For a reduced monic Gröbner basis, the normal form of a polynomial is unique: it is a ``canonical form'' in the sense that two polynomials are equivalent when their normal forms are equal. This can be used to simplify mathematical expressions with respect to polynomial relations. An example from the Dutch Mathematics Olympiad of 1991, which is described in detail in , section 14.7.
Let be real numbers such that
> siderels := [ a+b+c=3, a^2+b^2+c^2=9, a^3+b^3+c^3=24 ]: > polys := map( lhs - rhs, siderels );
Next, we compute the Gröbner basis of these polynomials with respect to the pure lexicographic ordering .
> G := gbasis( polys, [a,b,c], plex );
The normal form of turns out to be 69.
normalf( a^4+b^4+c^4, G, [a,b,c], plex );
So, this number is the answer to the question. In Maple, this method is actually carried out when simplification with respect to side relations is requested.
> simplify( a^4+b^4+c^4, siderels, [a,b,c] );