Generally rational polynomials in two variables

TL;DR

This paper investigates the properties of generally rational polynomials in two variables over algebraically closed fields, extending known results from characteristic zero to arbitrary characteristic.

Contribution

It generalizes the characterization of generally rational polynomials to fields of any characteristic, providing new algebraic criteria.

Findings

Characterization of generally rational polynomials in arbitrary characteristic

Existence of G in k(X,Y) such that k(F,G)=k(X,Y) for these polynomials

Extension of known results from characteristic zero to positive characteristic

Abstract

Let k be an algebraically closed field. A polynomial F in k[X,Y] is said to be "generally rational" if, for almost all c in k, the curve " F= c '' is rational. It is well known that, if char(k)=0, F is generally rational iff there exists G in k(X,Y) such that k(F,G)=k(X,Y). We give analogous results valid in arbitrary characteristic.