COMMUNICATIONS IN ALGEBRA

VANISHING POLYNOMIAL SUMS

Leonid N. Vaserstein^{1} and Ethel R. Wheland^{2}

1 Department of Mathematics, The Pennsylvania State University University Park, PA 16802, USA E-mail: vstein@math.psu.edu

2 Department of Mathematics and Computer Science The University of Akron, Akron, OH 44325, USA E-mail: wheland@deneka.cs.uakron.edu

ABSTRACT

If a sum of univariate polynomials is zero, then there are restrictions on the multiplicities of zeros of the polynomials.

1 INTRODUCTION

The Euler-Fermat equations are

x_{1}^{n} + · · · . + x_{m}^{n} = x_{0}^{n}  (1.1)

with unknown x_{i}. In 1769 Euler conjectured that (1.1) has no positive integral solutions when m < n. This generalizes both his result for n = 3 and Fermat’s Last Theorem. However his conjecture was refuted for m+1 = n = 5 [LP] and then for m+1 = n = 4 [E]. It is still unknown whether (1.1) has solutions with m + 1 = n ≥ 6 or with m = 3 = n − 2. On the other hand, (1.1) has positive integral solutions for m ≥ G(n) where G(n) is a number related with the Waring problem, whose best known bound for large n is G(n) ≤ n(log n + log log n + O(1)) [Wo].

In this paper, we are looking for solutions in F [t] rather than in the integers. Here F is a field of characteristic p ≥ 0 and F [t] is the polynomial ring in a variable t with coefficients in F.

When char(F) = 0, it is known that (1.1) has no nontrivial solutions in F [t] when m < (n/8)^{1/2} [NS]. Nontrivial here means primitive, indecomposable, nonconstant. A solution to (1.1) (or to the more general equation (1.2) below) is called primitive if gcd(x_{0}, x_{1}, ..., x_{m}) = 1. We call it indecomposable if no partial sum on the left hand side is 0. We call it nonconstant if not all x_{i} belong to F.

When p = char(F) ≠ 0, given any solution of (1.1) we can obtain solutions of a similar equation with the same m but n replaced by np, by taking the p-th power. So in the case of finite characteristic we can have nontrivial solutions of (1.1) with a small m and arbitrarily large n. Moreover, it is known that (1.1) may have nontrivial solutions with m = 3 and all n (including all n coprime with p) [P]. Furthermore, even for any fixed x_{0}, (1.1) with m = 3 may have nontrivial solutions with arbitrarily large n coprime with p [V2]. See Section 5.

Returning to the case p = 0, we consider equations more general than (1.1), the generalized Fermat equations

c_{0}x_{0}^{n_{0}} + · · · . + c_{m}x_{m}^{n_{m}} = 0,  (1.2)

in m + 1 unknowns x_{i} with given n_{i} ≥ 2 and nonzero c_{i} ∈ F. Our goal is to prove that (1.2) has no nontrivial solutions in F [t] when n_{i} are ‘large’ in a certain sense.

It is stated in [NS, p.486] where F = C, the complex numbers, that when m ≤ (n/8)^{1/2}, the equation (1.1) has no solutions in F [t] with nonconstant x_{i} which are pairwise coprime. Actually, the method of [NS] works when we relax the above condition to the following condition: the solution must be be primitive, indecomposable and nonconstant. Moreover, this method, involving the Wronskian of x_{i}, works to prove the following: Theorem 1.3. Let char(F) = 0. Then (1.2) has no nontrivial solutions in F [t] when ∑ 1/n_{j} ≤ 1/(m − 1) and 0 ≠ c_{j} ∈ F. In particular, (1.1) has no nontrivial solutions in F [t] when n ≥ m^{2} − 1.

Thus, this relaxes the condition m ≤ (n/8)^{1/2} of [NS] to the condition m ≤ (n+1)^{1/2}. Proposition 5.2 of [DG] is our Theorem 1.3 with the redundant condition that all x_{i} are pairwise coprime.

On the other hand, it is shown in [NS, p.486] (with attribution to Molluzzo) that (1.1) has a primitive nonconstant solution with an arbitrary x_{0} when m = [(4n+1)^{1/2}] and F = C. Moreover, this solution works for any algebraically closed field F of any characteristic. It is still unknown even for F = C whether (1.1) may have nontrivial solutions when (n + 1)^{1/2} < m < [(4n + 1)^{1/2}].

We prove Theorem 1.3 after improving upon results of [Vo], [BM], and [Z] generalizing the (abc)-theorem (see the next section). An interesting question outside the scope of the present paper is to establish a bound on the number of nontrivial solutions of (1.2) with given c_{i} ∈ F [t] for ‘large’ n_{i}. Results in this direction are known when char(F) = 0 (see [B], [BS], [Mu1], [Mu2]) or char(F) ≠ 0 and m = 2 [Vo2].

2 THE (abc)-THEOREM AND ITS GENERALIZATIONS

For any nonzero polynomial f = f(t) ∈ F [t] let ν(f) denote the number of α ∈ F̄, an algebraic closure of F, whose multiplicity in f is not divisible by p. When p = char(F) = 0, ν(f) is just the number of distinct zeros of f. Theorem 2.1. If a + b = c in A = F [t] with p = 0, gcd(a, b) = 1 and (abc)′ ≠ 0 then deg(c) ≤ ν(abc) − 1.

