Application

Suppose v₁, ..., v_n R[X,Y] is a set of polynomials. Then

v₁(x,y) = ··· = v_n(x,y) = 0

is a set of equations with unknown x, y R. Now, for any polynomial f R[X,Y], we also have f(x,y) = 0.

The reason is that f ({v₁, ..., v_n}) can be written as r₁v₁ + r₂ v₂ + ··· + r_nv_n for suitable r₁, ..., r_n R[X,Y], so that

f(x,y) = r₁v₁(x,y) + r₂v₂(x,y) + ··· + r_nv_n(x,y) = 0.

This means that we can try and derive a lot of "easier" equations from the given one as a first step to solve the set of equations.

For example, suppose we have v₁ = X²Y - 1 and v₂ = XY² - 1, so that the system of equations is

x²y = 1,         xy² = 1.

Then also f = Yv₁ - Xv₂ = X - Y belongs to the ideal generated by v₁ and v₂, and so we also have x = y.

Substituting this result in v₁(x,y) = 0, we find x³ = 1, which is readily solved.

Of course, ad hoc methods may lead to the same result here. The indicated method however is part of an algorithm that works in all cases to bring the set of equations in a better form.