ABC ESTIMATE, INTEGRAL POINTS, AND GEOMETRY OF P^{n} MINUS HYPERPLANES

Julie Tzu-Yueh Wang Institute of Mathematics Academia Sinica Nankang, Taipei 11529 Taiwan, R.O.C.

January 22, 1998

Abstract. Let K be a field and H be a set of hyperplanes in P^{n}(K). When K is a function field, we show that the following are equivalent. (a) H is nondegenerate over K. (b) The height of the (S, H)-integral points of P^{n}(K) − H is bounded.

(c) P_{K}^{n} − H is an abc variety. When K is a number field and H is nondegenerate over K, we establish an explicit bound on the number of (S, H)-integral points of P^{n}(K) − H. Finally, we discuss the geometric properties of holomorphic maps into P^{n}(ℂ) omitting a set of hyperplanes with moving targets.

0. Introduction

Let F be a number field and H be a set of hyperplanes in P^{n}(F). Let S be a finite set of valuations of F including all the archimedean valuations. When H is in general position and the number of hyperplanes in H is at least 2n + 1, Ru and Wong [RW] proved that the number of the (S, H)-integral points is finite; later the author [Wa2] provided an explicit bound on the number. Ru then found a necessary and sufficient condition on H such that the number of the (S, H)-integral points of P^{n}(F) − H is finite; he also showed that this is a necessary and sufficient condition of Brody hyperbolicity. However, an explicit bound on the number of the (S, H)-integral points was not obtained in [Ru].

Let C be an irreducible nonsingular projective algebraic curve of genus g defined over an algebraically closed field k of characteristic p ≥ 0. Let K be the function field of C and H be a set of hyperplanes in P^{n}(K). Let S be a set consisting of finitely many points of C. When p = 0, the author [Wa2] showed that if H is in general position and the number of hyperplanes in H is at least 2n + 1 then the height of the (S, H)-integral points is bounded and the bound is a linear function of |S|. When p > 0, the author [Wa3] showed that if H is in general position and

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the number of hyperplanes in H is at least 2n + 2 then under certain condition the height of the (S, H)-integral points is bounded and the bound is a linear function of |S|.

Recently, motivated by the abc theorem for function fields (cf. [Ma], [BM], [Vol]

[Wa1] and [No]), Buium defined abc varieties and proved that any affine open subset of an abelian variety over function fields (of characteristic 0) with trace zero is an abc variety.(cf. [Bu]) The definition of abc varieties is closely related to the (S, D)- integral points of a projective space V deleting a very ample divisor D. It turns out that the previous results on function fields done by the author are all theorems about abc varieties.

In the geometric case, as mentioned before that Ru gave a necessary and sufficient condition for P^{n}(ℂ) − H to be Brody hyperbolic. A more general question to consider is when the hyperplanes in H are moving, i.e. the coefficients of the linear forms corresponding to H are holomorphic functions. In [Wa4], the author applied the method in [Wa2] and obtained a generalization of the Picard’s theorem with moving targets.

In this paper, we will improve the number field result in [Ru] by giving an explicit bound on the number of the (S, H)-integral points. In the function field case of zero characteristic, we will show that the condition on H given in [Ru] is also necessary and sufficient for the height of the (S, H)-integral points to be bounded; and is also a necessary and sufficient condition for P_{K}^{n} − H to be an abc variety. Therefore, we will prove that P_{K}^{n} − H is an abc variety if and only if the height of the (S, H)- integral points of P^{n}(K) − H is bounded. Finally, in the geometric case we deal with the situation when the coefficients of the linear forms corresponding to the hyperplanes in H are holomorphic functions.

Ackowlegements. The author wishes to thank Jing Yu and Min Ru for helpful suggestions.

1. abc varieties and (S, D)-integral points

In this section we will restrict ourselves to function fields. However, the definition of abc varieties and (S, D)-integral points can be easily adapted to number fields.

Let C be an irreducible nonsingular projective algebraic curve of genus g defined over an algebraically closed field k of characteristic p ≥ 0. Let K be the function field of C. Given a point P ∈ C, we denote by v_{P} the normalized valuation associated to P. For elements f_{0}, ..., f_{n} of K, not all zeros, we define the height as

h(f_{0}, ..., f_{n}) := ∑_{P ∈C} − min{v_{P} (f_{0}), ..., v_{P} (f_{n})}.

P^{n} MINUS HYPERPLANES 3

For an element f of K, we define the height as

h(f) := ∑_{P ∈C} − min{0, v_{P} (f)}.

