NEW IDEAS FOR RESOLUTION OF SINGULARITIES IN ARBITRARY CHARACTERISTIC

TOHSUKE URABE

Dedicated to Professor Takuo Fukuda on his sixtieth birthday

Abstract. I have succeeded in showing that any two-dimensional hypersurface singularities of germs of varieties in any characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties. The new proof for the two-dimensional case depends on new ideas. Ideas are essentially different from Abhyankar’s one in [1] and Lipman’s one in [7]. It seems to be possible to generalize the new proof into higher dimensional case, if we add several ideas further. In this article I try to explain my new ideas rather than partial result I explained at the conference.

1. Introduction

At the conference I explained my partial result claiming that that any hypersurface singularities of germs of varieties in positive characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties, if we have a valuation ring of iterated divisor type associated with the germ. However, the existence of a certain valuation ring may be a strong condition, and there were several difficulties to remove this condition.

After coming back to Japan, through intensive study, I have succeeded in showing that any two-dimensional hypersurface singularities of germs of varieties in any characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties. The new proof for the two-dimensional case depends on new ideas. Ideas are essentially different from Abhyankar’s one in [1] and Lipman’s one in [7]. I got new ideas while I was checking Section eight of Abhyankar [1]. Section eight of [1] contains a lot of ideas, but it has nothing to do with the main result in [1]. Adding my own idea of coordinate change into special coordinates, I achieved a new proof. The origin of my idea of special coordinate change is the concept of “reductive” in Urabe [10], [11].

It seems to be possible to generalize the new proof into higher dimensional case, if we add several ideas further. I am now working in this generalization. I need much time.

In this article I try to explain my new ideas rather than partial result I explained at the conference.

Throughout this paper we fix the ground field k. We assume that k is an infinite field.

Date: May 31, 2000.

1991 Mathematics Subject Classification. Primary 14B05; Secondary 32S05, 13H99.

Key words and phrases. hypersurface, singularity, resolution of singularities, monoidal transformation, normal crossing, Weierstrass preparation theorem.

Let Γ be a totally ordered abelian group. Often in this article we associate a special element ∞ called the infinity with Γ. We assume that the element ∞ satisfies the following two conditions:

1. ∞ ≥ ∞, ∞ + ∞ = ∞, and n∞ = ∞n = ∞ for every positive integer n.

2. For every element γ ∈ Γ γ ≠ ∞, γ < ∞ and γ + ∞ = ∞ + γ = ∞.

By ℤ, ℚ and ℝ we denote the ring of integers, the rational number field and the real number field respectively. By ℤ_{0}, ℚ_{0} and ℝ_{0} we denote the set of non-negative integers, the set of non-negative rational numbers and the set of non-negative real numbers respectively.

The number of elements in a finite set X is denoted by ♯ X.

2. Ring theory

In this section we review several important points in the ring theory and explain non-standard concepts.

By a ring we mean a commutative ring with the identity. We use standard terminologies in the commutative ring theory. (Matsumura [8]. Zariski-Samuel [14].)

Let A be a ring. By A^{∗} we denote the set of invertible elements of A, i.e.,

A^{∗} = {a ∈ A | There exists an element b ∈ A with ab = 1}.

A ring with the unique maximal ideal is called a local ring. For a local ring A by M(A) we denote the maximal ideal of A. A local ring A is the disjoint union of A^{∗} and M(A).

Let n be a non-negative integer, and x_{1}, x_{2}, …, x_{n} n of variables. We consider the formal power series ring k[[x]] = k[[x_{1}, x_{2}, …, x_{n}]] over k. Let P = (x_{1}, x_{2}, …, x_{n}) be an ordered parameter system of k[[x]]. An element P = (x_{1}, x_{2}, …, x_{n}) ∈ k[[x]]^{n} such that the set {x_{1}, x_{2}, …, x_{n}} is a parameter system of the ring k[[x]] is called an ordered parameter system of k[[x]]. Any element f ∈ k[[x]] can be written uniquely in the form

f = ∑_{α∈ℤ_{0}^{n}} coeff(f, P, α)x^{α},

where α = (α_{1}, α_{2}, …, α_{n}) ∈ ℤ_{0}^{n}, α_{1}, α_{2}, …, α_{n} are non-negative integers, coeff(f, P, α) ∈ k for any α ∈ ℤ_{0}^{n}, and x^{α} = x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ x_{n}^{α_{n}}.

The set of exponents of f with respect to P is denoted by ex(f, P).

ex(f, P) = {α ∈ ℤ_{0}^{n} | coeff(f, P, α) ≠ 0}.

Definition 2.1. 1. Let P = (x_{1}, x_{2}, …, x_{n}) be an ordered parameter system of the ring k[[x]] and f ∈ k[[x]]. We say that f has a normal crossing with respect to P, if there are non-negative integers β_{1}, β_{2}, …, β_{n} with

\frac{f}{x_{1}^{β_{1}}x_{2}^{β_{2}} ⋯ x_{n}^{β_{n}}} ∈ k[[x]]^{∗}.

2. An element f ∈ k[[x]] is said to have a normal crossing, if there is an ordered parameter system P of k[[x]] such that f has a normal crossing with respect to P.

Lemma 2.2. Let P be an ordered parameter system of k[[x]], and f ∈ k[[x]], g ∈ k[[x]].

1. The following two conditions are equivalent:

(a) The product fg has a normal crossing with respect to P.

(b) Both f and g have a normal crossing with respect to P.

2. If the product fg has a normal crossing, then both f and g have a normal crossing.

For any A = (A_{1}, A_{2}, …, A_{n}), B = (B_{1}, B_{2}, …, B_{n}) ∈ ℝ^{n}, we denote (A, B) = ∑_{i=1}^{n} A_{i}B_{i}.

Let w = (w_{1}, w_{2}, …, w_{n}) ∈ ℤ_{0}^{n}, and P = (x_{1}, x_{2}, …, x_{n}) an ordered parameter system of k[[x]]. We will define a map

ord_{w}^{P} : k[[x]] → ℤ_{0} ∪ {∞}

associated with w and P. The value ord_{w}^{P}(f) ∈ ℤ_{0} ∪ {∞} for f ∈ k[[x]] is called the order of f with weight w with respect to P. Let f ∈ k[[x]] be an element. If f = 0, then we put ord_{w}^{P}(f) = ∞. Assume f ≠ 0. Since f ≠ 0, the set ex(f, P) is not empty. We define

ord_{w}^{P}(f) = min{(w, α) | α ∈ ex(f, P)} ∈ ℤ_{0}.