This well-known theorem (see [Ma], [L]) implies that the Fermat equation x^{n}+y^{n} = z^{n} has no nontrivial solutions in A = F [t] when p = 0 and n ≥ 3 (this was known before [NS], but in [NS] it was proved for n ≥ 32). It also has other applications [Mu1].

For a, b, c as in Theorem 2.1, one can ask whether there are restrictions on N = ν(abc), s = deg(c) other than the restrictions s ≤ N − 1 given by the theorem and the obvious restrictions 0 ≤ s and N ≥ 2. It can be shown by examples that the answer is “no” when char(F) = 0, while the complete description of all possible pairs (s, N) in the case of p ≠ 0 seems to be a difficult open problem.

Since we can switch a or b with -c in Theorem 2.1, its conclusion can be written as follows: max(deg(a), deg(b), deg(c)) ≤ ν(abc) − 1. It follows from our Theorem 3.1 that min(deg(a), deg(b), deg(c)) ≤ ν(abc) − 2. On the other hand, it is clear that

ν(abc) ≤ deg(a) + deg(b) + deg(c)) ≤ 3 max(deg(a), deg(b), deg(c)).

When char(F) = 0, it can be shown by examples that for any given integers M, s, N such that 0 ≤ s ≤ M ≤ N − 1 and s ≤ N − 2 ≤ s + 2M − 2 there are a, b, c ∈ F [t] such that deg(a) = s, deg(b) = deg(c) = M, a + b = c, and ν(abc) = ν(a) + ν(b) + ν(c) = N. A more general open problem is to describe all possible 6-tuples (deg(a), ν(a), deg(b), ν(b), deg(c), ν(c)). This seems to be a difficult problem even when char(F) = 0.

Here is our generalization of Theorem 2.1 in the case char(F) = 0: Theorem 2.2. Let m ≥ 2, y_{j} ∈ A = F [t], y_{1} + ⋯ + y_{m} = y_{0}, gcd(y_{1}, ..., y_{m}) = 1, char(F) = 0, not all y_{j} are constants, and no nonempty subsum of y_{1}, ..., y_{m} vanishes. Then:

(a) deg(y_{0}) < (m − 1)(∑_{j=0}^{m} ν(y_{j})),

(b) deg(y_{0}) ≤ (ν(y_{0}...y_{m}) − 1)m(m − 1)/2.

We will obtain the theorem in Sections 3 and 4 below as a corollary of more precise results. Namely, we will prove the theorem first in the case when y_{1}, ..., y_{m} are linearly independent over F (see Theorem 3.1 below which is an improved version of Theorem 2.2 in this case) and then in the case when y_{1}, ..., y_{m} are linearly dependent over F (see Theorem 4.1 below which is a more precise version of Theorem 2.2(b) and Corollary 4.8 below which is a more precise version of Theorem 2.2(a)).

Part (b) was first stated in [Ma], first proved in [BM, Theorem B] (with ν(y_{0}...y_{m}) instead of ν(y_{0}...y_{m}) − 1) and then proved in a more precise form in [Z]. In the case when y_{1}, ..., y_{m} are linearly independent over F, Theorem 2.2 is a particular case (with g = 0) of results of [BM] and [Vo1] (although part(a) is not stated there explicitly). It is unknown whether the number m(m − 1)/2 in (b) can be replaced by a smaller number when m ≥ 4. It is proved in [BB] that the correct number must be at least 2m − 3 and conjectured (and proved for m = 3) that the number is 2m − 3.

In the case m = 2 then both (a) and (b) coincide with Theorem 2.1 with p = 0, because in this case we have ν(y_{0}y_{1}y_{2}) = ν(y_{0}) + ν(y_{1}) + ν(y_{2}) and 1 = m − 1 = m(m − 1)/2.

Note also that (a) implies Theorem 1.3. Indeed, by obvious symmetry of the equation y_{1} + ⋯ + y_{m} = y_{0}, the number deg(y_{0}) in Theorem 2.2 can be replaced by deg(y_{j}) for any j and hence by M = max_{j} deg(y_{j}). Dividing (a) by M, we obtain that

1 < (m − 1) ∑_{i=0}^{m} ν(y_{i})/M ≤ (m − 1) ∑_{j=0}^{m} ν(y_{j})/ deg(y_{j}).

The number deg(y_{j})/ν(y_{j}) is the arithmetic mean of multiplicities of zeros of y_{i} while (m + 1)/ ∑_{i=0}^{m} ν(y_{i})/ deg(y_{i}) is the harmonic average of deg(y_{j})/ν(y_{j}). Corollary 2.3. Under the conditions of Theorem 2.2, the harmonic mean of deg(y_{j})/ν(y_{j}) is less than m^{2} − 1.

This corollary generalizes Theorem 1.3. (Applying the corollary to the case y_{0} = −c_{0}x^{n_{0}}, y_{i} = c_{j}x_{j}^{n_{j}} for j ≥ 1, we obtain Theorem 1.3.)

3 IMPROVING UPON THEOREM 2.2 IN THE LINEARLY INDEPENDENT CASE

In this section we assume that char(F) = 0. Our main goal is to prove Theorem 2.2 in the case when y_{1}, ..., y_{m} are linearly independent over F. In fact we state and prove a more precise version of Theorem 2.2, Theorem 3.1, involving the orders ord_{α}(y_{j}) We also obtain some known results as corollaries of Theorem 3.1.

