Homogeneous Cyclotomic Polynomials and Rationality of Curves

Let k be a field of arbitrary characteristic. Suppose that f1, . . . , fn are polynomials in k[t] and di = deg(fi). We prove that, if gcd of d1, . . . , dn is 1, then k(f1, . . . , fn) = f(t), or equivalently, the morphism ψ = (f1, . . . , fn) : Ak → A n k is proper and birational onto its image. By combining this result with the epimorphism theorem of Abhyankar and Moh, we prove that, if f and g are nonlinear polynomials in k[t] with relatively prime degrees, then the curve C ⊂ A given parametrically by x = f(t), y = g(t) is a rational curve not isomorphic to the affine line. To prove these results, we introduce homogeneous cyclotomic polynomials in the bivariate polynomial ring k[s, t] and explore their properties.