Lemma 2.3. 1. If f, g ∈ k[[x]], then ord_{w}(fg) = ord_{w}(f) + ord_{w}(g). 2. If f, g ∈ k[[x]], then ord_{w}(f + g) ≥ min{ord_{w}(f), ord_{w}(g)}.

Let E denote the field of fractions of k[[x]]. We can define an extended map

ord_{w}^{P} : E → ℤ ∪ {∞}

by putting ord_{w}^{P}(h) = ord_{w}^{P}(f) − ord_{w}^{P}(g), where h ∈ E, and f, g ∈ k[[x]] are elements satisfying h = f/g and g ≠ 0.

Lemma 2.4. 1. The map ord_{w}^{P} : E → ℤ ∪ {∞} associated with an element w ∈ ℤ_{0}^{n} and an ordered parameter system P is a valuation of the field E with the value group ℤ.

2. ord_{w}^{P}(f) ≥ 0 for every f ∈ k[[x]].

3. Let m be an integer with 0 ≤ m ≤ n. Assume w_{i} = 1 for every 1 ≤ i ≤ m, and w_{i} = 0 for every m < i ≤ n. Let Q = (y_{1}, y_{2}, …, y_{n}) be another ordered parameter system of k[[x]]. If the ideal generated by x_{1}, x_{2}, …, x_{m} and the ideal generated by y_{1}, y_{2}, …, y_{m} coincide, then we have ord_{w}^{P} = ord_{w}^{Q}.

When w_{i} = 0 or 1 for every 1 ≤ i ≤ n, we write

ord_{w}^{P} = ord_{x_{i_{1}} ,x_{i_{2}} ,... ,x_{im}} = ord_{ℑ},

where i_{1}, i_{2}, …, i_{m} are mutually different integers such that {i_{1}, i_{2}, …, i_{m}} = {i ∈ ℤ_{0} | 1 ≤ i ≤ n, w_{i} = 1}, and ℑ denotes the ideal generated by x_{i_{1}}, x_{i_{2}}, …, x_{i_{m}}. Moreover, if ℑ = M(k[[x]]) is the maximal ideal, then we write ord = ord_{ℑ}. The value ord(f) is called the order of f.

Let f = ∑_{α∈ℤ_{0}^{n}} coeff(f, P, α)x^{α} ∈ k[[x]] be a non-zero element. Let T = {α ∈ ex(f, P) | (w, α) = ord_{w}^{P}(f)}. Putting

in_{w}^{P}(f) = ∑_{α∈T} coeff(f, P, α)x^{α},

we call in_{w}^{P}(f) the w-initial polynomial of f with respect to P. For f = 0 ∈ k[[x]] we define in_{w}^{P}(0) = 0.

We can check that in_{w}^{P}(fg) = in_{w}^{P}(f)in_{w}^{P}(g) for any f, g ∈ k[[x]].

Assume that w_{i} = 0 for 1 ≤ i ≤ m and w_{i} > 0 for m < i ≤ n. One sees that in_{w}^{P}(f) ∈ k[[x_{1}, x_{2}, …, x_{m}]][x_{m+1}, x_{m+2}, …, x_{n}].

We say that f is a w-homogeneous polynomial with respect to P, if f = in_{w}^{P}(f). If w_{i} > 0 for every 1 ≤ i ≤ n, then any w-homogeneous polynomial contains only a finite number of terms, and it is a polynomial with coefficients in k.

When w_{i} = 0 or 1 for every 1 ≤ i ≤ n, we write

in_{w}^{P} = in_{x_{i_{1}} ,x_{i_{2}} ,... ,x_{im}}^{P} ,

where i_{1}, i_{2}, …, i_{m} are mutually different integers such that {i_{1}, i_{2}, …, i_{m}} = {i ∈ ℤ_{0} | 1 ≤ i ≤ n, w_{i} = 1}. In the case where w_{i} = 1 for every 1 ≤ i ≤ n, we write in_{x_{1},x_{2},... ,x_{n}}^{P} = in^{P} and a w-homogeneous polynomial is called a homogeneous polynomial.

Let e_{1}, e_{2}, …, e_{n} denote the standard basis of ℝ^{n}. Let S be a subset of ℝ^{n}. The set

I_{S} = {i ∈ ℤ_{0} | 1 ≤ i ≤ n, A + re_{i} ∈ S for any A ∈ S and for any r ∈ ℝ_{0}.}

is called the set of infinite directions of S.

Lemma 2.5. Let M be a positive integer and T be a subset of (\frac{1}{M} ℤ_{0})^{n}. Let S denote the minimum convex subset of ℝ^{n} containing T + ℤ_{0}^{n}.

1. The subset S is a convex polyhedron with a finite number of faces.

2. I_{S} = {1, 2, …, n}.

3. Any vertex of S belongs to (\frac{1}{M} ℤ_{0})^{n}.

4. Assume S ≠ ∅. For every w = (w_{1}, w_{2}, …, w_{n}) ∈ ℤ_{0}^{n} the set Δ(w, S) = {A ∈ S | (w, A) = min{(w, B) | B ∈ S}} is a face of S.

5. For any face F of S, there is an element w ∈ ℤ_{0}^{n} with F = Δ(w, S).

6. Assume S ≠ ∅. Let w = (w_{1}, w_{2}, …, w_{n}) ∈ ℤ_{0}^{n} and F = Δ(w, S). We have w_{i} = 0, if and only if, i ∈ I_{F}.

7. A face F of S is compact, if and only if, I_{F} = ∅.

Let P = (x_{1}, x_{2}, …, x_{n}) be an ordered parameter system of k[[x]], and f ∈ k[[x]]. The minimum convex subset of ℝ^{n} containing ex(f, P) + ℤ_{0}^{n} is called the Newton polyhedron of f, and is denoted by Γ_{+}(f, P). For any subset T of ℝ^{n}

ps_{T}^{P} (f) = \{ \begin{matrix}∑_{α∈ex(f,P)∩T} coeff(f, P, α)x^{α} ∈ k[[x]] & if ex(f, P) ∩ T ≠ ∅, \\ 0 ∈ k[[x]] & if ex(f, P) ∩ T = ∅. \end{matrix}

is called the partial sum of f over T with respect to P.

Lemma 2.6. 1. Γ_{+}(f, P) = ∅ if and only if f = 0.

