ON RESULTANT CRITERIA AND FORMULAS FOR THE INVERSION OF A POLYNOMIAL MAP

Abstract

Concerning the inversion of a polynomial map F: K ² ↦ K ² over an arbitrary field K, it is natural to consider the following questions: (1) Can we find a necessary and sufficient criterion in terms of resultants for F to be invertible with polynomial ((2) resp. rational) inverse such that, this criterion gives an explicit formula to compute the inverse of F in this case? MacKay and Wang [5] gave a partial answer to question (1), by giving an explicit expression of the inverse of F, when F is invertible without constant terms. On the other hand, Adjamagbo and van den Essen [3] have fully answered question (2) and have furnished a necessary and sufficient criterion which relies on the existence of some constants λ₁, λ₂ in K ^*. We improve this result by giving an explicit relation between λ₁, λ₂ and constants of the Theorem of MacKay and Wang [5].

Concerning question (2), Adjamagbo and Boury [2] give a criterion for rational maps which relies on the existence of two polynomials λ₁, λ₂. We also improve this result, by expliciting the relations between these λ₁, λ₂ and the coefficients of F. This improvement enables us, first to give an explicit proof of the corresponding Theorem of Abhyankhar [1], and secondly, to give a counter example where these λ₁, λ₂ are not in K ^*, contrary to claim of Yu [6].

Acknowledgments