A proof of A. Gabrielov’s rank theorem
Copyright policy )A proof of A. Gabrielov’s rank theorem
This article contains a complete proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung theorem. We include, furthermore, new extensions of the rank theorem (concerning the Zariski main theorem and elimination theory) to commutative algebra.
- Sorbonne University France
- Sorbonne Paris Cité France
- University of Paris France
- French National Centre for Scientific Research France
- UNIVERSITE PARIS DESCARTES France
Primary 13J05, 32B05, Secondary 12J10, 13A18, 13B35, 14B05, 14B20, 30C10, 32A22, 32S45, Analytic algebras and generalizations, preparation theorems, Commutative Algebra (math.AC), Modifications; resolution of singularities (complex-analytic aspects), [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], Mathematics - Algebraic Geometry, Polynomials and rational functions of one complex variable, FOS: Mathematics, rank of an analytic map, Complex Variables (math.CV), local analytic geometry, Algebraic Geometry (math.AG), Power series rings, Abhyankar-Jung's theorem, Completion of commutative rings, Mathematics - Complex Variables, Weierstrass preparation theorem, Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Mathematics - Commutative Algebra, formal power series, Nevanlinna theory; growth estimates; other inequalities of several complex variables, Valued fields, Formal neighborhoods in algebraic geometry
Primary 13J05, 32B05, Secondary 12J10, 13A18, 13B35, 14B05, 14B20, 30C10, 32A22, 32S45, Analytic algebras and generalizations, preparation theorems, Commutative Algebra (math.AC), Modifications; resolution of singularities (complex-analytic aspects), [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], Mathematics - Algebraic Geometry, Polynomials and rational functions of one complex variable, FOS: Mathematics, rank of an analytic map, Complex Variables (math.CV), local analytic geometry, Algebraic Geometry (math.AG), Power series rings, Abhyankar-Jung's theorem, Completion of commutative rings, Mathematics - Complex Variables, Weierstrass preparation theorem, Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Mathematics - Commutative Algebra, formal power series, Nevanlinna theory; growth estimates; other inequalities of several complex variables, Valued fields, Formal neighborhoods in algebraic geometry
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