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 770#770 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 190#190. The main idea comes from the following observation: Let 195#195 be an integer matrix, 771#771 a lattice basis of the integer kernel of 195#195. Assume that all components of 772#772 are positive. Then 773#773 i.e., the ideal on the right is already saturated w.r.t. all variables. The algorithm starts by finding a lattice basis 585#585 of the kernel of 190#190 such that 774#774 has no zero component. Let 775#775 be the set of indices 57#57 with 776#776. Multiplying the components 777#777 of 585#585 and the columns 777#777 of 190#190 by 778#778 yields a matrix 195#195 and a lattice basis 771#771 of the kernel of 195#195 that fulfill the assumption of the observation above. It is then possible to compute a generating set of 750#750 by applying the following “variable flip” successively to 779#779: Let 431#431 be an elimination ordering for 126#126. Let 780#780 be the matrix obtained by multiplying the 57#57-th column of 190#190 by 778#778. Let 781#781 be a Groebner basis of 782#782 w.r.t. 431#431 (where 126#126 is neither involved in 783#783 nor in 784#784). Then 785#785 is a generating set for 750#750.

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