Lines have the rather special property that they can be parametrized by polynomial functions x(t), y(t), indeed by functions x(t) = x0 + x1t, y(t) = y0 + y1t. Likewise, the parabola y2 = x can be parametrized as x(t) = t2, y(t) = t. However, this pattern soon breaks down.

Example 8.1 We claim that the conic x2 + y2 = 1 in ℝ2cannot be polynomially parametrized, in the sense that there do not exist nonconstant polynomials x = x(t), y = y(t) with x2+y2 = 1. (We suppress the variable t for concision.) Suppose that were possible. Differentiating this identity with respect to t, we would obtain a second identity x′x+y′y = 0, where the dash denotes differentiation with respect to t. Think of these identities as two linear equations in x, y. Thus we obtain xz = y′, yz = -x′ where z = xy′ - x′y. Write d, e, f for the degrees of the polynomials x(t), y(t), z(t). Assume z(t) is not zero, as a polynomial in t. Then, taking degrees in these relations, we obtain d + f = e - 1, e + f = d - 1 and hence f = -1, which is impossible. Thus z(t) is zero, implying that x′(t), y′(t) are zero, and hence that x(t), y(t) are constant, which is the required contradiction.

If we replace ‘polynomial’ functions by ‘rational’ functions, we obtain a much more useful concept, fundamental to the theory of algebraic curves.