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C.6.2.2 The algorithm of Pottier

The algorithm of Pottier (see [Pot94]) starts by computing a lattice basis $v_1,
\ldots, v_r$ for the integer kernel of $A$using the LLL-algorithm ( system). The ideal corresponding to the lattice basis vectors

\begin{displaymath}I_1=<x^{v_i^+}-x^{v_i^-}\vert i=1,\ldots,r> \end{displaymath}

is saturated - as in the algorithm of Conti and Traverso - by inversion of all variables: One adds an auxiliary variable $t$ and the generator $t\cdot x_1\cdot\ldots\cdot x_n -1$ to obtain an ideal $I_2$ in $K[t,x_1,\ldots,x_n]$ from which one computes $I_A$ by elimination of $t$.