A Jacobian criterion for separability

Abstract

Contents . 1. Introduction and Notation ; 2. A Jacobian Criterion for Separability ; 3. A Jacobian Criterion for Extension of Derivations ; 4. Explicit Formulas for the Separability Idempotent ; 4.1. 1-variable, 1-relation case and a theorem of Euler, 4.2. 1-variable, ¦ I ¦-relation case, 4.3. Symmetric expansions and n -algebras, 4.4. n -variable, n -relation case where the base ring contains the rational number field, 4.5. n -variable, ¦ I ¦-relation case where the base ring contains the rational number field, 4.6. n -variable, n -relation case where 2 is a unit, the relations are quadratic and the Jacobian determinant is in the base ring; 5. Applications to the Jacobian Problem ; 5.1. Brief history, 5.2. Interpretations of the Jacobian condition, 5.3. Consequences of flatness, 5.4. Equivalent conditions under separability, 5.5. Pseudo-inverse polynomials, 5.6. Counter-examples, 5.7. Quadratic case.