First of all, for any integer d ≥ 1 and any sequence k_{1}, ...., k_{d} of integers we will define its “diversity” Δ(k_{1}, ..., k_{d}), which is an integer between 0 and (d − 1)d/2. Namely, we consider the set X of all sequences l_{1}, ...., l_{d} of distinct integers such that l_{j} ≥ k_{j} for all js. Then Δ(k_{1}, ..., k_{d}), our diversity of k_{1}, ...., k_{d}, is the maximal value of (d − 1)d/2 − ∑_{j=1}^{d}(l_{j} − k_{j}). In other words, we compute the minimal total increase to make all k_{j} distinct and then subtract it from its maximal possible value, (d − 1)d/2. It is clear that 0 ≤ Δ(k_{1}, ..., k_{d}) ≤ (d − 1)d/2. (To see the first inequality, note that {k_{j} + j} ∈ X when {k_{j}} is nondecreasing.) It is also clear that Δ(k_{1}, ..., k_{d}) = 0 if and only if {k_{j}} is a constant sequence and that Δ(k_{1}, ..., k_{d}) = (d − 1)d/2 if and only if all k_{j} are distinct. Here are two other easy upper bounds for Δ(k_{1}, ..., k_{d}):

Δ(k_{1}, ..., k_{d}) ≤ ∑_{j=1}^{d} min(j − 1, k_{j} − k_{1})

when k_{1} ≤ ... ≤ k_{d} and

Δ(k_{1}, ..., k_{d}) ≤ (d − 1)d/2 − ∑_{s}(m_{s} − 1)m_{s}/2,

where m_{s} is the total number of js such that k_{j} = s.

Now let y_{1}, ..., y_{d} ∈ F [t] be linearly independent over F, and we set y_{0} = y_{1} + ⋯ + y_{d}. For every α ∈ F̄ let k_{0} ≤ ... ≤ k_{d} be the numbers ord_{α}(y_{j}) (0 ≤ j ≤ d) written in nondecreasing order. We set

Δ_{α}(y_{1}, ..., y_{d}) = Δ(k_{1}, ..., k_{d}).

Similarly, let k_{0} ≤ ... ≤ k_{d} be the numbers {ord_{∞}(y_{j}) = − deg(y_{j}) : 0 ≤ j ≤ d} written in nondecreasing order. We set

Δ_{∞}(y_{1}, ..., y_{d}) = Δ(k_{1}, ..., k_{d}).

Thus, Δ_{α}(y_{1}, ..., y_{d}) is defined for all α ∈ P^{1}(F̄) = F̄ ∪ {∞}. For almost all α (all but finitely many) ord_{α}(y_{j}) = 0 hence Δ_{α}(y_{1}, ..., y_{d}) = 0. Theorem 3.1. Let m ≥ 2, y_{j} ∈ A = F [t], y_{1} + ⋯ + y_{m} = y_{0}, gcd(y_{1}, ..., y_{m}) = 1, char(F) = 0, and assume that y_{1}, ...., y_{m} are linearly independent over F. Then

deg(y_{0}) ≤ −(m − 1)m + ∑_{α∈P^{1}(F̄)} Δ_{α}(y_{1}, ..., y_{m})

≤ −(m − 1)m/2 + (m − 1) min(ν(y_{0}...y_{m})m/2, ∑_{j=0}^{m} ν(y_{j})).

Note that Δ_{α}(y_{1}, ..., y_{m}) ≠ 0 if and only if either α is a zero of y_{0}...y_{m} (recall that gcd(y_{1}, ..., y_{m}) = 1) or α = ∞ and not all deg(y_{j}) are the same. Therefore

deg(y_{0}) ≤ (card(S) − 2)m(m − 1)/2

when S is a finite subset in P^{1}(F̄) and all y_{j} are units outside S. Thus, our Theorem 3.1 contains Theorem 4 of [Vo] (which is essentially the same as Corollary 1 of [BM]) with g = 0. Moreover Δ_{α}(y_{1}, ..., y_{m}) ≤ (m − 1)m/2 − (l − 2)(l − 1)/2 where l is the number of js with ord_{α}(y_{j}) taking a fixed value. (In the case when this value is not minimal, (l − 2)(l − 1)/2 can be replaced by (l − 1)l/2.) Therefore our Theorem 3.1 contains Theorem A of [BM] with g = 0. Moreover, the number S of places α where not all ord_{α}(y_{j}) are 0 in [BM] and [Vo] can be replaced by the number S′ ≤ S of places α where not all ord_{α}(y_{j}) are the same.

Theorem 3.1 gives Theorem 2.2 in the case when the polynomials y_{1}, ...., y_{m} in Theorem 2.2 are linearly independent (just use that Δ_{α}(y_{1}, ..., y_{m}) ≤ m(m − 1)/2 for all α to get Theorem 2.2 (b) and that Δ_{α}(y_{1}, ..., y_{m}) ≤ (m − 1){#j :ord_{α}(y_{j}) ≠ 0}).

After Theorem 3.1 is proved we will obtain some known results as its consequences, and then, in the next section, we use it to prove an improved version of Theorem 2.2, Theorem 4.1.

To prove Theorem 3.1, we introduce orders of finite-dimensional F -subspaces of F [t]. Let V be an F -vector subspace of A = F [t] of dimension d. Assume that 1 ≤ d < ∞.