We now recall the definitions of (S, D)-integral points (cf. [Voj]) and abc varieties (cf. [Bu]). Let V be a projective variety defined over K. Let D be a very ample effective divisor on V and let 1 = ϕ_{0}, ϕ_{1}, ..., ϕ_{N} be a basis of the vector space:

L(D) = {f|f is a rational function on V such that f = 0 or (f) ≥ −D}.

Then ϕ = (ϕ_{0}, ...., ϕ_{N}) defines a morphism from V to P^{N}; and τ → (ϕ_{1}(τ), ..., ϕ_{N}(τ)) is an embedding of V (K) − D into K^{N}.

Definition. A point τ of V (K) − D is said to be an (S, D)-integral point if v_{P} (ϕ_{i}(τ)) ≥ 0, 1 ≤ i ≤ N, for every P ∉ S.

Following [Bu] we may define height and conductor as following:

\begin{matrix}h_{ϕ}(τ) = h(ϕ(τ)) = h(ϕ_{0}(τ), ..., ϕ_{N}(τ)), \\ Cond_{ϕ}(τ) = \{P ∈ C : min\{v_{P} (ϕ_{1}(τ)), …, v_{P} (ϕ_{N}(τ))\} < 0\}, \end{matrix}

cond_{ϕ}(τ) = |Cond_{ϕ}(τ)|.

Definition. We say that V − D satisfies the abc estimate over K if

h_{ϕ}(τ) << cond_{ϕ}(τ) + O(1), for every τ ∈ V (K) − D,

where “ << ” means the inequality holds up to multiplication with a positive constant.

Remark. This definition does not depend on the choice of the embedding.(cf. [Bu]) When there is no confusion, we will omit the subscript ϕ.

Definition. V_{K} − D is an abc variety if it satisfies the abc estimate over every finite extension L of K.

In this paper, we only consider the case when V = P_{K}^{n} and D is a set of hyperplanes in P^{n}(K). Let H be a set of q distinct hyperplanes in P^{n}(K) and let L_{i}, 1 ≤ i ≤ q, be the linear forms corresponding to H. Then we can fix an embedding from P^{n}(K) − H to K^{N} in the following form

(x_{0}^{q}/ ∏_{i=1}^{q} L_{i}(x), …, x_{n}^{q}/ ∏_{i=1}^{q} L_{i}(x), …),

where each coordinate of the embedding is in the form x_{0}^{i_{0}}x_{1}^{i_{1}} … x_{n}^{i_{n}}/ ∏_{i=1}^{q} L_{i}(x) with ∑_{j=0}^{n} i_{j} = q. Let

ϕ = (1, x_{0}^{q}/ ∏_{i=1}^{q} L_{i}(x), …, x_{n}^{q}/ ∏_{i=1}^{q} L_{i}(x), …).

Suppose that η is a point in P^{n}(K) and is represented by (f_{0}, ..., f_{n}). Then from the definition of height

h(f_{0}^{q}, ..., f_{n}^{q}) ≤ h(∏_{i=1}^{q} L_{i}(f_{0}, ..., f_{n}), f_{0}^{q}, ..., f_{n}^{q}, ...) = h(ϕ(η)).

On the other hand, if L_{i} = ∑_{j=0}^{n} a_{ij}x_{j} then v_{P} (L_{i}(f_{0}, ..., f_{n})) ≤ min{v_{P} (f_{0}), ..., v_{P} (f_{n})}+ min{v_{P} (a_{i0}), ..., v_{P} (a_{in})}. Therefore

h(ϕ(η)) = h(∏_{i=1}^{q} L_{i}(f_{0}, ..., f_{n}), f_{0}^{q}, ..., f_{n}^{q}, ...) ≤ qh(f_{0}, ..., f_{n}) + ∑_{i=1}^{q} h(a_{i0}, ..., a_{in}).

Together we have

qh(f_{0}, ..., f_{n}) ≤ h(ϕ(η)) ≤ qh(f_{0}, ..., f_{n}) + O(1).

Proposition 1. Let H be a set of q distinct hyperplanes in P^{n}(K). If P_{K}^{n} − H is an abc variety, then the height of the (S, H)-integral points of P^{n}(K)−H is bounded linearly in |S|.

Proof. Let τ be an (S, H)-integral point of P^{n}(K) − H and be represented by (f_{0}, ..., f_{n}). Then from the definitions of Cond_{ϕ} and (S, H)-integral points we have

cond_{ϕ}(τ) ≤ |S|.

P_{K}^{n} − H is an abc variety, hence

h_{ϕ}(τ) << cond_{ϕ}(τ) + O(1).

Since qh(f_{0}, ..., f_{n}) ≤ h_{ϕ}(τ), we have

h(f_{0}, ..., f_{n}) << |S| + O(1).

Remark. This proposition is true for number fields.

