Use normal forms when you want to reduce an element modulo an ideal, compare two expressions modulo an ideal, or test membership by checking whether the remainder is zero. In Macaulay2 the operator
RingElement % Ideal computes this remainder using a Gröbner basis.
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 Gröbner 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];
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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
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i3 : f = x^3+y^3+z^3
3 3 3
o3 = x + y + z
o3 : R
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i4 : f % I
3
o4 = a - 3a*b + 3c
o4 : R
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