For any α ∈ F̄, we can choose a basis v_{1}, ...., v_{d} of V such that ord_{α}(v_{1}) < ... <ord_{α}(v_{d}). These numbers ord_{α}(v_{i}) are independent of the choice of such a basis. We define ord_{α}(V) by

ord_{α}(V) = −(d − 1)d/2 + ∑_{i=1}^{d}ord_{α}(v_{j}).

Clearly, ord_{α}(V) ≥ 0. For any basis y_{1}, ...., y_{d} of V we have

ord_{α}(V) ≥ −(d − 1)d/2 + ∑_{i=1}^{d}ord_{α}(y_{i}).

This is because by addition operations with the polynomials y_{j} we can make all ord_{α}(y_{j}) distinct while increasing them or keeping them the same. To get a better bound, we can replace one of y_{j} with minimal ord_{α}(y_{j}) by y_{0} = y_{1} + ⋯ + y_{d}. This proves a more precise inequality:

Lemma 3.2. Let V be as above and y_{1}, ...., y_{d} a basis of V over F. Then, for all α ∈ F̄,

ord_{α}(V) ≥ −Δ(ord_{α}(y_{1}), ...,ord_{α}(y_{d})) + ∑_{j=1}^{d}ord_{α}(y_{j}) and

ord_{α}(V) ≥ −Δ_{α}(y_{1}, ..., y_{d}) + ∑_{j=1}^{d}ord_{α}(y_{j}) − min(ord_{α}(y_{0}), ...,ord_{α}(y_{d})).

Also we can choose a basis v_{1}, ...., v_{d} of V such that deg(v_{1}) < ... < deg(v_{d}). The numbers deg(v_{i}) are independent of the choice of such a basis. We set

deg(V) = −d(d − 1)/2 + ∑_{i=1}^{d} deg(v_{i})

and

ord_{∞}(V) = −d(d − 1)/2 + ∑_{i=1}^{d} ord_{∞}(v_{i}),

where ord_{∞}(v_{i}) = − deg(v_{i}).

Clearly, deg(V) ≥ 0 and deg(V)+ord_{∞}(V) = −d(d − 1). For any basis v_{1}, ...., v_{d} of V we have deg(V) ≤ −d(d − 1)/2 + ∑_{i=1}^{d} deg(v_{i}). Moreover, we have the following analogue of Lemma 3.2:

Lemma 3.3. The conclusions of Lemma 3.2 also hold for α = ∞.

Proof. It is similar to that of Lemma 3.2. Also Lemma 3.3 can be reduced to Lemma 3.2 by a change of variable.

Proposition 3.4. deg(V) = ∑_{α∈F̄}ord_{α}(V).

Proof. We pick a basis v_{1}, ...., v_{d} of V and consider the Wronskian D = det(v_{i}^{(j)}). It is a nonzero polynomial in F [t], and it is determined by V up to a nonzero scalar factor. Choosing v_{i} as in the definition of deg(V), we see that deg(D) = deg(V). Indeed, writing the determinant D as a sum of d! terms, we see that every nonzero term has degree deg(V) and that the sum of coefficients in this degree is, up to a nonzero scalar factor equal to the product of leading terms in v_{i}, the determinant of the matrix (\binom{k_{i}}{j−1})_{1≤i,j≤d} which is, up to the factor (1!...(d − 1)!)^{d}, the Vandermonde determinant det(k_{i}^{j}) = ∏_{i<j}(k_{i} − k_{j}) ≠ 0 where k_{i} = deg(v_{i}).

On the other hand, choosing v_{i} as in the definition of ord_{α}(V), we see that ord_{α}(D) = ord_{α}(V) for all α ∈ F̄. Indeed writing the determinant D as a sum of d! terms, we see that every term is divisible by (x − α)^{ord_{α}(V)} and that the sum of the terms of order ord_{α}(V) is, up to a factor of order 0 equal to the product of v_{i}/(x − α)^{ord_{α}(v_{i})}, the determinant of the matrix (\binom{k_{i}}{j−1})_{1≤i,j≤d} which is, up to the factor (1!...(d − 1)!)^{d}, the Vandermonde determinant det(k_{i}^{j}) = ∏_{i<j}(k_{i} − k_{j}) where k_{i} =deg(v_{i}).

Now deg(V) = deg(D) = ∏_{α∈F̄}ord_{α}(D) = ∏ord_{α}(V).

Remark. When d = 1, the proposition says that deg(D) = ∑_{α∈F̄}ord_{α}(D) for every nonzero polynomial D ∈ F [t].

Remark. As a corollary of the proposition, ∑_{α}ord_{α}(V) < ∞.

Now we are ready to prove Theorem 3.1. Let d_{0} ≤ d_{1} ≤ ... ≤ d_{m} be the degrees of y_{j}(0 ≤ j ≤ m) written in nondecreasing order. Then d_{m−1} = d_{m} ≥ m. By Lemma 3.3 with d = m,

deg(V) = −m(m − 1) − ord_{∞}(V) ≤ −m(m − 1) + Δ_{∞}(y_{1}, ..., y_{m}) − d_{m} + ∑_{j=0}^{m} d_{j}.

On the other hand, since gcd(y_{1}, ...., y_{m}) = 1, for any α ∈ F̄ we have

min(ord_{α}(y_{1}), ...,ord_{α}(y_{m})) = 0

and, by Lemma 3.2 with d = m,

ord_{α}(V) ≥ −Δ_{α}(y_{1}, ..., y_{m}) + ∑_{j=1}^{m} ord_{α}(y_{j}).