2. Assume f ≠ 0. For every w = (w_{1}, w_{2}, …, w_{n}) ∈ ℤ_{0}^{n}, ord_{w}^{P}(f) = min{(w, A) | A ∈ Γ_{+}(f, P)}.

3. Assume that a subset T ⊂ ℝ^{n} and an element w = (w_{1}, w_{2}, …, w_{n}) ∈ ℤ_{0}^{n} satisfies T ∩ Γ_{+}(f, P) = Δ(w, Γ_{+}(f, P)), if f ≠ 0. We have in_{w}^{P}(f) = ps_{T}^{P} (f).

4. Let T be a subset of ℝ^{n}, and a_{1}, a_{2}, ⋯, ∈ k, f_{1}, f_{2}, ⋯ ∈ k[[x]] elements such that the sum ∑ a_{i}f_{i} exists in k[[x]]. We have ps_{T}^{P} (∑ a_{i}f_{i}) = ∑ a_{i}ps_{T}^{P} (f_{i}).

5. Let Γ be a convex subset of ℝ^{n} such that ex(f, P) ⊂ Γ, and F a face of Γ. For every positive integer j we have ps_{jF}^{P} (f^{j}) = ps_{F}^{P} (f)^{j}.

6. Let Γ be a convex subset of ℝ^{n}, F a face of Γ, f, g ∈ k[[x]], and r, s ∈ ℝ_{0}. Assume that ex(f, P) ⊂ rΓ and ex(g, P) ⊂ sΓ. We have ps_{(r+s)F}^{P} (fg) = ps_{rF}^{P} (f)ps_{sF}^{P} (g).

Theorem 2.7 (Weierstrass division theorem). Assume n > 0. Let P = (x_{1}, x_{2}, …, x_{n}) be an ordered parameter system of the ring k[[x]], and w = (w_{1}, w_{2}, …, w_{n}) ∈ ℤ_{0}^{n}.

We assume that w_{i} > 0 for every 1 ≤ i ≤ n − 1 and w_{n} = 1. We write k[[x′]] = k[[x_{1}, x_{2}, …, x_{n−1}]]. Let f ∈ k[[x]]. Assume that in_{w}^{P}(f)(0, 0, …, 0, 1) ≠ 0, where in_{w}^{P}(f)(0, 0, …, 0, 1) denotes the value of the w-initial polynomial of f at x_{1} = x_{2} = ⋯ = x_{n−1} = 0, x_{n} = 1. Let L = ord_{w}^{P}(f).

For every h ∈ k[[x]] there exists a unique set of an element e ∈ k[[x]] and L numbered elements H_{1}, H_{2}, …, H_{L} ∈ k[[x′]] satisfying

h = ef + ∑_{j=1}^{L} H_{j}x_{n}^{L−j}.

Corollary 2.8 (Weierstrass preparation theorem). We assume the same assumptions as in the above theorem.

1. There exists a unique set of an element e ∈ k[[x]] and L numbered elements G_{1}, G_{2}, …, G_{L} ∈ k[[x′]] satisfying

ef = x_{n}^{L} + ∑_{j=1}^{L} G_{j}x_{n}^{L−j}.

2. e ∈ k[[x]]^{∗}, and ord_{w}^{P}(G_{j}) ≥ j for every 1 ≤ j ≤ L.

Proof. It follows from the special case of Theorem 2.7 where h = x_{n}^{L}.

Lemma 2.9. Let f ∈ k[[x]] be a non-zero homogeneous polynomial with respect to an ordered parameter system P = (x_{1}, x_{2}, …, x_{n}).

1. There exist n elements a_{1}, a_{2}, …, a_{n−1} ∈ k with f(a_{1}, a_{2}, ⋯, a_{n−1}, 1) ≠ 0, where f(a_{1}, a_{2}, ⋯, a_{n−1}, 1) denotes the value of f at x_{1} = a_{1}, x_{2} = a_{2}, …, x_{n−1} = a_{n−1}, x_{n} = 1.

2. Assume that f(a_{1}, a_{2}, ⋯, a_{n−1}, 1) ≠ 0 for a_{1}, a_{2}, …, a_{n−1} ∈ k. Let y_{i} = x_{i} − a_{i}x_{n} for 1 ≤ i ≤ n − 1, and y_{n} = x_{n}. Then Q = (y_{1}, y_{2}, …, y_{n}) is an ordered parameter system of k[[x]], and the value of f at y_{1} = y_{2} = ⋯ = y_{n−1} = 0, y_{n} = 1 is not zero.

Remark . Recall that we have assumed that k is an infinite field. Lemma 2.9 is the essential reason why it is difficult to treat the case of a finite field in the resolution theory.

Lemma 2.10. Let E be a field, v : E → Γ ∪ {∞} a valuation, t a transcendental element over E, L a positive integer, and G_{1}, G_{2}, …, G_{L}, H_{1}, H_{2}, …, H_{L} ∈ E 2L elements satisfying

t^{L} + ∑_{j=1}^{L} G_{j}t^{L−j} = ∏_{j=1}^{L}(t + H_{j}).

We have

min\{\frac{1}{j}v(G_{j}) | 1 ≤ j ≤ L\} = min\{v(H_{j}) | 1 ≤ j ≤ L\}.

Let P = (x_{1}, x_{2}, …, x_{n}) be an ordered parameter system of k[[x]], w = (w_{1}, w_{2}, …, w_{n}) ∈ ℤ_{0}^{n}, L an positive integer, and G = (G_{1}, G_{2}, …, G_{L}) ∈ k[[x]]^{L}. We define

ord_{w}^{P}(G) = min\{\frac{1}{j}ord_{w}^{P}(G_{j}) | 1 ≤ j ≤ L\}.

The value ord_{w}^{P}(G) ∈ ℤ_{0} ∪ {∞} for G ∈ k[[x]]^{L} is called the order of G with weight w with respect to P. Let

Ḡ_{j} = \{ \begin{matrix}in_{w}^{P}(G_{j}) & if ord_{w}^{P}(G_{j}) = jord_{w}^{P}(G), \\ 0 & if ord_{w}^{P}(G_{j}) > jord_{w}^{P}(G). \end{matrix}

We define in_{w}^{P}(G) = (Ḡ_{1}, Ḡ_{2}, …, Ḡ_{L}). We call in_{w}^{P}(G) the w-initial polynomial of G with respect to P.