2. Further results in function fields

Let F be a number field and let H be a set of hyperplanes in P^{n}(F). Ru gave a necessary and sufficient condition on H such that P^{n}(F) − H has only finitely many (S, H)-integral points. We will show in this section that for a function field K this is a necessary and sufficient condition for P_{K}^{n} − H to be an abc variety; and also a necessary and sufficient condition such that the height of the (S, H)-integral points of P^{n}(K) − H to be bounded.

We recall some definitions and results from [Ru].

Notation. Let L be a set of linear forms in n + 1 variables which are pairwise linearly independent. We denote by (L)_{F} the vector space generated by the linear forms in L over F.

Definition. Let F be a field and H be a set of hyperplanes in P^{n}(F). We let L be the set of linear forms corresponding to H. (We note here that all linear forms in L are pairwise linearly independent over F.) H is said to be nondegenerate over F if dim(L)_{F} = n + 1 and for each proper nonempty subset L_{1} of L

(L_{1})_{F} ∩ (L − L_{1})_{F} ∩ L ≠ ∅.

Remark. If H is in general position and the number of hyperplanes of H is no less than 2n + 1, then H is nondegenerate over F.

Definition. Let F be a field and H be a set of hyperplanes in P^{n}(F). Let V be a subspace of P^{n}(F). V is called H-admissible if V is not contained in any hyperplane in H.

Proposition(Ru). Let H be a set of hyperplanes in P^{n}(F). Then H is nondegeneate over F if and only if for every H-admissible subspace V of P^{n}(F) of projective dimension greater than or equal to one, H ∩ V contains at least three distinct hyperplanes which are linearly dependent over F.

We will need the following version of the abc theorem [Br-Ma] for function fields of characteristic 0.

Theorem (Brownawell-Masser). Let the characteristic of K be zero. If f_{0}, ..., f_{n} are S-units and f_{0} + ⋯ + f_{n} = 1, then either some proper subsum of f_{0} + ⋯ + f_{n} vanishes or

h(f_{0}, ..., f_{n}) ≤ \frac{n(n + 1)}{2} max\{0, 2g − 2 + |S|\}.  (1)

We also need the following version of abc theorem [Wa1] for function fields of positive characteristic.

Theorem (Wang). Let the characteristic of K be a positive integer p. Suppose that f_{0}, ..., f_{n+1} are S-units of K. If f_{0} + ⋯ + f_{n} = f_{n+1} and f_{0}, ..., f_{n} are linearly independent over K^{p^{m}} for some positive integer m, then

h(f_{0}, ..., f_{n}) ≤ \frac{n(n + 1)}{2}p^{m−1} max\{0, 2g − 2 + |S|\}.  (2)

The main results in this section are the following.

Theorem 1. Let K be the function field of a nonsingular projective algebraic curve C which is defined over an algebraically closed field k with zero characteristic. Let S be a set consisting of finitely many points of C such that there exist nonconstant S-units. Let H be a set of hyperplanes in P^{n}(K). Then the following are equivalent

(a) H is nondegenerate over K.

(b) P_{K}^{n} − H is an abc variety.

(c) The height of the (S, H)-integral points of P^{n}(K) − H is bounded linearly in |S|.

(d) The height of the (S, H)-integral points of P^{n}(K) − H is bounded.

Theorem 2. Let K be the function field of a nonsingular projective algebraic curve C which is defined over an algebraically closed field k with characteristic p > 0. Let S be a set consisting of finitely many points of C. Let L_{i} = X_{i}, 0 ≤ i ≤ n, and L_{n+1+i} = ∑_{j=0}^{n} a_{ij}X_{j}, 0 ≤ i ≤ n, where a_{ij} are elements of K. Let H be the set of 2n+2 hyperplanes defined by L_{i}, 0 ≤ i ≤ 2n + 1. Let S_{n} be the permutation group of {0, 1, 2, .., n}. If H are in general position, i.e. any n + 1 linear forms corresponding to H are linearly independent, and the set {Π_{i=0}^{n}a_{iσ(i)}| σ ∈ S_{n}} is linearly independent over k, then P_{K}^{n} − H is an abc variety.