Switching, if neccessary, −y_{0} with a polynomial y_{j} such that y_{j}(α) ≠ 0, we have

ord_{α}(V) ≥ −Δ_{α}(y_{1}, ..., y_{m}) + ∑_{j=0}^{m} ord_{α}(y_{j}).

Taking the sum over all α ∈ F̄ and using Proposition 3.1 and the fact that deg(y_{j}) = ∑ord_{α}(y_{j}) for each j, we obtain

deg(V) = ∑_{α∈F̄}ord_{α}(V) ≥ − ∑_{α∈F̄} Δ_{α}(y_{1}, ..., y_{m}) + ∑_{j=0}^{m} d_{j}.

Comparing these upper and lower bounds for deg(V) and cancelling ∑_{j=0}^{m} d_{j}, we obtain the conclusion of Theorem 3.1.

Here are some consequences of Theorem 3.1. Corollary 3.5 (cf. Theorem on p.480 of [NS]). Let n ≥ 3 and ∑_{j=1}^{m} x_{j}^{n} = t in F [t] with char(F) = 0. Then m^{2} − m > n. Proof. Without loss of generality, we can assume that F is algebraically closed and that x_{1}^{n}, ..., x_{m}^{n} are linearly independent (otherwise, t is a sum of a smaller number of n-th powers). Also gcd(x_{1}^{n}, ..., x_{m}^{n}) = 1 because this is an n-th power dividing t and n ≥ 2. Let M = max(deg(x_{1}), ..., deg(x_{m})). By Theorem 3.1,

Mn ≤ −(m − 1)m/2 + (m − 1)(ν(t) + ∑_{j=1}^{m} ν(x_{j}))

≤ −(m − 1)m/2 + (m − 1)(1 + Mm) < Mm(m − 1),

because ν(y_{j}) = ν(x_{j}) ≤ deg(x_{j}) ≤ M, ν(t) = 1, and m ≥ 3 when n ≥ 3.

Remark. The condition n ≥ 3 is necessary, and it is implicit in the theorem on p.480 of [NS], see p.479 or 481 in [NS]. Corollary 3.6. Let n ≥ 2 and ∑_{j=1}^{m} x_{j}^{n} = tx_{0}^{n} in F [t] with x_{0} ≠ 0 and char(F) = 0. Then m^{2} > n + 1. Proof. Without loss of generality, we can assume that F is algebraically closed, that x_{1}^{n}, ..., x_{m}^{n} are linearly independent, and that gcd(x_{1}, ..., x_{m}) = 1. Let

M = max(deg(x_{1}), ..., deg(x_{m})).

By Theorem 3.1, Mn < (M(m + 1 + 1))(m − 1) − m(m − 1)/2, hence n < m^{2} − 1 when n ≥ 3. When n = 2, notice that m ≥ 2.

Remark. This improves upon the theorem on p.483 of [NS] which asserts that m^{2} > n/8.

Theorem 3.7. Let n ≥ 2 and x_{1}^{n} + ⋯ + x_{m}^{n} = x_{0} in F [t] with char(F) = 0. Assume that m ≥ 1 and x_{1}^{n}, ..., x_{m}^{n} are linearly independent over F. Then

deg(x_{0}) ≥ M(n − m^{2} + m) + (m − 1)m/2,

where M := max(deg(x_{1}), ..., deg(x_{m})). Proof. Without loss of generality, we can assume that F is algebraically closed, that m ≥ 2 (the case m = 1 is trivial), that gcd(x_{1}, ..., x_{m}) = 1 (the general case follows easily from the primitive one), that n > m(m − 1)/2 (otherwise there is nothing to prove), that x_{1}^{n}, ..., x_{m}^{n} are linearly independent over F (otherwise the number m can be reduced), and that M = deg(x_{m}). Using Theorem 3.1 with y_{0} = −x_{m}^{n}, y_{m} = −x_{0}, y_{j} = x_{j}^{n} for j ≤ m − 1 and the fact that

Δ_{α}(y_{1}, ..., y_{m}) ≤ (m − 1)(∑_{j=0}^{m−1}sign(ord_{α}(y_{j}))+ord_{α}(y_{m}) for all α ∈ F̄, we obtain that

nM = deg(y_{0}) ≤ deg(x_{0}) + ∑_{j=0}^{m−1} ν(y_{j}) − m(m − 1)/2

≤ deg(x_{0}) + (m − 1)mM − m(m − 1)/2,

hence deg(x_{0}) ≥ M(n − m^{2} + m) + m(m − 1)/2.

Remark. Our theorem is close to the theorem on p.481 of [NS] and has the same conclusion as the second conclusion of that theorem. The condition that n ≥ 2 is implicit in [NS] (without this condition the theorem is not true). We relaxed the condition of [NS] that all x_{j} are nonconstant and nonlinear to the condition that not all of them are constant. We imposed an additional condition that x_{1}^{n}, ..., x_{m}^{n} are linearly independent over F to prevent counterexamples to the theorem of [MS] like the following two:

(i) m = 2, any n ≥ 2, any M ≥ 1, x_{1} = t^{M}, x_{2} = −t^{M}, x_{0} = 0; (ii) m = 2, any n ≥ 3, any M ≥ 1, x_{1} = t^{M}, x_{2} = t^{M}, x_{0} = 2t^{M}. The condition is missing in [NS] by mistake. When F = F̄, this condition is weaker

than the condition that x_{0} is not a sum of a smaller than m number of n-th powers in F [t]. It could also be replaced by the condition that x_{0} is not divisible by any nonconstant n-th power in F [t].

