An algorithm to find a coordinate’s mate

Abstract

In th is paper an algorithm is given to decide if a given polynom ial in two variables w ith coefficients in a finitely generated K-algebra is a coordinate and if so, find a m ate for th is polynomial.

Triangulable locally nilpotent derivations in dimension three

“…Indeed, it is algorithmically possible to check whether a given polynomial in two variables is a coordinate, see e.g. [1,4,11,26]. We will use the algorithm given in [11], but it is worth mentioning that from the complexity point of view the algorithm given in [26] is the most efficient as reported in [22].…”

The fourteenth problem of Hilbert for polynomial derivations

“…Proposition 3.1. Assume that k contains two nonzero elements a and b such that 2 1 , bx 3 It is known ( [51], [15]) that if k is a field then every Dedekind k-subalgebra of k[X] is a polynomial ring in one variable over k. As a consequence of this fact we obtain…”

UFDs with commuting linearly independent locally nilpotent derivations

“…In case n = 2 this problem was solved by the famous Abhyankar-Moh theorem [1] which states that f is a coordinate if and only if I(∂ x f, ∂ y f ) = (1) and the curve defined by f = 0 has one place at infinity. Since then, various solutions have been given to this problem, see, e.g., [2][3][4][5]7,16,18]. For n 3 the problem remains open so far, and a well-known Conjecture of Abhyankar and Sathaye states that any time K[x]/f is K-isomorphic to K [n−1] , f is a coordinate in K [x].…”