following way: First note that 348#348 induces a positive grading w.r.t. which the ideal 751#751 is homogeneous corresponding to our lattice basis. We use the following lemma: Let 253#253 be a homogeneous ideal w.r.t. the weighted reverse lexicographical ordering with weight vector 348#348 and variable order 769#769. Let 190#190 denote a Groebner basis of 253#253 w.r.t. this ordering. Then a Groebner basis of 770#770 is obtained by dividing each element of 190#190 by the highest possible power of 305#305. From this fact, we can succesively compute 771#771 in the 57#57-th step we take 126#126 as the smallest variable and apply the lemma with 126#126 instead of 305#305. This procedure involves 17#17 Groebner basis computations. Actually, this number can be reduced to at most 772#772(see [HoSh98]), and each computation -- except for the first one -- proves to be simple and fast in practice.

User manual for Singular version 4.4.1, 2025, generated by texi2html.