Application

Suppose v₁, ..., v_n R[X,Y] is a set of polynomials and consider the corresponding set of equations (cf. the application of the theorem on the previous page).

v₁(x,y) = ··· = v_n(x,y) = 0

with unknown x, y R.

If 1 belongs to the ideal generated by the v_i, then there are no solutions.

For then 1 can be written as r₁v₁ + r₂v₂ + ··· + r_nv_n for suitable r₁, ..., r_n R[X,Y], so that the existence of a solution (x,y) R² would lead to

1 = 1(x,y) = r₁v₁(x,y) + r₂v₂(x,y) + ··· +r_nv_n(x,y) = 0,

a contradiction.

For example, suppose we have

v₁ = X²Y - 1,         v₂ = XY² - 1,         v₃ = X - Y - 1.

Then also f = Yv₁ - Xv₂ - v₃ = 1, and so the system

x²y = 1,           xy² = 1,          x - y = 1

has no solutions.