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 |

- Gröbner bases
- RingElement % Ideal -- normal form of ring elements and matrices