The Abhyankar-Jung Theorem

2011/03/13 by Parusinski, Adam, Rond, Guillaume · 1 citation
#26E10 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary: 13F25. Secondary: 13J15

paper · doi:10.48550/arxiv.1103.2559

Abstract

We show that every quasi-ordinary Weierstrass polynomial P(Z) = Z^d+a₁ (X) Z^d-1+...+a_d(X) ∈ \K[[X]][Z] , X=(X₁,..., X_n), over an algebraically closed field of characterisic zero \K, and satisfying a₁=0, is ν-quasi-ordinary. That means that if the discriminant Δ_P ∈ \K[[X]] is equal to a monomial times a unit then the ideal (a_i^d!/i(X))_i=2,...,d is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of \K[[X]] and the function germs of quasi-analytic families.

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• Quasi-ordinary hypersurfaces, multiplier ideals and local tropicalizations

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