Introduction. Let k be an algebraically closed field of characteristic zero. Let β[#,;y] be a polynomial ring of two variables and let Al= Spec(k[x,y]). Embed Al into the projective plane Pf as the complement of a line /«,. Let /^k[x,y] be an irreducible polynomial, let FΛ be the curve on Al defined by f=a for every a^k and let CΛ be the closure of FΛ in Pf. Then the set Λ(/): — {CΛ; a^kU (°°)} is a linear pencil on Pi defined by/, where Cco^dL, d being the degree of/. The set Λ0(/):— {FΛ; a^k} is called the linear pencil on Al defined by /. The polynomial / is called generically rational when the general members of Λ(/) (or Λ0(/)) are irreducible rational curves. Since the algebraic function field k(x, y) of one variable over the subfield k(f) then has genus 0, Tsen's theorem says that / is generically rational if and only if / is a field generator in the sense of Russell [9, 10], i.e., there is an element g^k(x,y) such that ^(tfjj)=&(/£). If / is a generically rational polynomial, we can associate with / a non-negative integer n, where n-\-1 is the number of places at infinity of a general member FΛ of Λ0(/), i.e., the number of places of FΛ whose centers lie outside Al. If d^ 1, the pencil Λ(/) has base points situated outside Al. Let φ: W-+PI be the shortest succession of quadratic transformations with centers at the base points (including infinitely near base points) of Λ(/) such that the proper transform Λ' of Λ(/) by φ has no base points. Then the linear pencil Λ' defines a morphism p: W—*Pk9 whose general fibers are the proper transforms of general members of Λ(/); thence they are nonsingular rational curves by virtue of Bertini's theorem. Moreover, W contains in a canonical way an open subset isomorphic to Al. A generically rational polynomial / is said to be of simple type if the morphism p has n+l cross-sections contained in the boundary set W-Al (cf. Definition 1.8, below). If n=0, a generically rational polynomial / is sent to one of the coordinates x, y of Al by a biregular automorphism of Al (cf. Abhyankar-Moh's theorem [1, 4]); hence/ is of simple type. If w=l, a generically rational polynomial is always of simple type (cf. Theorem 2.3, below). However, if n>l, a generi-