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normal forms

Let $R = k[x_1, ..., x_n]$ be a polynomial ring over a field k, and let $I \subset{} R$ be an ideal. Let $\{g_1, ..., g_t\}$ be a Groebner basis for $I$. For any $f \in{} R$, there is a unique `remainder' $r \in{} R$ such that no term of $r$ is divisible by the leading term of any $g_i$ and such that $f-r$ belongs to $I$. This polynomial $r$ is sometimes called the normal form of $f$.

For an example, consider symmetric polynomials. The normal form of the symmetric polynomial f with respect to the ideal I below writes f in terms of the elementary symmetric functions a,b,c.

i1 : R = QQ[x,y,z,a,b,c,MonomialOrder=>Eliminate 3];
i2 : I = ideal(a-(x+y+z), b-(x*y+x*z+y*z), c-x*y*z)

o2 = ideal (- x - y - z + a, - x*y - x*z - y*z + b, - x*y*z + c)

o2 : Ideal of R
i3 : f = x^3+y^3+z^3

      3    3    3
o3 = x  + y  + z

o3 : R
i4 : f % I

      3
o4 = a  - 3a*b + 3c

o4 : R

See also