2010/01/25 by Vivien Ripoll, Ripoll, Vivien
Mathematics · #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR) #math.AC #math.CO #math.GR
paper · pdf · doi:10.48550/arxiv.1001.4470
Version 3: 12 pages, corrected typos, updated references
arxiv created 2010/12/23 · arxiv updated 2010/12/24
Let A be a polynomial algebra with complex coefficients. Let B be a finite extension ring of A which is also a polynomial algebra. We describe the factorisation of the Jacobian J of the extension into irreducibles. We also introduce the notion of a well-ramified extension and define its discriminant polynomial D. In the particular case where A is the ring of invariants of B under the action of a group (i.e., a Galois extension), this framework corresponds to the classical invariant theory of complex reflection groups. In the more general case of a well-ramified extension, we explain how the pair (D,J) behaves similarly to a Galois extension. This work can be viewed as the first step towards a possible invariant theory of "virtual reflection groups".