
C.6.2.1 The algorithm of Conti and Traverso
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The algorithm of Conti and Traverso (see [CoTr91])
computes $I_A$ via the
extended matrix $B=(I_m|A)$,
where $I_m$ is the $m\times m$ unity matrix. A lattice basis of $B$ is
given by the set of vectors $(a^j,-e_j)\in Z\!\!\! Z^{m+n}$, where $a^j$
is the $j$-th row of $A$ and $e_j$ the $j$-th coordinate vector. We
look at the ideal in $K[y_1,\ldots,y_m,x_1,\ldots,x_n]$ corresponding to
these vectors, namely
$$ I_1=<y^{a_j^+}- x_j y^{a_j^-} | j=1,\ldots, n>.$$
We introduce a further variable $t$ and adjoin the binomial $t\cdot
y_1\cdot\ldots\cdot y_m -1$ to the generating set of $I_1$, obtaining
an ideal $I_2$ in the polynomial ring $K[t,
y_1,\ldots,y_m,x_1,\ldots,x_n]$. $I_2$ is saturated w.r.t. all
variables because all variables are invertible modulo $I_2$. Now $I_A$
can be computed from $I_2$ by eliminating the variables
$t,y_1,\ldots,y_m$.

Because of the big number of auxiliary variables needed to compute a
toric ideal, this algorithm is rather slow in practice. However, it has
a special importance in the application to integer programming
(see Integer programming).

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