A NOTE ON WRONSKIANS AND THE ABC THEOREM IN FUNCTION FIELDS OF PRIME CHARACTERISTIC

Julie Tzu-Yueh Wang Institute of Mathematics Academia Sinica Nankang, Taipei 11529 Taiwan, R.O.C. May 14, 1998

Abstract. We provide a sharp bound for the order sequence of Wronskians. We also give another proof of the truncated second main theorem over function fields which is a generalization of the ABC theorem due to Mason, Voloch, Brownawell and Masser, Noguchi and the author.

1. Introduction

Let C be an irreducible nonsingular projective algebraic curve of genus g defined over an algebraically closed field k of characteristic p. Let K be the function field of C. Suppose that t is a local parameter of a point P of C, i.e. v_{P} (t) = 1. K is then a finite separable field extension of k(t).(cf. [Si], Chapter II, Proposition 1.4) Recall that the Hasse derivatives D_{t}^{(i)} is defined on k(t) by D_{t}^{(i)}(t^{j}) = \binom{j}{i}t^{j−i}, and then can be naturally extended to K(cf. [S-V]). The Hasse derivatives satisfy the following:

Proposition 1.1. (a) D_{t}^{(i)}(zw) = ∑_{j=0}^{i} D_{t}^{(j)}(z)D_{t}^{(i−j)}(w), ∀ z, w ∈ K.

(b) D_{t}^{(i)}D_{t}^{(j)} = \binom{i+j}{i}D_{t}^{i+j}.

Let now p > 0. Following Garcia and Voloch’s notation in [G-V], we let K_{m} = {x ∈ K|D_{t}^{(i)}(x) = 0 for 1 ≤ i < p^{m}} and let K_{∞} = {x ∈ K|D_{t}^{(i)}(x) = 0 for i ≥ 1}.

Proposition 1.2. For all m ≥ 1, K_{m} is a field. Moreover, D_{t}^{(p^{m})} is a derivation on K_{m} and K_{m+1} = {x ∈ K_{m}|D_{t}^{(p^{m})}(x) = 0}.

Proof. See [G-V], Proposition 1.□

Typeset by AMS-TEX 2 JULIE TZU-YUEH WANG

Remark. K_{m} = kK^{p^{m}} and K_{∞} = k (See [G-V]).

Throughout the paper x_{0},...,x_{n} will denote elements of K.

Garcia and Voloch also proved the following (cf. [G-V])

Theorem A. x_{0},...,x_{n} are linearly independent over K_{m} if and only if there exist integers 0 = ϵ_{0} < ⋯ < ϵ_{n} < p^{m} with det (D_{t}^{(ϵ_{i})}(x_{j})) ≠ 0

In [Wa1], the author gave an upper bound for each ϵ_{i} by showing the following:

Theorem B. If x_{0}, ..., x_{n} are linearly independent over K_{m}, then there exist integers 0 = ϵ_{0} < ϵ_{1} < ... < ϵ_{n} with ϵ_{i} ≤ ip^{m−1} such that

det(D_{t}^{(ϵ_{i})}x_{j})_{0≤i,j≤n} ≠ 0.

This bound is the best possible when n < p. For example,

Example. Let K = k(t) and p > n. Let x_{i} = t^{ip^{(m−1)}}, 0 ≤ i ≤ n, where m is a positive integer. Then x_{0}, …, x_{n} are linearly independent over K^{p^{m}}, and ϵ_{i} = ip^{(m−1)}.

When n ≥ p, this bound is clearly not the best possible. We will see later that the field extension degree [K : K^{p^{α}}] = p^{α}. Therefore, when p^{α} ≤ n < p^{α+1} it only makes sense to consider the case where x_{0}, …, x_{n} are linearly independent over K^{p^{m}} for m ≥ α + 1. In Section 2, we will locate the best possible bound for each ϵ_{i} in the general case. The bound on ϵ_{i} is closely related to the theory of Weierstrass points in positive characteristic.(cf. [G-V], [S-V], etc.)

