A resultant criterion and formula for the inversion of a rational map in two variables (Q1196788)

The authors prove the following resultant criterion of birationality of the rational mapping \(F=(F_ 1,F_ 2)\in K(X_ 1,X_ 2)^ 2\), \(F\notin K(X_ i)^ 2\), in two variables over a field \(K\): if \(F_ i=P_ i/Q_ i\), \(P_ i\), \(Q_ i\in K[X_ 1,X_ 2]\), \(P_ i,Q_ i\) are relatively prime, then \(F\) is birational if and only if there exist relatively prime nonzero \(R_ i,S_ i\in K[Y_ 1,Y_ 2]\), \((R_ i,S_ i)\notin K^ 2\), and \(\lambda_ i\in K[X_ i]\backslash\{0\}\) such that, for \(j\neq i\), \(\text{Res}_{X_ j}(P_ 1-Y_ 1Q_ 1,P_ 2-Y_ 2Q_ 2)=\lambda_ i(X_ i)(S_ iX_ i-R_ i)\). This result generalizes the one of \textit{K. Adjamagbo} and \textit{A. van den Essen} [J. Pure Appl. Algebra 64, No. 1, 1-6 (1990; Zbl 0704.13009)] for polynomial mappings. Moreover, there are corollaries of the theorem concerning: formulas for the inverse, degree of the inverse and algorithms for testing the birationality.