The minimum convex subset of ℝ^{n} containing (⋃_{j=1}^{L}(1/j)ex(G_{j}, P)) + ℤ_{0}^{n} is called the Newton polyhedron of G, and is denoted by Γ_{+}(G, P). For any subset T of ℝ^{n}

ps_{T}^{P} (G) = (ps_{T}^{P} (G_{1}), ps_{2T}^{P} (G_{2}), …, ps_{jT}^{P} (G_{j}), …, ps_{LT}^{P} (G_{L})) ∈ k[[x]]^{L}

is called the partial sum of G over T with respect to P.

Lemma 2.11. 1. Γ_{+}(G, P) = ∅ if and only if G = 0.

2. For every integer j with 1 ≤ j ≤ L we have Γ_{+}(G_{j}, P) ⊂ jΓ_{+}(G, P).

3. Assume G ≠ 0. For every w = (w_{1}, w_{1}, …, w_{n}) ∈ ℤ_{0}^{n}, ord_{w}^{P}(G) = min{(w, A) | A ∈ Γ_{+}(G, P)}.

4. Assume that a subset T ⊂ ℝ^{n} and an element w = (w_{1}, w_{1}, …, w_{n}) ∈ ℤ_{0}^{n} satisfies T ∩ Γ_{+}(G, P) = Δ(w, Γ_{+}(G, P)), if G ≠ 0. We have in_{w}^{P}(G) = ps_{T}^{P} (G).

3. Essential definitions

Let n be a non-negative integer, and x_{1}, x_{2}, …, x_{n} n of variables. We consider the formal power series ring k[[x]] = k[[x_{1}, x_{2}, …, x_{n}]] over k in the former half of this section. Let P = (x_{1}, x_{2}, …, x_{n}) be an ordered parameter system of k[[x]], x_{n+1} a transcendental element over k[[x]], L an positive integer, and G = (G_{1}, G_{2}, …, G_{L}) ∈ k[[x]]^{L} an element.

Definition 3.1. Let Δ be a face of the Newton polyhedron Γ_{+}(G, P). We say that Δ is first erasable, if there exists an element E ∈ k[[x]] satisfying

x_{n+1}^{L} + ∑_{j=1}^{L} ps_{jΔ}^{P}(G_{j})x_{n+1}^{L−j} = (x_{n+1} + E)^{L}. (3.1)

Lemma 3.2. 1. For every E ∈ k[[x]] there exists a unique element Ĝ = (Ĝ_{1}, Ĝ_{2}, …, Ĝ_{L}) ∈ k[[x]]^{L} satisfying

x_{n+1}^{L} + ∑_{j=1}^{L} G_{j}x_{n+1}^{L−j} = (x_{n+1} + E)^{L} + ∑_{j=1}^{L} Ĝ_{j}(x_{n+1} + E)^{L−j}. (3.2)

2. Let G_{0} = Ĝ_{0} = 1. We consider equalities below.

(3.3)  G_{j} = ∑_{i=0}^{j} \binom{L − i}{L − j}Ĝ_{i}E^{j−i},

(3.4)  Ĝ_{j} = ∑_{i=0}^{j} \binom{L − i}{L − j}G_{i}(−E)^{j−i},

The following three conditions are equivalent:

(a) The equality 3.2 holds.

(b) The equality 3.3 holds for every 1 ≤ j ≤ L.

(c) The equality 3.4 holds for every 1 ≤ j ≤ L.

Theorem 3.3. 1. Let Δ be a first erasable face of the Newton polyhedron Γ_{+}(G, P),

and Δ_{1} a face with Δ_{1} ⊂ Δ. Then, the face Δ_{1} is first erasable.

2. There is an element E ∈ k[[x]] satisfying three conditions below. Let Ĝ ∈ k[[x]]^{L} be the element satisfying the relation 3.2 in Lemma 3.2.

(a) Every face of Γ_{+}(Ĝ, P) is not first erasable.

(b) Γ_{+}(Ĝ, P) ⊂ Γ_{+}(G, P).

(c) ex(E, P) ⊂ Γ_{+}(G, P) − Γ_{+}(Ĝ, P)

3. Let E ∈ k[[x]] be an arbitrary element, and Ĝ ∈ k[[x]]^{L} the element satisfying the relation 3.2 in Lemma 3.2. Assume that every face of Γ_{+}(Ĝ, P) is not first erasable.

(a) Γ_{+}(E, P) ⊂ Γ_{+}(G, P), Γ_{+}(Ĝ, P) ⊂ Γ_{+}(G, P)

(b) If a face Δ of Γ_{+}(G, P) is not first erasable, then Δ ∩ Γ_{+}(Ĝ, P) ≠ ∅.

(c) If Γ_{+}(Ĝ, P) = Γ_{+}(G, P), then Γ_{+}(G, P) has no first erasable face.

Definition 3.4. Let M be a positive integer and T a subset of (\frac{1}{M} ℤ_{0})^{n}, S the minimum convex subset of ℝ^{n} containing T + ℤ_{0}^{n}.

1. We say that S is of Weierstrass type, if n > 0 and the convex polyhedron π(S) in ℝ^{n−1} has only one vertex, where π : ℝ^{n} → ℝ^{n−1} denotes the projection satisfying π(A_{1}, A_{2}, …, A_{n−1}, A_{n}) = (A_{1}, A_{2}, …, A_{n−1}).

We assume below that S is of Weierstrass type.

2. Let A′ be the unique vertex of π(S). There exists a unique vertex A of S with A′ = π(A). We call A the characteristic vertex of S.

3. We write A = (A′, A_{n}), where A′ ∈ ℝ^{n−1} and A_{n} ∈ ℝ. The domain D = ℝ^{n−1} × {r ∈ ℝ | r < A_{n}} is called the characteristic domain of S.

4. The map ρ : D → ℝ^{n−1} from D to ℝ^{n−1} defined by

ρ(B) = \frac{1}{A_{n} − B_{n}} (B′ − A′)

is called the characteristic map of S, where B = (B′, B_{n}), B′ ∈ ℝ^{n−1}, B_{n} ∈ ℝ, and B_{n} < A_{n}.

Remark . When n = 1 or 2, S is of Weierstrass type, if and only if, S is not empty.

Lemma 3.5. Let M be a positive integer and T be a subset of (\frac{1}{M} ℤ_{0})^{n}. Assume that the minimum convex subset S of ℝ^{n} containing T + ℤ_{0}^{n} is of Weierstrass type. Let A = (A′, A_{n}), D, and ρ : D → ℝ^{n−1} denote the characteristic vertex, the characteristic domain, and the characteristic map of S respectively. Let V be the set of vertices of S.