The first conclusion of the theorem of [NS] is that deg(x_{0}) ≥ n−(m−1)m/2. However, this conclusion is stated twice on the same page as deg(x_{0}) ≥ n − (m − 1)m, which follows from our theorem because either M = 0 and m = 1 or M ≥ 1. Also the theorem implies that deg(x_{0}) ≥ n − (m − 1)m when either n ≥ m^{2} − m or M = 1. By the way, when M = 1 it is easy to prove that in fact deg(x_{0}) ≥ n − m + 1, and that this bound cannot be improved when n ≥ m − 1 and F = F̄.

Theorem 2.2(a) with linearly independent y_{1}, ..., y_{m} (which we have obtained from Theorem 3.1) allows us now to obtain Theorem 1.3 in the following two cases (in general, Theorem 1.3 follows from Theorem 2.2 which we will obtain from Theorem 4.4 below). Corollary 3.8. The equation (1.1) has no nonconstant indecomposable primitive solutions when n ≥ m^{2} − 1. Proof. Consider such a solution of (1.1). Let M = max(deg(x_{1}), ..., deg(x_{m})). If x_{1}^{n}, ..., x_{m}^{n} are linearly independent, then Theorem 3.1 via Theorem 2.2 (a) gives that Mn < (m + 1)M(m − 1).

If x_{1}^{n}, ..., x_{m}^{n} are linearly dependent, we consider a linear relation of minimal length. Permuting terms, we can assume that this relation is c_{1}x_{1}^{n} + ⋯ + c_{d+1}x_{d+1}^{n} = 0. If not all terms here are colinear, we divide this equality by the gcd and pass to F̄ to conclude that n < d^{2} − 1 < m^{2} − 1. Otherwise, we also pass to F̄ and replace all terms by one term of the form x^{′n} and after dividing by the gcd we obtain an equation of the form (1.1) with not all terms constant and with the same n but m replaced by m − d < m, hence we are done by induction on m.

Corollary 3.9. The equation (1.2) has no nonconstant indecomposable solutions with pairwise coprime terms when ∑ 1/n_{j} ≤ 1(m − 1).

Proof. Again either the terms are linearly independent and we are done by Theorem 3.1 or we can obtain a shorter equation of the same form form (passing, if necessary, to F̄). Here we cannot divide by the gcd because this produces nonconstant coefficients, but we do not need to do this because the terms are pairwise coprime.

4 IMPROVING THEOREM 2.2 IN GENERAL

Now we drop the condition about linear independence and prove a more precise version of Theorem 2.2 involving the dimension d ≤ m of F y_{1} + ⋯ + F y_{m} and another number d′ ≤ d which is defined below.

First we introduce two definitions which make sense for any sequence (family, multiset) y_{0}, ..., y_{m} of vectors in an F -vector space. Following [BM], a nonempty subset I ⊂ {0, 1, ..., m} is called minimal if the corresponding family {y_{j} : j ∈ I} of vectors is linearly dependent over F but {y_{j} : j ∈ J} is linearly independent over F for any proper subset J ⊂ I. A linear relation is called minimal if the corresponding family of vectors is minimal.

Clearly, card(I) ≤ d + 1 for any minimal set I. We define d′ + 1 to be the maximal cardinality of the minimal sets. Thus, d′ ≤ d = dim(F y_{0} + ⋯ + F y_{m}).

In general not every index j needs to be an element of a minimal set. The condition y_{0} = y_{1} + ⋯ + y_{m} in Theorem 2.2 implies that every j is contained in a minimal set. Namely, we can take a basis not involving j, write y_{j} with respect to this basis, and drop zero terms to obtain such a set. This condition, together with the condition of no vanishing subsums of Theorems 2.2 and 4.1, implies that all linear relations between y_{0}, y_{1}, ..., y_{m} follow from minimal relations (Lemma 4 of [BM]).

We call a family (multiset) y_{0}, ..., y_{m} of vectors irreducible if for any partition I ∪ I′ = {0, 1, ..., m} into nonempty parts, the corresponding vector subspaces ∑_{j∈I} F y_{j} and ∑_{j∈I′} F y_{j} have a nonzero intersection. For example, every minimal set gives an irreducible family of vectors. Under the conditions of Theorem 2.2, the family y_{0}, ..., y_{m} is irreducible. It can be shown that for every irreducible family z_{0}, ..., z_{m} of vectors there are nonzero c_{j} ∈ F such that y_{j} = c_{j}z_{j} satisfy the conditions of Theorem 2.2. So the conditions of the next two theorems are not really more general than those of Theorem 2.2 (but the conclusions are stronger).

Theorem 4.1. Let m ≥ 2, y_{j} ∈ F [t], gcd(y_{1}, ..., y_{m}) = 1, and y_{0}, y_{1}, ..., y_{m} be an irreducible family. Let d be the dimension of F y_{1} + ⋯ + F y_{m} over F and let d′ ≤ d be as above. Then:

(a) deg(y_{0}) ≤ (ν(y_{0}...y_{m}) − 1)(d′ − 1)d/2.

