THE ABHYANKAR JUNG THEOREM

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THE ABHYANKAR-JUNG THEOREM ADAM PARUSI?SKI, GUILLAUME ROND Abstract. We show that every quasi-ordinary Weierstrass polynomial P (Z) = Zd + a1(X)Zd?1 + · · ·+ ad(X) ? K[[X]][Z], X = (X1, . . . , Xn), over an algebraically closed field of characterisic zero K, such that a1 = 0, is ?-quasi-ordinary. That means that if the dis- criminant ∆P ? K[[X]] is equal to a monomial times a unit then the ideal (a d!/i i (X))i=2,...,d is monomial and generated by one of ad!/ii (X). 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. 1. Introduction Let K be an algebraically closed field of characteristic zero and let P (Z) = Zd + a1(X1, . . . , Xn)Z d?1 + · · ·+ ad(X1, . . . , Xn) ? K[[X]][Z](1) be a unitary polynomial with coefficients formal power series in X = (X1, .

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Complex analysis Ordinary

THE ABHYANKAR-JUNG THEOREM

ADAM PARUSIŃSKI, GUILLAUME ROND

Abstract. We show that every quasi-ordinary Weierstrass polynomial P ( Z ) = Z d + a 1 ( X ) Z d − 1 + ∙ ∙ ∙ + a d ( X ) ∈ K [[ X ]][ Z ] , X = ( X 1 , . . . , X n ) , over an algebraically closed ﬁeld of characterisic zero K , such that a 1 = 0 , is ν -quasi-ordinary. That means that if the dis-criminant Δ P ∈ K [[ X ]] is equal to a monomial times a unit then the ideal ( a id ! /i ( X )) i =2 ,...,d is monomial and generated by one of a id ! /i ( X ) . 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.

1. Introduction Let K be an algebraically closed ﬁeld of characteristic zero and let (1) P ( Z ) = Z d + a 1 ( X 1  . . .  X n ) Z d − 1 + ∙ ∙ ∙ + a d ( X 1  . . .  X n ) ∈ K [[ X ]][ Z ] be a unitary polynomial with coeﬃcients formal power series in X = ( X 1  . . . X n ) . Such a polynomial P is called quasi-ordinary if its discriminant Δ P ( X ) equals X 1 α 1 ∙ ∙ ∙ X nα n U ( X ) , with α i ∈ N and U (0) 6 = 0 . We call P ( Z ) a Weierstrass polynomial if a i (0) = 0 for all i = 1  . . .  d . We show the following result. Theorem 1.1. Let K be an algebraically closed ﬁeld of characteristic zero and let P ∈ K [[ X ]][ Z ] be a quasi-ordinary Weierstrass polynomial such that a 1 = 0 . Then the ideal ( a id ! /i ( X )) i =2 ,...,d is monomial and generated by one of a id ! /i ( X ) . The latter condition is equivalent to P being ν -quasi-ordinary in the sense of Hironaka, [H1], [L], and satisfying a 1 = 0 . Being ν -quasi-ordinary is a condition on the Newton polyhedron of P that we recall in Section 3 below. Thus Theorem 1.1 can be rephrased as follows. Theorem 1.2 ([L] Theorem 1) . If P is a quasi-ordinary Weierstrass polynomial with a 1 = 0 then P is ν -quasi-ordinary. As noticed in [K-V], Luengo’s proof of Theorem 1.2 is not complete. We complete the proof of Luengo and thus we complete his proof of the Abhyankar-Jung Theorem. Theorem 1.3 (Abhyankar-Jung Theorem) . Let K be an algebraically closed ﬁeld of charac-teristic zero and let P ∈ K [[ X ]][ Z ] be a quasi-ordinary Weierstrass polynomial such that the discriminant of P satisﬁes Δ P ( X ) = X 1 α 1 ∙ ∙ ∙ X rα r U ( X ) , where U (0) 6 = 0 , and r ≤ n . Then 1 1 there is q ∈ N \ { 0 } such that P ( Z ) has its roots in K [[ X 1 q  ... X r q  X r +1  ... X n ]] . 2000 Mathematics Subject Classiﬁcation. Primary: 13F25. Secondary: 13J15, 26E10. 1

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