Singular Manual: Di Biase and Urbanke

| #### C.6.2.4 The algorithm of Di Biase and Urbanke Like the algorithm of Hosten and Sturmfels, the algorithm of Di Biase and Urbanke (see [DBUr95]) performs up to 772#772 Groebner basis computations. It needs no auxiliary variables, but a supplementary precondition; namely, the existence of a vector without zero components in the kernel of 191#191. The main idea comes from the following observation: Let 196#196 be an integer matrix, 773#773 a lattice basis of the integer kernel of 196#196. Assume that all components of 774#774 are positive. Then 775#775 i.e., the ideal on the right is already saturated w.r.t. all variables. The algorithm starts by finding a lattice basis 587#587 of the kernel of 191#191 such that 776#776 has no zero component. Let 777#777 be the set of indices 57#57 with 778#778. Multiplying the components 779#779 of 587#587 and the columns 779#779 of 191#191 by 780#780 yields a matrix 196#196 and a lattice basis 773#773 of the kernel of 196#196 that fulfill the assumption of the observation above. It is then possible to compute a generating set of 752#752 by applying the following “variable flip” successively to 781#781: Let 433#433 be an elimination ordering for 126#126. Let 782#782 be the matrix obtained by multiplying the 57#57-th column of 191#191 by 780#780. Let 783#783 be a Groebner basis of 784#784 w.r.t. 433#433 (where 126#126 is neither involved in 785#785 nor in 786#786). Then 787#787 is a generating set for 752#752.

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