1. V − {A} ⊂ D.

2. S′ = ρ(S ∩ D) coincides with the minimum convex subset of ℝ^{n−1} containing ρ(V − {A}) + ℤ_{0}^{n−1}.

3. S′ = ∅, if and only if, S has only one vertex.

4. Let M′ be the least common multiple of 1, 2, …, MA_{n}. ρ(V −\{A\}) ⊂ (\frac{1}{M′} ℤ_{0})^{n−1}.

5. Let F be a face of S with A ∈ F and F ∩ D ≠ ∅. Then, F′ = ρ(F ∩ D) is a face of S′. Moreover, the following holds:

(a) dim F = dim F′ + 1.

(b) I_{F} ∩ {1, 2, …, n − 1} = I_{F′}.

(c) The following two conditions are equivalent: (i) n ∈ I_{F}. (ii) If B′ = (B_{1}, B_{2}, …, B_{n−1}) ∈ F′, and if i ∈ {1, 2, …, n − 1} − I_{F′},

then B_{i} = 0.

(d) The following conditions are equivalent: (i) F is compact. (ii) F − {A} ⊂ D. (iii) F′ is compact. (iv) F = ρ^{−1}(F′) ∪ {A}.

6. For every face F′ of S′ there exists a unique face F of S satisfying A ∈ F, F ∩ D ≠ ∅ and ρ(F ∩ D) = F′.

In what follows, we write k[[x′]] = k[[x_{1}, x_{2}, …, x_{n−1}]], P′ = (x_{1}, x_{2}, …, x_{n−1}), and x′^{α′} = x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ x_{n−1}^{α_{n−1}} for α′ = (α_{1}, α_{2}, …, α_{n−1}) ∈ ℤ_{0}^{n−1}.

For any ordered parameter system Q = (y_{1}, y_{2}, …, y_{n}) of k[[x]], G ∈ k[[x]] and α ∈ ℝ^{n} − ℤ_{0}^{n}, we define coeff(G, Q, α) = 0.

For a point B ∈ ℝ^{n} and a subset T ⊂ ℝ^{n} we put

Cone(B, T) = {B + t(C − B) | C ∈ T, t ∈ ℝ_{0}},

and call it the cone over T with vertex B.

Definition 3.6. Assume that the Newton polyhedron Γ_{+}(G, P) is of Weierstrass type. Let A = (A′, A_{n}), D, and ρ : D → ℝ^{n−1} denote the characteristic vertex, the characteristic domain, and the characteristic map of Γ_{+}(G, P) respectively.

We say that a face Δ of Γ_{+}(G, P) is second erasable, if A ∈ Δ and Δ ∩ D ≠ ∅, and if there exist elements E ∈ k[[x]] and F ∈ k[[x′]] satisfying the following two conditions. Δ′ = ρ(Δ ∩ D) denotes the corresponding face of Γ_{+}(G, P)′ = ρ(Γ_{+}(G, P) ∩ D):

1. ex(E, P) ⊂ Cone(A, Δ) − {A}, ex(F, P′) ⊂ Δ′

2.

x_{n+1}^{L} + ∑_{j=1}^{L} ps_{jΔ}^{P}(G_{j})x_{n+1}^{L−j} = (3.5)

(x_{n+1} + E)^{L} + ∑_{j=1}^{L} coeff(G_{j}, P, jA)x′^{jA′}(x_{n} + F)^{jA_{n}}(x_{n+1} + E)^{L−j}.

A face Δ′ of Γ_{+}(G, P)′ is said to be second erasable, if the corresponding face Δ of Γ_{+}(G, P) is second erasable.

Theorem 3.7. Assume that the Newton polyhedron Γ_{+}(G, P) is of Weierstrass type. Let A = (A′, A_{n}), D, and ρ : D → ℝ^{n−1} denote the characteristic vertex, the characteristic domain, and the characteristic map of Γ_{+}(G, P) respectively.

1. Let Δ and Δ_{1} be faces of Γ_{+}(G, P) with A ∈ Δ_{1} ⊂ Δ and Δ_{1} ∩ D ≠ ∅. If Δ is second erasable, then also Δ_{1} is second erasable.

2. There exist elements E ∈ k[[x]] and F ∈ M(k[[x′]]) satisfying the following conditions. Let Ĝ ∈ k[[x]]^{L} be the element satisfying the relation 3.2 in Lemma 3.2, and P̂ = (x_{1}, x_{2}, …, x_{n−1}, x_{n} + F). P̂ is an ordered parameter system of k[[x]]:

(a) The Newton polyhedron Γ_{+}(Ĝ, P̂) is of Weierstrass type. Its characteristic vertex coincides with A.

(b) Every face Δ of Γ_{+}(Ĝ, P̂) is not second erasable.

(c) Any vertex of Γ_{+}(Ĝ, P̂) belonging to D is not first erasable.

(d) For every integer j with 1 ≤ j ≤ L coeff(Ĝ_{j}, P̂, jA) = coeff(G_{j}, P, jA)

(e) ρ(Γ_{+}(Ĝ, P̂) ∩ D) ⊂ ρ(Γ_{+}(G, P) ∩ D).

(f) ex(E, P̂) ⊂ D. ρ(ex(E, P̂)) ⊂ ρ(Γ_{+}(G, P) ∩ D).

(g) ex(F, P′) ⊂ ρ(Γ_{+}(G, P) ∩ D) − ρ(Γ_{+}(Ĝ, P̂) ∩ D).

The unique closed point of an affine scheme Spec(k[[x]]) is denoted by s_{0}. The completion of the local ring O_{Σ, s} of a scheme Σ at a point s ∈ Σ is denoted by Õ_{Σ, s}. Let σ : Σ → Spec(k[[x]]) be a morphism of schemes. A k-valued point s ∈ Σ with σ(s) = s_{0} is called a point lying over s_{0}. For any point s ∈ Σ lying over s_{0} we have a ring homomorphism k[[x]] → Õ_{Σ, s} defined by pulling-back a function. We denote it by σ^{∗} : k[[x]] → Õ_{Σ, s}.

Definition 3.8. Assume that an ordered parameter system P = (x_{1}, x_{2}, …, x_{n}) of k[[x]] is given.