Furthermore, if deg(y_{1}) = ... = deg(y_{m}), then:

(b) deg(y_{0}) ≤ (ν(y_{0}...y_{m}) − 2)(d′ − 1)d/2.

Proof. First we consider the special case when d = m. Since the family y_{0}, y_{1}, ..., y_{m} is irreducible, d′ = d = m and y_{0} = ∑_{j=1}^{m} c_{j}y_{j} with 0 ≠ c_{j} ∈ F. Applying Theorem 3.1, we obtain that

deg(y_{0}) ≤ deg(y_{m}) ≤ −(m − 1)m + ∑_{α∈P^{1}(F̄)} Δ_{α}(c_{1}y_{1}, ..., c_{m}y_{m}).

Now we use that Δ_{α}(c_{1}y_{1}, ..., c_{m}y_{m}) ≤ (m − 1)m/2 for all α ∈ P^{1}(F̄) and that in the case deg(y_{1}) = ⋯ = deg(y_{m}) we have

Δ_{∞}(c_{1}y_{1}, ..., c_{m}y_{m}) = min(deg(y_{m}) − deg(y_{0}), m − 1) ≤ deg(y_{m}) − deg(y_{0})

to obtain the theorem.

Now we consider the general case. Note that y_{j} ≠ 0 for all js because of the irreducibility condition. We proceed by induction on m. If m = 2, then either d = 2 and we are done by the special case above or d = 1 and all y_{j} are constants, so both our conclusions say that 0 ≤ 0 which is true. Assume now that d < m ≥ 3.

By the irreducibility condition, we can write y_{0} as a linear combination of y_{1}, ..., y_{m}. Dropping js with zero coefficients, we obtain a minimal I_{0} ⊂ {0, 1, ..., m} containing 0 with 2 ≤ card(I_{0}) ≤ d′ +1 ≤ d+1 ≤ m. The corresponding family {y′_{j} : j ∈ I_{0}} is an irreducible family. Let {y_{j} : j ∈ I_{1}} be an irreducible family containing I_{0} and with maximal possible card(I_{1}) ≤ m. Set m′ = card(I_{1}) − 1, V_{1} = ∑_{j∈I_{1}} F y_{j}, d_{1} = dim(V_{1}), and z_{1} = gcd(V_{1}).

By the induction hypothesis,

\begin{aligned}deg(y_{0}/z_{1}) & ≤ (ν( ∏_{j∈I_{1}}(y_{j}/z_{1}) − e)d_{1}(d′ − 1)/2 \\ & ≤ (ν(y_{0}y_{1}...y_{m}) − e)d_{1}(d′ − 1)/2 \end{aligned}  (4.2)

where e = 2 when deg(y_{1}) = ... = deg(y_{m}), and e = 1 otherwise.

When z_{1} = 1, e.g., d′ = d, we are done. In general, we need an upper bound for deg(z_{1}) to finish the proof. Let I_{2} be the complement of I_{1} in {0, 1, ..., m} and V_{2} = ∑_{j∈I_{2}} F y_{j}. By the irreducibility condition, V_{1} ∩ V_{2} ≠ 0.

Let 0 ≠ z = ∑_{j∈I′} c_{j}y_{j} ∈ V_{1} ∩ V_{2} be a vector with the least number of nonzero coefficients when written as a linear combination of {y_{j} : j ∈ I_{2}} with minimal possible number card(I′) of nonzero coefficients, where I′ ⊂ I_{2} and c_{j} ∈ F. This relation is minimal, hence I_{1} ∪ I′ is irreducible. By the maximality of I_{1}, we have I′ = I_{2}. Thus, m_{2} := card(I_{2}) = dim(I_{2}) = d − d_{1} = m − m_{1} ≤ d′. Set z_{2} = gcd(V_{2}). Note that gcd(z_{1}, z_{2}) = 1.

Applying Theorem 3.1, we obtain that

deg(z_{1}) ≤ deg(z/z_{2}) ≤ max(deg(y_{j}/z_{2}) : j ∈ I_{2})

≤ −(m_{2} − 1)m_{2} + ∑_{α∈P^{1}(F̄)} Δ_{α}(c_{j}y_{j}/z_{2} : j ∈ I_{2}). (4.3)

Now we use different upper bounds for

Δ_{α}(c_{j}y_{j}/z_{2} : j ∈ I_{2}) = Δ_{α}(c_{j}y_{j} : j ∈ I_{2})

in (4.3) for different α ∈ P^{1}(F̄). When α is not a zero of (z/z_{2})y_{0}...y_{m} nor α = ∞, we have Δ_{α}(c_{j}y_{j} : j ∈ I_{2}) = 0.

When α is a zero of y_{0}...y_{m}, we use that

Δ_{α}(c_{j}y_{j} : j ∈ I_{2}) ≤ (m_{2} − 1)m_{2}/2 ≤ (d − d_{1})(d′ − 1)/2.

We use the same bound when α = ∞, unless deg(y_{1}) = ... = deg(y_{m}) in which case we use the following more precise bound:

Δ_{∞}(c_{j}y_{j} : j ∈ I_{2}) = min(m_{2} − 1, max(deg(y_{j}) : j ∈ I_{2}) − deg(z/z_{2}))

≤ max(deg(y_{j}) : j ∈ I_{2})− deg(z_{1}).