In [Wa1], the author used the previous theorem and some results of [S-V] on Weierstrass points to prove the truncated second main theorem of function fields of any characteristic which is a generalization of the ABC Theorem due to Mason, Voloch, and Brownawell and Masser. The truncated second main theorem has many applications in function field Diophantine geometry. For example, it was used by the author to study the S-integral points on projective spaces minus hyperplanes.(cf. [Wa1], [Wa2], [Wa3], and [Wa4]) In Section 3, we will include another proof which is in the flavor of Nevanlinna theory and only involves Wronskians. We should also mention here that Noguchi(cf. [No]) also used techniques from Nevanlinna theory to show a truncated second main theorem for function fields of higher dimension with characteristic zero.

WRONSKIANS AND ABC THEOREM 3

Ackowlegements. This note is motivated by some questions about the ABC theorem and the theory of Weierstrass points that arose during the ABC workshop of the 1998 Arizona Winter School. The author would like to thank Professor W. McCallum for suggestions on the proof of Lemma 2.1. The author also wishes to thank Professor J. F. Voloch for some helpful comments.

2. Wronskians

We first formulate some results to understand the structure of the vector space spanned by {x_{0}, ..., x_{n}} over K^{p^{γ}}, γ ≥ 1. Lemma 2.1. Let u and m be nonnegative integers. Then the field extension degree of [K^{p^{m}}, K^{p^{m+u}}] is p^{u}. Furthermore, {t^{ip^{m}}| 0 ≤ i ≤ p^{u} − 1} is a set of basis for the vector space K^{p^{m}} over K^{p^{m+u}}. Proof. To show the field extension degree of [K^{p^{m}}, K^{p^{m+u}}] is p^{u}, we first prove that [K : K^{p}] = p. It then follows similarly that [K^{p^{m}} : K^{p^{m+1}}] = p. Therefore [K^{p^{m}}, K^{p^{m+u}}] = p^{u}.

Since the function field K is a finite separable field extension over k(t), there exists an element y ∈ K such that K = k(t, y). Furthermore, we may assume that y satisfies a monic minimal polynomial f(Y) ∈ k(t)[Y]. Then we have

\begin{matrix}[k(t, y) : k(t)][k(t) : k(t^{p})] \\ = [k(t, y)] : k(t^{p})] \end{matrix}

= [k(t, y) : k(t^{p}, y^{p})][k(t^{p}, y^{p}) : k(t^{p})].

Because that [k(t) : k(t^{p})] = p, it remains to show that [k(t, y) : k(t)] = [k(t^{p}, y^{p}) : k(t^{p})].

Suppose that f(Y) = Y^{r} + a_{n−1}(t)Y^{r−1} + ⋯ + a_{0}(t) = f̃(t, Y), where a_{i}(t) ∈ k(t). We raise each coefficient of a_{i}(t) to its p−th power and denote it by a_{i}^{p}(t). Let f̃^{p}(t, y) = Y^{r} + a_{n−1}^{p}(t)Y^{r−1} + ⋯ + a_{0}^{p}(t). Then f̃^{p}(t^{p}, y^{p}) = (f̃(t, y))^{p} = 0. Therefore, [k(t^{p}, y^{p}) : k(t^{p})] ≤ r. Since k is an algebraically closed field, k = k^{p}. Therefore if f̃^{p}(t^{p}, Y^{p}) is reducible over k(t^{p}), then f(t, Y) will be reducible in k(t)[Y] which gives a contradiction. Hence [k(t^{p}, y^{p}) : k(t^{p})] = r = [k(t, y) : k(t)].

Similarly, one can show that [K^{p^{m}} : K^{p^{m+1}}] = p. Therefore [K^{p^{m}}, K^{p^{m+u}}] = p^{u}. One can easily see that det (D_{t}^{(ip^{m})}t^{jp^{m}}) _{0≤i,j≤p^{u}−1} ≠ 0, so the second part of

the assertion follows directly from the Theorem A and the first assertion. □

4 JULIE TZU-YUEH WANG

Lemma 2.2. Suppose p^{α} ≤ n < p^{α+1} for some nonnegative integer α, and let β be a nonnegative integer less than or equal to α. If x_{0}, …, x_{n} are linearly independent over K^{p^{α+s}} for some positive integer s, then the dimension of the vector space spanned by x_{0}, …, x_{n} over K^{p^{β+s}} is strictly greater than p^{β}.

