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

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] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. Crossref], Web of Science ®] , Google Scholar] 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] Adjamagbo, K. and van den Essen, A. 1990. A Resultant Criterion and Formula for the Inversion of a Polynomial Map in Two Variables. J. of Pure and Appl. Algebra., 64: 1–6. North-Holland Google Scholar] 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] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. Crossref], Web of Science ®] , Google Scholar].

Concerning question (2), Adjamagbo and Boury 2] Adjamagbo, K. and Boury, P. 1992. A Resultant Criterion and Formula for the Inversion of a Rational Map in Two Variables. J. of Pure and Appl. Algebra., 79: 1–13. North-Holland Google Scholar] 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] Abhyankar, S. S. 1990. Algebraic Geometry for Scientists and Engineers. Math. Surveys and Monographs., 5: 267–273.  Google Scholar], and secondly, to give a counter example where these λ₁, λ₂ are not in K ^*, contrary to claim of Yu 6] Yu, J.-T. 1993. Computing Minimal Polynomials and the Inverse via GCP. Comm. Algebra, 21(No.7): 2279–2294.  Google Scholar].