Proof of Theorem 1. We first show that (a) implies (b). Let L = {L_{1}, ..., L_{q}} be the set of linear forms corresponding to H. Let τ be a point of P^{n}(K) − H and be represented by (f_{0}, ..., f_{n}). Denote by l_{i} = L_{i}(f_{0}, ..., f_{n}). Let S_{H} be a set consisting of finitely many points of C such that every coefficient of each linear form L_{i} has no zero or pole outside S_{H}. Therefore, v_{P} (l_{i}) ≥ min{v_{P} (f_{0}), ..., v_{P} (f_{n})} for P ∉ S_{H}. On the other hand, from the definition of Cond(τ), we have qv_{P} (f_{j}) ≥ ∑_{i=1}^{q} v_{P} (l_{i}), 0 ≤ j ≤ n, for every P ∉ Cond(τ). Therefore

v_{P} (l_{i}) = min{v_{P} (f_{0}), ..., v_{P} (f_{n})}, for P ∉ Cond(τ) ∪ S_{H}, 1 ≤ i ≤ q. (1)

Suppose that the set {L_{i_{1}}, ..., L_{i_{m}}} is linearly dependent over K and every proper subset of {L_{i_{1}}, ..., L_{i_{m}}} is linearly independent over K. Then we have a linear equation

a_{i_{1}}L_{i_{1}}(X) + ⋯ + a_{i_{m}}L_{i_{m}}(X) ≡ 0,  (2)

where a_{i_{j}} ∈ K^{×}. We call equation (2) a minimal relation. Since elements of L are linear forms in n + 1 variables and are pairwise linearly independent over K, we have 3 ≤ m ≤ n + 2. It is clear that up to a nonzero factor in K there are only finitely many such minimal relations for the set L. Throughout the proof we will fix a finite set of minimal relations representing all minimal relations for L up to a nonzero factor in K. Without loss of generality, we can enlarge the size of S_{H}. Therefore, we will assume that every coefficient of the minimal relations in this finite set has no zero or pole outside of S_{H}. Let S_{τ} = Cond(τ) ∪ S_{H}. We now consider equation (2). After rearranging the index, we may assume that

a_{1}L_{1}(X) + ⋯ + a_{m}L_{m}(X) ≡ 0,  (3)

where a_{i}, 1 ≤ i ≤ m, is an S_{τ} -unit. Then we have the following equation.

a_{1}l_{1} + ⋯ + a_{m}l_{m} = 0.

After rearranging the index we may assume that a_{1}l_{1} + ⋯ + a_{u}l_{u} = 0 with u ≤ m and no proper subsum of a_{1}l_{1} + ⋯ + a_{u}l_{u} vanishes. Therefore

\frac{a_{2}l_{2}}{a_{1}l_{1}} + ⋯ + \frac{a_{u}l_{u}}{a_{1}l_{1}} = −1.  (4)

By (1), \frac{l_{i}}{l_{1}} , 1 ≤ i ≤ q, is an S_{τ} unit. Hence by the theorem of Brownawell and Masser we have

h(\frac{a_{i}l_{i}}{a_{1}l_{1}} ) ≤ \frac{u(u − 1)}{2} max\{0, 2g − 2 + |S_{τ} |\}, 1 ≤ i ≤ u.  (5)

From the definition of height,

h(\frac{l_{i}}{l_{1}} ) ≤ h(\frac{a_{i}l_{i}}{a_{1}l_{1}} ) + h(\frac{a_{i}}{a_{1}} ).  (6)

The coefficients a′_{i}s in the representing set of minimal relations only depend on L and the number is finite. Therefore inequalities (5) and (6) imply

h(\frac{l_{i}}{l_{1}} ) < \frac{n(n + 1)}{2}|S_{τ} | + O_{1},  (7)

where 1 ≤ i ≤ u and O_{1} only depends on L and can be determined effectively. From now O_{i} always represents a constant which only depends on L and can be determined effectively. If the dimension of the vector space spanned by L_{1}, ..., L_{u} over K is n + 1, then after a linear transformation one can show that (cf. [Wa2])

h(f_{0}, ..., f_{n}) ≤ \frac{n^{2}(n + 1)}{2}|S_{τ} | + O_{2}.  (8)

If the dimension of the vector space spanned by L_{1}, ..., L_{u} over K is less than n + 1, then the set {L_{1}, ..., L_{u}} is a proper subset of L. Since H is nondegenerate,

(L_{1}, ..., L_{u})_{K} ∩ (L_{u+1}, ..., L_{q})_{K} ∩ L ≠ ∅.  (9)

Suppose that L_{i} ∈ (L_{1}, ..., L_{u})_{K} ∩ (L_{u+1}, ..., L_{q})_{K}. If 1 ≤ i ≤ u, then after rearranging the linear forms we have L_{i} = a_{u+1}L_{u+1} + a_{u+2}L_{u+2} ⋯ + a_{w}L_{w}, where a_{j} ≠ 0 and is assumed previously to be an S_{τ} -unit. Similarly, after rearranging the index, we have an equation

l_{i} = a_{u+1}l_{u+1} + a_{u+2}l_{u+2} ⋯ + a_{ν}l_{ν}, ν ≤ w,

where no proper subsum of the equation vanishes. Therefore we have

h(\frac{l_{u+1}}{l_{i}}) ≤ \frac{n(n + 1)}{2}|S_{τ} | + O_{3}.  (10)