Proof. Suppose r + 1 is the dimension of the vector space spanned by x_{0}, …, x_{n} over K^{p^{β+s}}. Without loss of generality, we can assume that x_{0}, …, x_{r} are linearly independent over K^{p^{β+s}}. By Lemma 2.1, {t^{ip^{β+s}}| 0 ≤ i ≤ p^{α−β} −1} is a set of basis for the vector space K^{p^{β+s}} over K^{p^{α+s}}. Then the set

{t^{ip^{β+s}}x_{j}| 0 ≤ i ≤ p^{α−β} − 1, 0 ≤ j ≤ r}

is linearly independent over K^{p^{α+s}}. Since x_{j}, r + 1 ≤ j ≤ n, is a linear combination of x_{0}, …, x_{r} over K^{p^{β+s}}, the dimension of the vector space spanned by {t^{ip^{β+s}}x_{j}| 0 ≤ i ≤ p^{α−β} − 1, 0 ≤ j ≤ r} over K^{p^{α+s}} is no less than the dimension of the vector space spanned by {x_{0}, …, x_{n}} over K^{p^{α+s}}. Therefore, p^{α−β}(r + 1) ≥ n + 1 > p^{α}. Hence r + 1 > p^{β}. □

We will formulate an upper bound on ϵ_{i} according to the dimension of each vector space spanned by {x_{0}, …, x_{n}} over K^{p^{γ}} for γ ≥ 0. Later, we will offer some examples to show that this formulation is necessary.

Theorem 1. Suppose that p^{α} ≤ n < p^{α+1} and m ≥ α + 1. Let the dimension of the vector space spanned by x_{0}, …, x_{n} over K^{p^{γ}} be l_{γ} + 1 for 0 ≤ γ ≤ m. Let 0 = γ_{0} < γ_{1} < ⋯ < γ_{u} ≤ m be a sequence of integers between 0 and m such that 0 = l_{γ_{0}} < l_{γ_{1}} < ⋯ < l_{γ_{u}} = n and l_{γ_{δ}} = l_{γ_{δ}+1} = ⋯ = l_{γ_{δ+1}−1}. Then x_{0}, …, x_{n} are linearly independent over K^{p^{m}} if and only if there exist integers 0 = ϵ_{0} < ϵ_{1} ⋯ < ϵ_{n} with

p^{γ_{δ}−1} + s − 1 ≤ ϵ_{l_{γ_{δ−1}} +s} ≤ min{sp^{γ_{δ}−1}, p^{γ_{δ}} − 1}, 1 ≤ s ≤ l_{γ_{δ}} − l_{γ_{δ−1}},

such that det (D_{t}^{(ϵ_{i})}x_{j})_{0≤i,j≤n} ≠ 0.

Remark 2.1. By Lemma 2.2, l_{m−α+i} ≥ p^{i}.

Remark 2.2. It is easy to see that in general one has ϵ_{i} ≤ ip^{m}. Therefore Theorem 1 is stronger than the Theorem B.

Remark 2.3. When the characteristic of k is 0, ϵ_{i} = i.

WRONSKIANS AND ABC THEOREM 5

Corollary. ϵ_{l_{γ_{δ−1}+1}} = p^{γ_{δ}−1}, for δ = 1, 2, ..., u.

Proof. It follows easily from the inequalities in Theorem 1. □

Proof of Theorem 1. The “ if ” part comes from Theorem A. Therefore, we only need to show the other direction.

Without loss of generality, we may assume that x_{0}, ..., x_{l_{γ_{δ}}} are linearly independent over K^{p^{γ_{δ}}}. By definition of γ_{δ}, x_{0}, ..., x_{l_{γ_{δ−1}+1}} are linearly dependent over K^{p^{γ_{δ} −1}}. Therefore, by Theorem A we have ϵ_{l_{γ_{δ−1}+1}} ≥ p^{γ_{δ}−1}. Hence, ϵ_{l_{γ_{δ−1}+s}} > ϵ_{l_{γ_{δ−1}+1}} + s − 1 ≥ p^{γ_{δ}−1} + s − 1. This proves one side of the inequality.

