On pencils of polynomial curves

Abstract

Let k be a field. An irreducible F ϵ k[ X, Y] is called a polynomial curve if the plane curve that it determines is rational and has one rational place at infinity. In characteristic zero, it is well-known that those F which satisfy F − λ is called a polynomial curve for infinitely many λ ϵ k are precisely the variables of k[X, Y]. This paper gives an analogous characterization in positive characteristic.