Hence,

h(\frac{l_{u+1}}{l_{1}} ) ≤ h(\frac{l_{u+1}}{l_{i}}) + h(\frac{l_{i}}{l_{1}} )
≤ n(n + 1)|S_{τ} | + O_{4}.  (11)

If i ≥ u + 1, after rearranging the index we may assume that i = u + 1. Then we have L_{u+1} = a_{i_{1}}L_{i_{1}} + ⋯ + a_{i_{w}}L_{i_{w}}, where {i_{1}, ..., i_{w}} is an index subset of {1, ..., u} and a_{i_{j}} is an S_{τ} -unit. Similarly, we have

h(\frac{l_{u+1}}{l_{i_{j}}}) ≤ \frac{n(n + 1)}{2}|S_{τ} | + O_{5}.

Therefore

h(\frac{l_{u+1}}{l_{1}} ) ≤ h(\frac{l_{u+1}}{l_{i_{j}}}) + h(\frac{l_{i_{j}}}{l_{1}} )
≤ n(n + 1)|S_{τ} | + O_{6}.  (12)

Hence, we have showed that h(\frac{l_{i}}{l_{1}} ) ≤ O_{7}|S_{τ} | + O(1) for 1 ≤ i ≤ u + 1. If (L_{1}, ..., L_{u+1}) = (L), then we are done. Otherwise, we can repeat the same argument. Since dim(L)_{K} = n + 1, after repeating the argument finitely many times we can find linear forms L_{1}, ..., L_{w} such that dim(L_{1}, ..., L_{w})_{K} = n + 1 and h(\frac{l_{i}}{l_{1}} ) ≤ O_{8}|S_{τ} |+O_{9}, 1 ≤ i ≤ w. Therefore, after a linear transformation (cf. [Wa2])

h(f_{0}, ..., f_{n}) ≤ O_{10}|S_{τ} | + O_{11} << cond(τ) + O(1).  (13)

It is clear from the proof that the abc estimate holds for every finite extension of K. This shows that P_{K}^{n} − H is an abc variety.

It follows from Proposition 1 that (b) implies (c). (c) implies (d) trivially. It remains to show that (d) implies (a). We follow the arguments in [Ru]. Assume that H is not nondegenerate over K. Then there exists an H-admissible subspace V of P^{n}(K) of projective dimension greater than or equal to 1 such that H∩V does not contain at least three distinct hyperplanes which are linearly dependent over K. After linear changing of basis we may assume that V = P^{m}(K), m ≤ n. Then H∩V contains exactly q distinct hyperplanes which are linearly independent over K and q ≤ m + 1. Without loss of generality we may assume that V = P^{n}(K) and H contains exactly q ≤ n + 1 distinct coordinate hyperplanes. Let f be a nonconstant S-unit. Then the point in P^{n}(K)−H represented by (1, f^{|S|^{r}}, ..., f^{|S|^{r}}) is an (S, H)- integral point and

h(1, f^{|S|^{r}}, ..., f^{|S|^{r}}) = |S|^{r}h(f) ≥ |S|^{r}.  (14)

This shows that (d) implies (a).

The proof in [Wa3] can be easily modified to show Theorem 2. For convenience of readers, we give an outline of the proof.

Proof of Theorem 2. Since k = ∩_{i=0}^{∞}K^{p^{i}} [GV], if the set {Π_{i=0}^{n}a_{iσ(i)}| σ ∈ S_{n}} is linearly independent over k, then there exists a positive integer m which depends only on a_{ij} such that the set {Π_{i=0}^{n}a_{iσ(i)}| σ ∈ S_{n}} is linearly independent over K^{p^{m}}.

Let (f_{0}, ..., f_{n}) represent a point τ in P^{n}(K) − H. Denote by l_{i} = L_{i}(f_{0}, ..., f_{n}). Let S_{H} be a finite subset of C such that every coefficient of each linear form has no zero or pole outside S_{H}. Therefore (1) gives

v_{P} (l_{i}) = min{v_{P} (f_{0}), ..., v_{P} (f_{n})} for P ∉ Cond(τ) ∪ S_{H}, 1 ≤ i ≤ q.  (15)

Since f_{i} = l_{i} ≠ 0, 0 ≤ i ≤ n, we may assume that f_{n} = 1. Let S_{τ} = Cond(τ) ∪ S_{H}.