For the other side of the inequality, we will prove it by induction. Since x_{0}, …, x_{l_{γ_{1}}} are linearly independent over K^{p^{γ_{1}}} , from Theorem B there exist 0 = ϵ_{0} < ϵ_{1} ⋯ < ϵ_{l_{γ_{1}}} with ϵ_{i} ≤ ip^{γ_{1}−1} such that det (D_{t}^{(ϵ_{i})}x^{j})_{0≤i,j≤l_{γ_{1}}} ≠ 0. Together with Theorem A, we conclude

ϵ_{i} ≤ min{ip^{γ_{1}−1}, p^{γ_{1}} − 1}, for 1 ≤ i ≤ l_{γ_{1}}.

The proof will be completed by two induction steps. First we show that if the theorem is true for n = l_{γ_{β}}, 0 ≤ β < u, then it is true for n = l_{γ_{β}} + 1. We then show that if the theorem is true for n = l_{γ_{β}} + s, then it is true for n = l_{γ_{β}} + s + 1, where 1 ≤ s < l_{γ_{β+1}} − l_{γ_{β}}.

Now suppose that the theorem is true for n = l_{γ_{β}} and the conclusion of the theorem does not hold for n = l_{γ_{β}} + 1. Then the vectors

(x_{j}, D_{t}^{(1)}x_{j}, D_{t}^{(2)}x_{j}, …, D_{t}^{(p^{γ_{β+1}} −1)}x_{j}), 0 ≤ j ≤ l_{γ_{β}} + 1

are linearly dependent over K. Then there exist a_{0}, …, a_{l_{γ_{β}} +1} ∈ K such that

∑_{j=0}^{l_{γ_{β}} +1} a_{j}D_{t}^{(i)}x_{j} = 0 for 0 ≤ i ≤ p^{γ_{β+1}−1}.  (2.1)

Since x_{0}, …, x_{l_{γ_{β}}} are linearly independent over K^{p^{γ_{β}}}, by the induction hypothesis we have that the l_{γ_{β}} + 1 vectors above, for 0 ≤ j ≤ l_{γ_{β}}, are linearly independent over K. Hence a_{l_{γ_{β}} +1} ≠ 0 and, without loss of generality, we can assume that a_{l_{γ_{β}} +1} = 1. We will show that a_{j} ∈ K^{p^{γ_{β+1}}} , for 0 ≤ j ≤ l_{γ_{β}}. Then we have x_{0}, …, x_{l_{γ_{β}} +1} are linearly dependent over K^{p^{γ_{β+1}}} which will give a contradiction that completes this step of induction proof.

6 JULIE TZU-YUEH WANG

To prove that a_{j} ∈ K^{p^{γ_{β+1}}} , by Proposition 1.2, it suffices to show that D_{t}^{(r)}a_{j} = 0 for r = 1, p, p^{2}, …, p^{γ_{β+1}−1} . We will prove it by induction. For r = 1, applying D_{t} to (2.1), we have

(i + 1) ∑_{j=0}^{l_{γ_{β}} +1} a_{j}D_{t}^{(i+1)}x_{j} + ∑_{j=0}^{l_{γ_{β}}} D_{t}a_{j}D_{t}^{(i)}x_{j} = 0.

It follows from (2.1) that

∑_{j=0}^{l_{γ_{β}}} D_{t}a_{j}D_{t}^{(i)}x_{j} = 0, for 0 ≤ i ≤ p^{γ_{β+1}−1} − 1.  (2.2)

Since the l_{γ_{β}} + 1 vectors above (i.e. for 0 ≤ j ≤ l_{γ_{β}}) are linearly independent over K we get D_{t}a_{j} = 0 for 0 ≤ j ≤ l_{γ_{β}}.

