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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 $n/2$ 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 $A$.

The main idea comes from the following observation:

Let $B$ be an integer matrix, $u_1,\ldots,u_r$ a lattice basis of the integer kernel of $B$. Assume that all components of $u_1$ are positive. Then

\begin{displaymath}I_B=<x^{u_i^+}-x^{u_i^-}\vert i=1,\ldots,r>, \end{displaymath}

i.e., the ideal on the right is already saturated w.r.t. all variables.

The algorithm starts by finding a lattice basis $v_1, \ldots, v_r$ of the kernel of $A$ such that $v_1$ has no zero component. Let $\{i_1,\ldots,i_l\}$ be the set of indices $i$ with $v_{1,i}<0$. Multiplying the components $i_1,\ldots,i_l$ of $v_1, \ldots, v_r$ and the columns $i_1,\ldots,i_l$ of $A$ by $-1$ yields a matrix $B$ and a lattice basis $u_1,\ldots,u_r$ of the kernel of $B$ that fulfill the assumption of the observation above. It is then possible to compute a generating set of $I_A$ by applying the following ``variable flip'' successively to $i=i_1,\ldots,i_l$:

Let $>$ be an elimination ordering for $x_i$. Let $A_i$ be the matrix obtained by multiplying the $i$-th column of $A$ by $-1$. Let

\begin{displaymath}\{x_i^{r_j} x^{a_j} - x^{b_j} \vert j\in J \}\end{displaymath}

be a Groebner basis of $I_{A_i}$ w.r.t. $>$ (where $x_i$ is neither involved in $x^{a_j}$ nor in $x^{b_j}$). Then

\begin{displaymath}\{x^{a_j} - x_i^{r_j} x^{b_j} \vert j\in J \}\end{displaymath}

is a generating set for $I_A$.
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