A note on higher derivations and ordinary points of curves
We prove the following theorem: Let k be an algebraically closed field of arbitrary characteristic. Let C denote a reduced curve in \(A^ n_ k\) and let p be a point of C. Let R denote the local ring of C at p and let \(\bar R\) denote the integral closure of R in its total quotient ring. Let \(M_ 1,...,M_ h\) be the branches of C at p. Then p is an ordinary point of C if and only if the following two conditions are satisfied: \((a) Der^ q_ k(R)\subseteq Der^ q_ k(\bar R)\) for all \(q\geq 1\); (b) For t a common uniformizing parameter of C in R, there exists an x in the maximal ideal of R such that \(\partial x/\partial t\) is a unit in \(\bar R\), and \((\partial x/\partial t)mod M_ i\neq(\partial x/\partial t)mod M_ j\) for all \(1\leq i<j\leq h.\)
scientific article; zbMATH DE number 3853223