1. Let i be an integer with 1 ≤ i ≤ n. We say that an element f ∈ k[[x]] is simple at level i with respect to P, if there is an element g ∈ M(k[[x_{1}, x_{2}, …, x_{i−1}]]) with f = x_{i} + g.

2. Let I ⊂ {1, 2, …, n} be a subset with ♯ I ≥ 2, and C ⊂ Spec(k[[x]]) the closed smooth subscheme of codimension ♯ I defined by the ideal in k[[x]] generated by {x_{i} | i ∈ I}. Let σ : Σ → Spec(k[[x]]) denote the blowing-up of the scheme Spec(k[[x]]) with center C. Let s ∈ Σ be an arbitrary point lying over s_{0}. Let σ^{∗} : k[[x]] → Õ_{Σ, s} be the induced ring homomorphism. An ordered parameter system P̃ = (ξ_{1}, ξ_{2}, …, ξ_{n}) of Õ_{Σ, s} is said to be induced by P, if there is an integer i ∈ I satisfying the following conditions:

(a) σ^{∗}x_{i} = ξ_{i}.

(b) σ^{∗}x_{j} = ξ_{j}, if j ∈ {1, 2, …, n} − I.

(c) σ^{∗}x_{j} = ξ_{i}ξ_{j}, if j ∈ I and j < i.

(d) σ^{∗}x_{j} = ξ_{i}(ξ_{j} + λ_{j}) for some λ_{j} ∈ k, if j ∈ I and j > i.

Remark . Depending on the subset I ⊂ {1, 2, …, n} and the point s ∈ Σ, the induced ordered parameter system P̃ of Õ_{Σ, s} exists, and is uniquely determined by the ordered parameter system P of k[[x]].

Let m be a positive integer. The lexicographic order on ℚ^{m} is a total order ≤ with the following properties: Let α = (α_{1}, α_{2}, …, α_{m}), β = (β_{1}, β_{2}, …, β_{m}) ∈ ℚ^{m}. α ≤ β, if and only if, α = β or α < β. α < β, if and only if, there is an integer i with 1 ≤ i ≤ m such that α_{j} = β_{j} for every j with 1 ≤ j < i and α_{i} < β_{i}. Expressions β ≥ α and β > α are equivalent α ≤ β and α < β respectively. We can check that with this lexicographic order ℚ^{m} is a totally ordered abelian group.

Lemma 3.9. Let α_{i} (i = 1, 2, 3, …) be a sequence of elements in ℤ_{0}^{m}. Assume that α_{i} ≥ α_{i+1} for every i, where ≥ is the lexicographic order. Then, there exists a number i_{0} such that α_{i} = α_{i_{0}} for every number i with i ≥ i_{0}.

In the part below of this article, we assume that n = 3, k[[x]] = k[[x_{1}, x_{2}, x_{3}]], and consider hypersurface singularities of dimension two.

Definition 3.10. 1. A quadruple (g, g′, h, P) satisfying the following conditions is called a first standard decomposition:

(a) The last term P = (x_{1}, x_{2}, x_{3}) is an ordered parameter system of the ring k[[x]].

(b) The first term g is an element of k[[x]] such that there exist a non-negative integer ℓ, and ℓ elements g_{1}, g_{2}, …, g_{ℓ} ∈ M(k[[x_{1}, x_{2}]]) satisfying g = x_{3}^{ℓ} + ∑_{j=1}^{ℓ} g_{j}x_{3}^{ℓ−j}.

(c) The second term g′ is an element of k[[x]], and is a product of a finite number of simple elements at level 3. In other words, there exist a nonnegative integer m, and m elements g′_{1}, g′_{2}, …, g′_{m} ∈ M(k[[x_{1}, x_{2}]]) satisfying g′ = ∏_{j=1}^{m}(x_{3} + g′_{j})

(d) The third term h is an element of k[[x]]. There is an invertible element u ∈ k[[x]]^{∗} such that h/u is a product of a finite number of simple elements at level less than 3.

2. Let f ∈ k[[x]] be an element with f ≠ 0. We say that a first standard decomposition (g, g′, h, P) is one of f, if f = gg′h holds.

3. If h has a normal crossing with respect to P, then we say that a first standard decomposition (g, g′, h, P) is of normal crossing type.

4. The multiplicity μ(g, g′, h, P) of a first standard decomposition (g, g′, h, P) is an element (ℓ, m) in the set Z_{0}^{2} equipped with the lexicographic order, where ℓ and m are integers appearing in the above conditions.

The Gauss symbol [r] of a real number r ∈ ℝ is defined to be

[r] = max{α ∈ ℤ | α ≤ r}.

Definition 3.11. 1. A sextuple (g, g′, h, G, H, P) satisfying the following conditions is called a second standard decomposition:

(a) The quadruple (g, g′, h, P) is a first standard decomposition. Let μ(g, g′, h, P) = (ℓ, m) and L = ℓ + m.

(b) The fifth term H is an element of k[[x′]]. There is an invertible element U ∈ k[[x′]]^{∗} such that H/U is a product of a finite number of simple elements at level less than 3.

(c) The fourth term G = (G_{1}, G_{2}, …, G_{L}) is an element in k[[x′]]^{L}. They satisfy

gg′ = x_{3}^{L} + ∑_{j=1}^{L} H^{j}G_{j}x_{3}^{L−j},

and [ord_{x_{2}}(G)] = [ord_{x_{1}}(G)] = 0.

2. Let f ∈ k[[x]] be an element with f ≠ 0. We say that a second standard decomposition (g, g′, h, G, H, P) is one of f, if f = gg′h holds.

3. If h and H have a normal crossing with respect to P, then we say that a second standard decomposition (g, g′, h, G, H, P) is of normal crossing type.

Remark . For any second standard decomposition (g, g′, h, G, H, P) we have G ≠ 0 and H ≠ 0 by definition.

4. Resolution theorems

Under definitions in the previous section, the proof for the resolution of a germ of a two-dimensional hypersurface singularity over an arbitrary infinite field k becomes a chain of propositions. By propositions below together with Lemma 3.9 we conclude that the resolution is always possible. Repeating blowing-up of the space and pulling-back the function finite times, any function is reduced to one with a normal crossing.

In this section we consider only the case n = 3. Let k[[x]] = k[[x_{1}, x_{2}, x_{3}]]. For an ordered parameter system P = (x_{1}, x_{2}, x_{3}) of k[[x]], we write P′ = (x_{1}, x_{2}). This P′ is an ordered parameter system of the subring k[[x_{1}, x_{2}]].

