Lemma 14.2.3. Let a be a word ordering on (X>, and let a be the restriction of a to (X>. Then also a is a word ordering.

Now we are ready to state and prove the main theorem of this section.

Theorem 14.2.4 (Computation of Elimination Ideals). LetL c X, let a be an elimination ordering for L, and let a be the restriction of a to (X>. Given a two-sided ideal I c K (X> and a a-Gröbner basis G of I, the set G = G n K(X> isaa-Gröbner basis of the elimination ideal ^{7} = I n K(X>.

Proof. It is clear that G is contained in 7 We have to show that the leading words of the polynomials in G generate the leading word ideal Lw_{5}(/). For f ϵ ^{7} \{0|,we have Lw_{a} (f) = Lw_{a} (f) ϵ (X). Since G is a a -Gröbner basis of I, we find words w_{1}, w_{2} ϵ (X> and g ϵ G such that ...

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