(15) then implies that l_{i} is an S_{τ} -unit for 0 ≤ i ≤ 2n + 1. Since L_{0}, ..., L_{2n+1} are in general position, a_{ij} ≠ 0. Without loss of generality, we let a_{in} = 1 for every n + 1 ≤ i ≤ 2n + 1. Then we have the following S_{τ} -unit equations

a_{i0}f_{0} + a_{i1}f_{1} + ⋯ + a_{i,n−1}f_{n−1} + 1 = l_{n+1+i}, 0 ≤ i ≤ n.

If a_{β0}f_{0}, a_{β1}f_{1}, ..., a_{β,n−1}f_{n−1}, and 1 are linearly independent over K^{p^{m}} for some 0 ≤ β ≤ n, then by the theorem of Wang, for 0 ≤ j ≤ n − 1,

h(a_{βj}f_{j}) ≤ h(a_{β0}f_{0}, a_{β1}f_{1}, ..., a_{β,n−1}f_{n−1}, 1)

≤ \frac{n(n + 1)}{2}p^{m−1} max\{0, 2g − 2 + |S_{τ} |\}.

From the definition of height we have the following abc estimate

\begin{aligned}h(f_{0}, ..., f_{n}) & ≤ h(a_{i0}f_{0}, a_{i1}f_{1}, ..., a_{i,n−1}f_{n−1}, a_{i,n}f_{n}) + h(\frac{1}{a_{i0}} , ..., \frac{1}{a_{in}}) \\ & ≤ \frac{n(n + 1)}{2}p^{m−1} max\{0, 2g − 2 + |S_{τ} |\} + ∑_{0≤i,j≤n} h(a_{ij}) \end{aligned}

≤ \frac{n(n + 1)}{2}p^{m−1} max\{0, 2g − 2 + cond(τ) + S_{H}\} + ∑_{0≤i,j≤n} h(a_{ij}).

Therefore, we only need to consider the case where each set {a_{i0}f_{0}, ..., a_{i,n}f_{n}}, 0 ≤ i ≤ n, is linearly dependent over K^{p^{m}}. The next lemma shows that this is impossible if {Π_{i=0}^{n}a_{iσ(i)}| σ ∈ S_{n}} is linearly independent over K^{p^{m}}. The theorem is then proved. □

Lemma. Let f_{i} and a_{ij}, 0 ≤ i, j ≤ n, be non-zero elements of a field E. If each set {a_{i0}f_{0}, a_{i1}f_{1}, ..., a_{i,n}f_{n}}, 0 ≤ i ≤ n, is linearly dependent over a subfield F of E, then the set {Π_{i=0}^{n}a_{iσ(i)}| σ ∈ S_{n}} is linearly dependent over F.

Proof. See [Wa3].

3. The explicit bound for number fields

The proof of Theorem 1 can be adapted to the number field case directly. However, the S-unit theorem for number fields only provides an explicit bound on the number of S-unit solutions. Therefore, our method can provide explicit bound on the number of (S, H)-integral points, but can not say anything about the abc estimate. Let F be a number field of degree d. Denote by M_{F} as the set of valuations of F and by M_{∞} as the set of archimedean valuations of F. We first recall the S-unit theorem by Schlickewei [Sc]:

Theorem (Schlickewei). Let a_{1}, ..., a_{n} be nonzero elements of F. Suppose that S is a finite subset of M_{F} of cardinality s, containing M_{∞}. Then the equation

a_{1}x_{1} + ⋯ + a_{n}x_{n} = 1  (16)

has no more than

(4sd!)2^{36nd!s^{6}}  (17)

solutions in S-units x_{1}, ..., x_{n} such that no proper subsum a_{i_{1}}x_{i_{1}} + ⋯ + a_{i_{m}}x_{i_{m}} vanishes.

When H is nondegenerate to provide an explicit bound on the number of (S, H)- integral points we can apply the same method in the proof of Theorem 1. Since it is completely parallel to the function field case, we will only reproduce the parts which need the S-unit theorem.

Following the first part of the proof of Theorem 1, we apply the S-unit theorem to equation (4) in the number field case. Then we showed that the number of the S-unit solutions (\frac{l_{2}}{l_{1}} , ..., \frac{l_{u}}{l_{1}} ) is no more than (4sd!)2^{36ud!s^{6}}. Therefore the number of \frac{l_{i}}{l_{1}} , 2 ≤ i ≤ u, which satisfies (4), is no more than

(4sd!)2^{36ud!s^{6}}.

