
C.6.1 Toric ideals
------------------

Let $A$ denote an $m\times n$ matrix with integral coefficients. For $u
\in Z\!\!\! Z^n$, we define $u^+,u^-$ to be the uniquely determined
vectors with nonnegative coefficients and disjoint support (i.e.,
$u_i^+=0$ or $u_i^-=0$ for each component $i$) such that
$u=u^+-u^-$. For $u\geq 0$ component-wise, let $x^u$ denote the monomial
$x_1^{u_1}\cdot\ldots\cdot x_n^{u_n}\in K[x_1,\ldots,x_n]$.

The ideal
$$ I_A:=<x^{u^+}-x^{u^-} | u\in\ker(A)\cap Z\!\!\! Z^n>\ \subset
K[x_1,\ldots,x_n] $$
is called a \bf toric ideal. \rm

The first problem in computing toric ideals is to find a finite
generating set: Let $v_1,\ldots,v_r$ be a lattice basis of $\ker(A)\cap
Z\!\!\! Z^n$ (i.e, a basis of the $Z\!\!\! Z$-module). Then
$$ I_A:=I:(x_1\cdot\ldots\cdot x_n)^\infty $$
where
$$ I=<x^{v_i^+}-x^{v_i^-}|i=1,\ldots,r> $$


The required lattice basis can be computed using the LLL-algorithm (see [Coh93]). For the computation of the saturation, there are various
possibilities described in the
section Algorithms.

* Algorithms::             Various algorithms for computing toric ideals.
* Buchberger algorithm::   Specializing it for toric ideals.

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