Assume that D_{t}^{(p^{ν})}a_{j} = 0 for ν = 0, 1, …, r − 1 < γ_{β+1} − 1 and for 0 ≤ j ≤ l_{γ_{β}}. Then we have a_{j} ∈ K^{p^{r}}. Therefore, D_{t}^{(b)}a_{j} = 0 for 1 ≤ b ≤ p^{r} − 1 and for 0 ≤ j ≤ l_{γ_{β}}. Apply D_{t}^{(b)}, 1 ≤ b ≤ p^{r} − 1, to (2.1) for i = p^{γ_{β+1}−1}, i.e. ∑_{j=0}^{l_{γ_{β}} +1} a_{j}D_{t}^{(p^{γ_{β+1}−1})}x_{j} = 0. Then we have

( \begin{matrix}p^{γ_{β+1}−1} + b \\ b \end{matrix} ) ∑_{j=0}^{l_{γ_{β}} +1} a_{j}D_{t}^{(p^{γ_{β+1}−1}+b)}x_{j} = 0.

Since ( \begin{matrix}p^{γ_{β+1}−1} + b \\ b \end{matrix} ) ≠ 0 mod p,

∑_{j=0}^{l_{γ_{β}} +1} a_{j}D_{t}^{(i)}x_{j} = 0, for 1 ≤ i ≤ p^{γ_{β+1}−1} + p^{r} − 1.  (2.3)

Applying the operator D_{t}^{p^{r}} to (2.1) for 0 ≤ i ≤ p^{γ_{β+1}−1} − 1, we then have

∑_{j=0}^{l_{β}} D_{t}^{(p^{r})}a_{j}D_{t}^{(i)}x_{j} = 0, for 1 ≤ i ≤ p^{γ_{β+1}−1} − 1.

Since the l_{γ_{β}} + 1 vectors above (i.e. for 0 ≤ j ≤ l_{γ_{β}}) are linearly independent over K we get D_{t}^{(p^{r})}a_{j} = 0 for j = 0, 1, …, l_{γ_{β}}. This completes the first step of the induction proof.

Next, we will show that if the theorem is true for n = l_{γ_{β}} + s, then it is true for n = l_{γ_{β}} + s + 1, 1 ≤ s < l_{γ_{β+1}} − l_{γ_{β}}. If (s + 1)p^{γ_{β+1}−1} ≥ p^{γ_{β+1}} − 1, then by Theorem A the theorem holds. Therefore, we may assume that (s + 1)p^{γ_{β+1}−1} < p^{γ_{β+1}} − 1. Suppose that the theorem is true for n = l_{γ_{β}} + s, and it does not hold for n = l_{γ_{β}} + s + 1, 1 ≤ s < l_{β+1} − l_{β}. Similar to the previous argument, we have

∑_{j=0}^{l_{γ_{β}} +s+1} a_{j}D_{t}^{(b)}x_{j} = 0 for 0 ≤ b ≤ (s + 1)p^{γ_{β+1}−1}.  (2.4)

Similarly, we can assume that a_{l_{γ_{β}} +s+1} = 1. It then suffices to show that D_{t}^{(p^{ν})}a_{j} = 0 for ν = 1, 2, 3, …, γ_{β+1} − 1. We will do it by induction. Suppose D_{t}^{(ν)}a_{j} = 0 for ν = 1, 2, 3, …, r − 1 < γ_{β+1} − 1. Applying the operator D_{t}^{(p^{r})} to (2.4) for 0 ≤ b ≤ sp^{γ_{β+1}−1}, we have

∑_{j=0}^{l_{γ_{β}} +s} D_{t}^{(p^{r})}a_{j}D_{t}^{(b)}x_{j} = 0 for 0 ≤ b ≤ sp^{γ_{β+1}−1}.  (2.5)

By the induction hypothesis, the l_{γ_{β}} + s + 1 vectors above (i.e. 0 ≤ j ≤ l_{γ_{β}} + s) are linearly independent over K. Therefore, D_{t}^{(p^{r})}a_{j} = 0 for j = 0, 1, …, l_{γ_{β}}. Inductively, we have D_{t}^{(ν)}a_{j} = 0 for ν = 1, 2, 3, …, p^{γ_{β+1}−1}. This completes the second step of the induction proof. Therefore the proof for this theorem is completed. □

We provide the following examples to show that the results is indeed the best possible and the decomposition on each K^{p^{γ}} is necessary.