Again if the dimension of the vector space spanned by L_{1}, ..., L_{u} over F is n+1, then without loss of generality we may assume that L_{1}, ..., L_{n+1} are linearly independent over F. Therefore the number of (S, H)-integral points is equal to the number of (1, \frac{l_{2}}{l_{1}} , ..., \frac{l_{n+1}}{l_{1}} ) which is bounded by

(4sd!)^{n}2^{36n(n+1)d!s^{6}}.  (18)

If the dimension of the vector space spanned by L_{1}, ..., L_{u} over F is less than n + 1, then we can repeat the method in Theorem 1 and establish the same bound (18) for the number of (S, H)-integral points. Therefore, together with Ru’s result (cf. [Ru]) we have the following

Theorem 3. Let F be a number field of degree d. Suppose that S is a finite subset of M_{F} of cardinality s, containing M_{∞}. Let H be a set of hyperplanes in P^{n}(F). H is nondegenerate if and only if the number of (S, H)-integral points of P^{n}(F)−H is finite. Furthermore, the number of (S, H)-integral points of P^{n}(F) − H is bounded by

(4sd!)^{n}2^{36n(n+1)d!s^{6}}.

4. A generalization of the Picard’s theorem

A complex space M is called Brody hyperbolic if every holomorphic curve f : ℂ → M is constant. Ru proved the following (cf. [Ru]).

Theorem (Ru). Let H be a set of hyperplanes in P^{n}(ℂ). Then P^{n}(ℂ) − H is Brody hyperbolic if and only if H is nondegenerate over ℂ.

In [Wa4] we extended the classical Picard’s theorem to the case where the coefficients of the linear forms corresponding to H are holomorphic functions. In this section we will improve the results by adapting the proof of Theorem 1.

First, we explain our notation and terminologies. Let L_{i}(z)(X) = ∑_{j=0}^{n} g_{ij}(z)X_{j}, 1 ≤ i ≤ q, z ∈ ℂ, where g_{ij} are holomorphic functions and for each i, g_{i0}, ..., g_{in} has no common zeroes . Denote by H_{i}(z) = {(x_{0}, ..., x_{n}) ∈ P^{n}(ℂ)| L_{i}(z)(x_{0} , ..., x_{n}) = 0} as the corresponding moving hyperplane of L_{i}(z), 1 ≤ i ≤ q, z ∈ ℂ, and let H(z) = {H_{1}(z), ..., H_{q}(z)}. Let f_{0},...,f_{n} be holomorphic functions without common zeroes. We say that a holomorphic map f represented by (f_{0}, ..., f_{n}) is a holomorphic map omitting H(z) if L_{i}(z)(f_{0}(z), ..., f_{n}(z)) ≠ 0 for each z ∈ ℂ and i = 1, ..., q.

Denote by Hol(ℂ) as the ring consisting of all holomorphic functions on ℂ, and Mero(ℂ) as the field consisting of all meromorphic functions on ℂ. One can identify H_{i} as a hyperplane in P^{n}(Hol(ℂ)). Then L = {L_{1}, ..., L_{q}} can be identified as a set of linear forms with holomorphic functions as coefficients. Suppose that the set {L_{i_{1}}, ..., L_{i_{m}}} is linearly dependent over Mero(ℂ) and any proper subset of {L_{i_{1}}, ..., L_{i_{m}}} is linearly independent over Mero(ℂ). Then we have a minimal relation

a_{i_{1}}L_{i_{1}}(X) + ⋯ + a_{i_{m}}L_{i_{m}}(X) ≡ 0,  (19)

where a_{i_{j}} is a nonzero holomorphic function and a_{i_{1}}, …, a_{i_{m}} has no common zeros.

Definition. H is said to be unitary related if every holomorphic function a_{i_{j}} which appears in any of the minimal relations (19) has no zero.

We also need the following Unit Theorem which is a consequence of the Borel’s Lemma.

Unit Theorem. Let u_{0}, ..., u_{m} be holomorphic functions without zeroes and u_{0} + ⋯ + u_{m} = 1. Suppose that no proper subsum u_{0} + ⋯ + u_{m} − 1 = 0 vanishes, then u_{0}, ..., u_{m} are all constants.

The main result in this section is the following.

Theorem 4. Let L_{i}(z)(X) = ∑_{j=0}^{n} g_{ij}(z)X_{j}, 1 ≤ i ≤ q, z ∈ ℂ, where g_{ij} are holomorphic functions. Denote by H_{i}(z) the corresponding moving hyperplane of L_{i}(z), 1 ≤ i ≤ q. Let H = {H_{1}, ..., H_{q}} be unitary related. Then H is nondegenerate over Mero(ℂ) if and only if there exist finitely many (n + 1) × (n + 1) invertible matrices with holomorphic functions as entries such that every holomorphic map omitting H(z) multiplied by one of the matrices is constant. In addition, this set of matrices depends only on the hyperplanes and can be determined effectively.

