Elimination of variables

Given Boolean generators $ G$ of an ideal $ I\subset \mathbb{Z}_2[x_1,\ldots,x_n,y_1,\ldots,y_m]/\langle x_1^2+x_1,\ldots,x_n^2+x_n,y_1,\ldots,y_m \rangle$ we would like to compute a generating system $ H$ of Boolean polynomials, where

$\displaystyle \langle H\rangle=\{p\in I\vert p$    can be represented by a (Boolean) polynomial in $\displaystyle \mathbb{Z}_2[y_1,\ldots,y_m]\}.$

This can be done as in the classical case despite the field equations using an elimination ordering for $ x_1, \ldots, x_n$

Definition 1 (Elimination orderings)   Let $ R=\mathbb{Z}_2[x_1,\ldots, x_n, y_1, \ldots y_m]$ . An ordering ``$ >$ '' is called an elimination ordering of $ x_1, \ldots, x_n$ , if $ x_i>t$ for every monomial $ t$ in  $ K[y_1, \ldots, y_m]$ and every $ i=1,\ldots,n$ .



Subsections

2010-09-14