Lemma 4.1. Let f ∈ k[[x]] be a non-zero element. Assume that there exist m + 1 elements f_{0}, f_{1}, …, f_{m} ∈ k[[x]] such that f = f_{0}f_{1} ⋯ f_{m}, ord(f_{i}) = 1 for 1 ≤ i ≤ m, and ℓ = ord(f_{0}). Then, there exists a first standard decomposition (g, g′, h, P) of f such that μ(g, g′, h, P) ≤ (ℓ, m), h ∈ k[[x]]^{∗}, and it is of normal crossing type.

Proof. It follows from Lemma 2.9 and Corollary 2.8.

Lemma 4.2. 1. Every non-zero element f ∈ k[[x]] has a first standard decom-

position of normal crossing type.

2. Every non-zero element f ∈ k[[x]] without a normal crossing has a second standard decomposition (g, g′, h, G, H, P) of normal crossing type such that the Newton polyhedron Γ_{+}(G, P′) has no first erasable face.

3. Let (g, g′, h, G, H, P) be a second standard decomposition, P = (x_{1}, x_{2}, x_{3}), μ(g, g′, h, P) = (ℓ, m), and L = ℓ + m. Assume that the Newton polyhedron Γ_{+}(G, P′) has no first erasable face. Then, there are elements Ĝ ∈ k[[x_{1}, x_{2}]]^{L}, Ĥ ∈ k[[x_{1}, x_{2}]] E ∈ M(k[[x_{1}, x_{2}]]), and F ∈ M(k[[x_{1}]]) satisfying the following conditions. Put P̂ = (x_{1}, x_{2} + F, x_{3} + E) = (x̂_{1}, x̂_{2}, x̂_{3}):

(a) Sextuple (g, g′, h, Ĝ, Ĥ, P̂) is a second standard decomposition.

(b) μ(g, g′, h, P̂) = (ℓ, m).

(c) ordx̂_{1}(Ĝ) = ord_{x_{1}}(G).

(d) ordx̂_{2}(inP̂ ; x̂_{1}(Ĝ)) ≤ ord_{x_{2}}(in_{x_{1}}^{P}(G)).

(e) The Newton polyhedron Γ_{+}(Ĝ, P̂′) has no first erasable face.

(f) The Newton polyhedron Γ_{+}(Ĝ, P̂′) has no second erasable face.

4. If ord_{x_{2}}(in_{x_{1}}^{P}(G)) ≥ 1 for a second standard decomposition (g, g′, h, G, H, P) with P = (x_{1}, x_{2}, x_{3}), then the Newton polyhedron Γ_{+}(G, P′) has two or more vertices.

Proof. 1. It follows from Lemma 4.1.

2. It follows from Theorem 3.3.2.

3. It folows from Theorem 3.7.2.

4. Easy.

Theorem 4.3. Let (g, g′, h, G, H, P) be a second standard decomposition of normal crossing type. We assume that the Newton polyhedron Γ_{+}(G, P′) has no first erasable face, and ord(G) < 1. We put f = gg′h.

1. If, moreover, ord_{x_{1}}(H) + ord_{x_{2}}(H) = 0, then there exists a first standard decomposition (ĝ, ĝ′, ĥ, P̂) of f such that μ(ĝ, ĝ′, ĥ, P̂) < μ(g, g′, h, P).

2. Assume that i = 1 or 2, and ord_{x_{i}}(H) > 0. Let σ : Σ → Spec(k[[x]]) be the blowing-up of the scheme Spec(k[[x]]) with center in the closed smooth subscheme defined by x_{i} = x_{3} = 0. For every point s ∈ Σ lying over s_{0} one of the following claims holds:

(a) The element σ^{∗}f ∈ Õ_{Σ, s} has a normal crossing.

(b) There exists a first standard decomposition (g̃, g̃′, h̃, P̃) of σ^{∗}f at the ring Õ_{Σ, s} such that μ(g̃, g̃′, h̃, P̃) < μ(g, g′, h, P).

(c) There exists a second standard decomposition (g̃, g̃′, h̃, G̃, H̃, P̃) of σ^{∗}f at the ring Õ_{Σ, s} satisfying the following conditions. We write P̃ = (x̃_{1}, x̃_{2}, x̃_{3}):

(i) It is of normal crossing type.

(ii) The Newton polyhedron Γ_{+}(G̃, P̃′) has no first erasable face.

(iii) ord(G̃) = ord(G) < 1.

(iv) μ(g̃, g̃′, h̃, P̃) ≤ μ(g, g′, h, P).

(v) ordx̃_{i}(H̃) = ord_{x_{i}}(H) − 1.

(vi) ordx̃_{j}(H̃) = ord_{x_{j}}(H), where j is the integer satisfying {1, 2} = {i, j}.

Proof. Claim 1 follows from Lemma 4.1.

Theorem 4.4. Let (g, g′, h, G, H, P) be a second standard decomposition of normal crossing type, and P = (x_{1}, x_{2}, x_{3}). We assume that the Newton polyhedron Γ_{+}(G, P′) has no first erasable face, ord(G) ≥ 1, and ord_{x_{2}}(in_{x_{1}}^{P′}(G)) < 1.

Let σ : Σ → Spec(k[[x]]) be the blowing-up of the scheme Spec(k[[x]]) with center in the closed smooth subscheme defined by x_{1} = x_{2} = 0. For every point s ∈ Σ lying over s_{0} there exists a second standard decomposition (g̃, g̃′, h̃, G̃, H̃, P̃) of σ^{∗}f at the ring Õ_{Σ, s} satisfying the following conditions:

1. It is of normal crossing type.

2. The Newton polyhedron Γ_{+}(G̃, P̃′) has no first erasable face.

3. μ(g̃, g̃′, h̃, P̃) ≤ μ(g, g′, h, P).

4. Let P̃′ = (x̃_{1}, x̃_{2}, x̃_{3}).

(ordx̃_{2}(inP̃′ ; x̃_{1}(G̃)), ordx̃_{1}(G̃)) < (ord_{x_{2}}(in_{x_{1}}^{P′}(G)), ord_{x_{1}}(G)),

where < is the lexicographic order.

Let (g, g′, h, G, H, P) be a second standard decomposition, and P = (x_{1}, x_{2}, x_{3}). By definition there is a unique set of an element u ∈ k[[x]]^{∗}, non-negative integers

α_{1}, α_{2}, and α_{2} elements h_{1}, h_{2}, …, h_{α_{2}} ∈ M(k[[x_{1}]]) satisfying

hH = ux_{1}^{α_{1}} ∏_{i=1}^{α_{2}}(x_{2} + h_{i}).