Proof. Since the proof is completely parallel to the proof of Theorem 1, we will only reproduce the necessary parts. Let f_{0},...,f_{n} be holomorphic functions on ℂ without common zeroes and (f_{0}, ..., f_{n}) represents a holomorphic map f into P^{n}(ℂ) omitting H(z). Let l_{i} = L_{i}(f_{0}, ..., f_{n}). From (19) we have (after rearranging the index) the following unit equation

1 + \frac{a_{2}l_{2}}{a_{1}l_{1}} + ⋯ + \frac{a_{u}l_{u}}{a_{1}l_{1}} = 0,  (20)

where no proper subsum vanishes. Then by the Unit Theorem, we have \frac{a_{i}l_{i}}{a_{1}l_{1}} , 1 ≤ i ≤ u, is constant. The argument in the proof of Theorem 1 shows that there exist (after rearranging the index) {L_{1}, ..., L_{w}} ⊂ L and holomorphic units b_{1}, ..., b_{w}, c_{1}, ..., c_{w} such that dim(L_{1}, ..., L_{w})_{K} = n + 1, and \frac{b_{i}l_{i}}{c_{i}l_{1}} , 1 ≤ i ≤ w, is a nonzero constant. In addition, b_{i} and c_{i} , 1 ≤ i ≤ w, are coefficients of some minimal relations as (19). Therefore there are only finitely many such holomorphic units. After rearranging the index, we may assume that L_{1}, ..., L_{n+1} are linearly independent over Mero(ℂ). Hence (f_{0}(z), ..., f_{n}(z)) multiplied by

( \begin{matrix}g_{10}(z), & \frac{b_{2}}{c_{2}} g_{20}(z), & … & \frac{b_{n+1}}{c_{n+1}} g_{n+1,0}(z) \\ ⋮ & ... & ... & ⋮ \\ g_{1n}(z), & \frac{b_{2}}{c_{2}} g_{2n}(z), & … & \frac{b_{n+1}}{c_{n+1}} g_{n+1,n}(z) \end{matrix} )  (21)

is constant in P^{n}(Hol(ℂ)). Since L_{1}, ..., L_{n+1} are linearly independent over Mero(ℂ) and b_{i}, c_{i} are units, the matrix in (21) is invertible. It is also clear from the proof that the matrices used in (21) can be determined effectively and the number of the matrices is finite.

Conversely, if H is not nondegenerate over Mero(ℂ), we assume that there exist finitely many (n + 1) × (n + 1) invertible matrices with holomorphic functions as entries such that every holomorphic map omitting H(z) multiplied by one of the matrices is constant. In the following proof, we refer to [Rub] for some basic results and definitions of Nevanlinna theory. Let f(z) be a holomorphic function such that its characteristic function grows much rapidly than the characteristic function of the entries of the above matrices, i.e.

T (r, a) = o(T (r, f)),

for every entry a(z) of the above matrices. Since H is not nondegenerate over Mero(ℂ), there exists an H-admissible subspace V of P^{n}(Mero(ℂ)) of projective dimension greater than or equal to one such that H∩V does not contain at least three distinct hyperplanes in P^{n}(Mero(ℂ)) which are linearly dependent over Mero(ℂ). We may assume, without loss of generality, that V = P^{n}(Mero(ℂ)). Let H = {H_{1}, ..., H_{q}}. Then q ≤ n+1 and H_{1}, ..., H_{q} are linearly independent over Mero(ℂ). We may assume that H_{1}, ..., H_{q} are the first q coordinate planes. Let exp^{[n]}f(z) is the function defined recursively by exp^{[1]}f(z) = e^{f(z)} and exp^{[j+1]}f(z) = e^{exp^{[j]}f(z)}. Then the holomorphic map represented by (1, exp^{[1]}f(z), exp^{[2]}f(z), ..., exp^{[n]}f(z)) omits H(z). If A = (a_{ij}) is one of the above matrices such that the product with

this holomorphic map is constant, then

m_{β} ∑_{j=0}^{n} a_{αj}(z)exp^{[j]}f(z) = m_{α} ∑_{j=0}^{n} a_{βj}(z)exp^{[j]}f(z),  (22)

where 0 ≤ α, β ≤ n, m_{α} and m_{β} are constant. Since T (r, a_{ij}) = o(T (r, f)) and

T (r, exp^{[j]}f) = o(T (r, exp^{[j+1]}f)), (22) implies that

m_{β}a_{αj} = m_{α}a_{βj},  (23)

for 0 ≤ j ≤ n. Therefore, the determinant of the matrix A is zero which contradicts

the assumption that A is invertible. Therefore the proof is completed.

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Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529 Taiwan, R.O.C.

   E-mail address: jwang@ math.sinica.edu.tw