Automorphisms from face polynomials via two Gröbner bases
It is now a well-known fact that a polynomial automorphism \(F = (F_ 1, \dots, F_ n) : k^ n \to k^ n\), where \(k\) is a field, \(n \geq 2\) is completely determined by its face polynomials \(F_ i |_{X_ j = 0}\), \(i,j=1,\dots,n\). Several proofs were given, by McKay and Wang and Li. Explicit formulae to reconstruct the automorphism for the case \(n=2\) were given by McKay and Wang and Adjamagbo and van den Essen. In J. Pure Appl. Algebra 80, No. 3, 327-336 (1992; Zbl 0763.14004), \textit{A. van den Essen} and the author show that these formulae do not always work in higher dimensions. Instead, an algorithm for reconstructing the inverse, based on a theorem from \textit{A. van den Essen} [Commun. Algebra 18, No. 10, 3183-3186 (1990; Zbl 0718.13008)], is given there. That algorithm involves \(n+1\) Gröbner basis computations. The main theorem presented here expresses the inverse of \(F\) in terms of the reduced Gröbner basis of an ideal constructed from face polynomials. Thus, it leads to an algorithm for the computation of \(F\) which involves only two Gröbner basis computations.
scientific article; zbMATH DE number 149934