1. We write δ(h, H, P) = ♯ {h_{i} | 1 ≤ i ≤ α_{2}} ∈ ℤ_{0}.

2. We say that (g, g′, h, G, H, P) is of regular type, if, either α_{2} = 0, or, α_{2} > 0 and h_{i} = 0 for some i with 1 ≤ i ≤ α_{2}.

3. We define ϵ(h, H, P) = min{ord(h_{i}) | 1 ≤ i ≤ α_{2}} if α_{2} > 0, and ϵ(h, H, P) = ∞ if α_{2} = 0.

Lemma 4.5. 1. The following three conditions are equivalent:

(a) The second standard decomposition (g, g′, h, G, H, P) is of normal crossing type.

(b) δ(h, H, P) ≤ 1 and (g, g′, h, G, H, P) is of regular type.

(c) ϵ(h, H, P) = ∞.

2. Assume that the Newton polyhedron Γ_{+}(G, P′) has no first erasable face, and

ord_{x_{2}}(in_{x_{1}}^{P′}(G)) < 1.

There exists a second standard decomposition (ĝ, ĝ′, ĥ, Ĝ, Ĥ, P̂) of gg′h satisfying the following conditions. We write P̂ = (x̂_{1}, x̂_{2}, x̂_{3}):

(a) μ(ĝ, ĝ′, ĥ, P̂) ≤ μ(g, g′, h, P)

(b) The Newton polyhedron Γ_{+}(Ĝ, P̂′) has no first erasable face.

(c) ordx̂_{2}(inP̂′ ; x̂_{1}(Ĝ)) = ord_{x_{2}}(in_{x_{1}}^{P′}(G)) < 1

(d) δ(ĥ, Ĥ, P̂) = δ(h, H, P)

(e) The second standard decomposition (ĝ, ĝ′, ĥ, Ĝ, Ĥ, P̂) is of regular type.

Theorem 4.6. Let (g, g′, h, G, H, P) be a second standard decomposition, and P = (x_{1}, x_{2}, x_{3}). We assume that the Newton polyhedron Γ_{+}(G, P′) has no first erasable face, ord_{x_{2}}(in_{x_{1}}^{P′}(G)) < 1, (g, g′, h, G, H, P) is not of normal crossing type, and it is of regular type. We put f = gg′h.

Let σ : Σ → Spec(k[[x]]) be the blowing-up of the scheme Spec(k[[x]]) with center in the closed smooth subscheme defined by x_{1} = x_{2} = 0. For every point s ∈ Σ lying over s_{0} there exists a second standard decomposition (g̃, g̃′, h̃, G̃, H̃, P̃) of σ^{∗}f at the ring Õ_{Σ, s} satisfying the following conditions. We write P̃ = (x̃_{1}, x̃_{2}, x̃_{3}):

1. The Newton polyhedron Γ_{+}(G̃, P̃′) has no first erasable face.

2. ordx̃_{2}(inP̃′ ; x̃_{1}(G̃)) < 1.

3. μ(g̃, g̃′, h̃, P̃) ≤ μ(g, g′, h, P).

4. One of the following three conditions holds:

(a) (g̃, g̃′, h̃, G̃, H̃, P̃) is of normal crossing type.

(b) δ(h̃, H̃, P̃) < δ(h, H, P).

(c) The following conditions hold:

(i) (g̃, g̃′, h̃, G̃, H̃, P̃) is not of normal crossing type. (ii) It is of regular type. (iii) δ(h̃, H̃, P̃) = δ(h, H, P) (iv) ϵ(h̃, H̃, P̃) = ϵ(h, H, P) − 1

Let (g, g′, h, G, H, P) be a second standard decomposition with ord_{x_{2}}(in_{x_{1}}^{P′}(G)) ≥ 1. The Newton polyhedron Γ_{+}(G, P′) has two or more vertices. Let A_{i} = (A_{1,i}, A_{2,i}) ∈ ℝ_{0}^{2} (i = 1, 2, …, s) be its vertices, where s is the number of vertices. We can assume that

\begin{aligned}A_{1,1} & < & A_{1,2} & < & ⋯ & < & A_{1,s}, \\ A_{2,1} & > & A_{2,2} & > & ⋯ & > & A_{2,s}. \end{aligned}

We write

e(G, P′) = −\frac{A_{1,1} − A_{1,2}}{A_{2,1} − A_{2,2}} .

Theorem 4.7. Let (g, g′, h, G, H, P) be a second standard decomposition, and P = (x_{1}, x_{2}, x_{3}). We assume that the Newton polyhedron Γ_{+}(G, P′) has no first erasable face, it has no second erasable face, and ord_{x_{2}}(in_{x_{1}}^{P′}(G)) ≥ 1. We write f = gg′h.

Let σ : Σ → Spec(k[[x]]) be the blowing-up of the scheme Spec(k[[x]]) with center in the closed smooth subscheme defined by x_{1} = x_{2} = 0. For every point s ∈ Σ lying over s_{0} there exists a second standard decomposition (g̃, g̃′, h̃, G̃, H̃, P̃) of σ^{∗}f at the ring Õ_{Σ, s} satisfying the following conditions. We write P̃ = (x̃_{1}, x̃_{2}, x̃_{3}):

1. The Newton polyhedron Γ_{+}(G̃, P̃′) has no first erasable face.

2. μ(g̃, g̃′, h̃, P̃) ≤ μ(g, g′, h, P).

3. It satisfies one of the following two conditions:

(a) ordx̃_{2}(inP̃′ ; x̃_{1}(G̃)) < ord_{x_{2}}(in_{x_{1}}^{P′}(G)).

(b) The following conditions holds:

(i) ordx̃_{2}(inP̃′ ; x̃_{1}(G̃)) = ord_{x_{2}}(in_{x_{1}}^{P′}(G)).

(ii) The Newton polyhedron Γ_{+}(G̃, P̃′) has no second erasable face. (iii) e(G̃, P̃′) = e(G, P′) − 1.

In the proof of above theorems the concept of induced ordered parameter systems in Definition 3.8.2 is effective.

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Department of Mathematical Sciences, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

E-mail address: urabe@mito.ipc.ibaraki.ac.jp
URL: http://mathmuse.sci.ibaraki.ac.jp/urabe/indexE.html