Example 1. Let K = k(t), p = 3, and n = 10. Let x_{i} = t^{i} for 0 ≤ i ≤ 9, and let x_{10} = t^{18}. Then x_{0}, ..., x_{10} are linearly independent over K^{27}. In this case, α = 2, m = 3, l_{0} = 0, l_{1} = 2, l_{2} = 8, and l_{3} = 9; γ_{0} = 0, γ_{1} = 1, γ_{2} = 2, and γ_{3} = 3. We have ϵ_{i} = i for 0 ≤ i ≤ 9, and ϵ_{10} = 18.

Example 2. Let K = k(t), p = 3, and n = 4. Let x_{0} = 1, x_{1} = t^{3}, x_{2} = t^{6}, x_{3} = t^{9}, x_{4} = t^{18}. Then x_{0}, ..., x_{4} are linearly independent over K^{27}. In this case, α = 1, m = 3, l_{0} = 0, l_{1} = 0, l_{2} = 2, and l_{3} = 4; γ_{0} = 0, γ_{1} = 2, and γ_{2} = 3. We have ϵ_{1} = 3, ϵ_{2} = 6, ϵ_{3} = 9, ϵ_{4} = 18.

Example 3. Let K = k(t), p = 3, and n = 3. Let x_{0} = 1, x_{1} = t^{3}, x_{2} = t^{9}, x_{3} = t^{18}. Then x_{0}, ..., x_{3} are linearly independent over K^{27}. In this case, α = 1, m = 3, l_{0} = 0, l_{1} = 0, l_{2} = 1, and l_{3} = 3; γ_{0} = 0, γ_{1} = 2, and γ_{2} = 3. We have ϵ_{1} = 3, ϵ_{2} = 9, ϵ_{3} = 18.

8 JULIE TZU-YUEH WANG

3. The Proof of the Truncated Second Main Theorem

In this section we will give another proof for the truncated second main theorem in the flavor of Nevanlinna theory. The method which we are going to use here is basically the argument from [B-M] and [La](p.220), but we replace the ordinary higher derivatives by the Hasse derivatives in order to deal with function fields of positive characteristic.

We will need some basic propositions. Proposition 3.1. (a) If g_{i} = ∑ a_{ij}x_{j} with (a_{ij}) ∈ GL_{n+1}(k), then

det(D_{t}^{ϵ_{i}}g_{j}) = det(a_{ij})det(D_{t}^{ϵ_{i}}x_{j}).

(b) If h ∈ K, then

det(D_{t}^{ϵ_{i}}hx_{j}) = h^{n+1}det(D_{t}^{ϵ_{i}}x_{j}).

(c) If x is another separating variable, then

det(D_{x}^{ϵ_{i}}x_{j}) = (\frac{dt}{dx})^{ϵ_{0}+ϵ_{1}...+ϵ_{n}}det(D_{t}^{ϵ_{i}}x_{j}).

Proof. See [S-V], Proposition 1.4.□ Proposition 3.2. Let t_{P} be a local parameter of a point P ∈ C. If x be a nonzero element in K, then

v_{P} (\frac{D_{t_{P}}^{i} x}{x}) ≥ \{ \begin{matrix}−i, & if v_{P} (x) ≠ 0 \\ 0, & if v_{P} (x) = 0. \end{matrix}

Proof. If v_{P} (x) = 0, we have v_{P} (D_{t_{P}} x) ≥ 0. Therefore v_{P} (\frac{D_{t_{P}}^{i} x}{x}) ≥ 0 if v_{P} (x) = 0.

If v_{P} (f) = m ≠ 0, then there exists η ∈ K with v_{P} (η) = 0 such that x = t_{P}^{m}η. By Proposition 1.1.a

\frac{D_{t_{P}}^{i} x}{x} = \frac{1}{x} ∑_{j=0}^{i} ( \begin{matrix}m \\ j \end{matrix} ) t_{P}^{m−j}D_{t_{P}}^{i−j}η

= \frac{t_{P}^{−i}}{η} ∑_{j=0}^{i} ( \begin{matrix}m \\ j \end{matrix} ) t_{P}^{i−j}D_{t_{P}}^{i−j}η.

