normal forms

Let R = k[x₁, ..., x_n] be a polynomial ring over a field k, and let I ⊂ R be an ideal. Let g₁, ..., g_t be a Groebner basis for I. For any f ∈ R, there is a unique ‘remainder’ r ∈ 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