Therefore v_{P} (\frac{D_{t_{P}}^{i} x}{x}) ≥ −i. □

WRONSKIANS AND ABC THEOREM 9

Proposition 3.3. Let L_{1}, ..., L_{q} be linear forms in n+1 variables with coefficients in k and in general position. Let x_{0}, ..., x_{n} be elements of K such that L_{i}(x_{0}, ..., x_{n}) ≢ 0. Denote by l_{i} = L_{i}(x_{0}, ..., x_{n}). If v_{P} (l_{1}) ≤ v_{P} (l_{2}) ≤ .... ≤ v_{P} (l_{q}), then v_{P} (l_{1}) =

v_{P} (l_{2}) = .... = v_{P} (l_{q−n}) = min_{0≤i≤n}{v_{P} (x_{i})}.

Proof. See [Wa], Proposition 4.2.□

Theorem 2. Suppose that x_{0}, ..., x_{n} are elements of K, and are linearly independent over kK^{p^{m}}. Suppose that L_{1}, ..., L_{q} are linear forms in n+1 variables with coefficients in k, and are in general position, i.e. any n+1 elements of {L_{i}} are linearly independent over k. Then there exists a sequence of numbers 0 = ϵ_{0} < ϵ_{1} ⋯ < ϵ_{n}, such that

(q − n − 1)h(x_{0}, ..., x_{n}) ≤ ∑_{i=1}^{q} ∑_{P ∉S} min_{1≤i≤q}{ϵ_{n}, v_{P} (L_{i}(x_{0}, ..., x_{n})) − min_{0≤j≤n}{v_{P} (x_{j})}}
+ (ϵ_{1} + ⋯ + ϵ_{n}) max{0, 2g − 2 + |S|}.  (3.1)

Furthermore, when the characteristic of p = 0, ϵ_{i} = i; when p > 0, the upper bound for ϵ_{i} is described in Theorem 1.

Proof.

If x_{0}, ..., x_{n} are linearly independent over kK^{p^{m}} (p ≥ 0), then by Theorem 1 and its remarks there exists a sequnce of bounded numbers 0 = ϵ_{0} < ⋯ < ϵ_{n} such that

det(D_{t}^{ϵ_{i}}x_{j}) ≠ 0.

Let L_{i}(x_{0}, ..., x_{n}) = ∑_{j=0}^{n} a_{ij}x_{j} := l_{i} for 1 ≤ i ≤ q. Let I = {i_{0}, ..., i_{n}} be an index

subset of {1, ..., q} and {u_{1}, ..., u_{q−n−1}} be the complement of I. Let

G = \frac{l_{1} … l_{q}}{det(D_{t}^{ϵ_{i}}x_{j})}.  (3.2)

By Proposition 3.1.a

det(D_{t}^{ϵ_{i}}l_{i_{j}}) = det(a_{ij})det(D_{t}^{ϵ_{i}}x_{j}).

Then we have the following formula:

l_{u_{1}} … l_{u_{q−n−1}} = c_{I}G\frac{det(D_{t}^{ϵ_{i}}l_{i_{j}})}{l_{i_{0}} … l_{i_{n}}}
= c_{I}G det(\frac{D_{t_{P}}^{ϵ_{i}} l_{i_{j}}}{∏_{i=0}^{n} l_{i_{j}}})(\frac{dt}{dt_{P}} )^{− ∑_{i=0}^{n} ϵ_{i}},  (3.3)

where c_{I} is a nonzero constant in k. Let e_{P} = − min_{0≤i≤n}{v_{P} (x_{i})}. By Proposition

3.1.b we also have

v_{P} (det(\frac{D_{t_{P}}^{ϵ_{i}} l_{i_{j}}}{l_{i_{j}}})) = v_{P} (det(\frac{D_{t_{P}}^{ϵ_{i}} L_{i_{j}}(t_{P}^{e_{P}} f_{0}, ..., t_{P}^{e_{P}} f_{n})}{∏_{i=0}^{n} L_{i_{j}}(t_{P}^{e_{P}} f_{0}, ..., t_{P}^{e_{P}} f_{n})})).  (3.4)

Assume that v_{P} (l_{1}) ≥ v_{P} (l_{2}) ≥ ⋯ ≥ v_{P} (l_{q}) for P ∈ C. Then by Proposition 3.3,

v_{P} (l_{n+1}) = ⋯ = v_{P} (l_{q}) = min{v_{P} (x_{i})} = −e_{P} . Let I = {1, ..., n + 1}. Then (3.3)

and (3.4) imply

(q − n − 1) min\{v_{P} (x_{i})\} = v_{P} (G) + v_{P} (det(\frac{D_{t_{P}}^{ϵ_{i}} (l_{i_{j}}t^{e_{p}})}{∏_{i=0}^{n} l_{i_{j}}t^{e_{P}}})) − ∑_{i=0}^{n} ϵ_{i}v_{P} (\frac{dt}{dt_{P}} ). (3.5)

By Proposition 3.2, we have the following two inequalities:

v_{P} (det(\frac{D_{t_{P}}^{ϵ_{i}} (l_{i_{j}}t^{e_{P}})}{∏_{i=0}^{n} l_{i_{j}}t^{e_{P}}})) ≥ − min_{0≤i≤n}\{ϵ_{i}, v_{P} (l_{i_{j}}) + e_{P} \}.  (3.6)

v_{P} (det(\frac{D_{t_{P}}^{ϵ_{i}} (l_{i_{j}}t^{e_{P}})}{∏_{i=0}^{n} l_{i_{j}}t^{e_{P}}})) ≥ ∑_{i=0}^{n} ϵ_{i}.  (3.7)

Let S be a finite set of points of C. Then (3.5), (3.6), (3.7) and the Riemann-Roch

theorem (cf. [Ma]) imply

(q − n − 1)h(x_{0}, ..., x_{n}) ≤ ∑_{P ∉S} ∑_{j=1}^{q} min{ϵ_{n}, v_{P} (l_{j}) + e_{P} }

+ (ϵ_{1} + ⋯ + ϵ_{n}) max{0, 2g − 2 + |S|}.  (3.8)

This completes the proof. □

References

[B-M] Brownawell, D and Masser, D., Vanishing Sums in Function Fields, Math. Proc. Cambridge

Philos. Soc. 100 (1986), 427-434. [G-V] Garcia, A and Voloch, J. F., Wronskians and Linear Independence in Fields of Prime

Charactereistic, Manuscripta Math. 59 (1987), 457-469.

[La] Lang, S., Introduction to Complex Hyperbolic Spaces, Springer-Verlag, 1987.

[Ma] Mason, R. C., Diophantine Equations over Function Fields LMS. Lecture Notes 96, Cambridge Univ. Press, 1984.

[No] Noguchi, J., Nevanlinna-Cartan Theory and a Diophantine Equation over Function Fields, J. rein angew. Math. 487 (1997), 61-83.

[S-V] Stöhr, K-O. and Voloch, J. F., Weierstrass Points and Curve over Function Fields, Proc.

London Math. Soc.(3) 52 (1986), 1-19.

[Si] Silverman, J. H., The Arithmetic of Elliptic Curves, Springer-Verlag, 1985.

[Vol] Voloch, J. F., Diagonal Equations over Function Fields, Bol. Soc. Brazil Math. 16 (1985), 29-39.

[Wa1] Wang, J., T.-Y., The Truncated Second Main Theorem of Function Fields, J. of Number Theory 58 (1996), 139-157.

[Wa2] Wang, J., T.-Y., S-integral points of P^{n} − {2n + 1hyperplanesingeneralposition} over number fields and function fields, Trans. Amer. Math. Soc. 348 (1996), 3379-3389.

[Wa3] Wang, J., T.-Y., Integral points of projective spaces omitting hyperplanes over function fields of positive characteristic, preprint 1997.

[Wa4] Wang, J., T.-Y., abc estimate, integral points, and geometry of P^{n} minus hyperplanes, preprint 1998.

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

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