Finally, when α is a zero of z/z_{2} but not a zero of y_{0}...y_{m}, we use that

Δ_{α}(c_{j}y_{j} : j ∈ I_{2}) ≤ ord_{α}(z/z_{2}) = ord_{α}(z)− ord_{α}(z_{1}). Substituting these bounds to (4.3), we obtain that deg(z_{1}) ≤ deg(z/z_{2}) ≤ (ν(y_{0}...y_{m}) − e)(d − d_{1})(d′ − 1)/2. Adding this to (4.2), we obtain the conclusions of the theorem. Remark. Our Theorem 4.1(a) improves the main result of [Z], (see (4) on page 88

there) with g = 0. The number μ in [Z] is our d, and we replaced μ(μ − 1)/2 in the upper bound of [Z] by d′(d − 1)/2 ≤ d(d − 1)/2. The number S in [Z] is our ν(y_{0}...y_{m}) + 1 when S ≠ 0; when S = 0 all terms are constant and ν(y_{0}...y_{m}) = 0. Our Theorem 4.1(b) shows that S in [Z] can be replaced by S −1 in the case when all degrees are the same. Our proof followed those of [BM] and [Z].

We neither stated nor proved extensions of our results to the case of arbitrary genus g although there are no difficulties in doing this. The number S in [BM], [Z] can be replaced by the number S′ ≤ S of places where not all ord_{α}(y_{j}) are the same.

Our next result is similar to Theorem B of [BM] (in the case g = 0). For any nonzero polynomials y_{0}, ..., y_{m} ∈ F [t] and any α ∈ P^{1}(F̄) let μ(α) + 1 be the number of js with minimal value of ord_{α}(y_{j}) (over 0 ≤ j ≤ m). Note that 1 ≤ μ(α) ≤ m. When m ≥ 1 and either y_{0} = y_{1} + ⋯ + y_{m} or a family y_{0}, ..., y_{m} is irreducible, we have 1 ≤ μ(α) for all α. When gcd(y_{0}, ..., y_{m}) = 1, the minimal value is 0 for all α ∈ F̄. For α = ∞, μ(∞) is the number of js with maximal deg(y_{j}).

Theorem 4.4. Let m ≥ 2, y_{j} ∈ F [t], gcd(y_{1}, ..., y_{m}) = 1, and y_{0}, y_{1}, ..., y_{m} be an irreducible family. Let d be the dimension of F y_{1} + ⋯ + F y_{m} over F and let d′ ≤ d be as above (i.e., d′ + 1 is the maximal cardinality of the minimal sets). Then

deg(y_{0}) ≤ (d′ − 1) ∑_{j=0}^{m} ν(y_{j}).

If either d = m or the index 0 belongs to a minimal set I_{0} with card(I_{0}) = d′, then deg(y_{0}) ≤ −d′(d′ − 1)/2 + (d′ − 1) ∑_{j=0}^{m} ν(y_{j}).

Proof. We follow closely the proof of Theorem 4.1.

Case 1 : d = m. In this case d′ = d = m and y_{0} = ∑_{j=1}^{m} c_{j}y_{j} with 0 ≠ c_{j} ∈ F. So our statement follows from Theorem 3.1 because

Δ_{α}(c_{1}y_{1}, ..., c_{m}y_{m}) ≤ m(m − 1)/2 for α = ∞ and

Δ_{α}(c_{1}y_{1}, ..., c_{m}y_{m}) ≤ (m − 1) ∑_{j=0}^{m} sign(ord_{α}(y_{j}))

for all α ∈ F̄.

Case 2: the index 0 belongs to a minimal set I_{0} with card(I_{0}) = d′.

We proceed by induction on m. If m = 2, then either d = 2 and we are done by Case 1 or d = 1 and all y_{j} are constants, so our conclusion is that 0 ≤ 0. Assume now that d < m ≥ 3.

Let I_{0}, I_{1}, I_{2}, d_{1}, m_{1}, z, z_{1}, z_{2}, m_{2} be as in the proof of Theorem 4.1 with the following additional condition: we choose I_{0} ∋ 0 with card(I_{0}) = d′, so card(I_{1}) ≥ d′.

By the induction hypothesis,

deg(y_{0}/z_{1}) ≤ −(d′ − 1)d′/2 + (d′ − 1) ∑_{j∈I_{1}} ν(y_{j}/z_{1}).  (4.5)

If z_{1} = 1, we are done because the right hand side in (4.5) is less than the right hand side in the conclusion of Theorem 4.4. Let now z_{1} ≠ 1. Then d′ ≥ d_{2} ≥ 2. We set I_{3} = I_{2} ∪{m+1} and y_{m+1} = z. Applying Theorem 3.1, we obtain that

deg(z_{1}) ≤ deg(z/z_{2}) ≤ d_{2} ∑_{j∈I_{3}} ν(y_{j}/z_{2}) ≤ (d′ − 1)ν(z/z_{2}) + (d′ − 1) ∑_{j∈I_{2}} ν(y_{j}). (4.6)

Now it remains to add (4.5) and (4.6) and to observe that ν(z/z_{2}) ≤ ν(z_{1}) and ∑_{j∈I_{1}} ν(y_{j}/z_{1}) + ν(z_{1}) ≤ ∑_{j∈I_{1}} ν(y_{j}). General case. Set M = max(deg(y_{1}),..., deg(y_{m})). We pick a β ∈ F̄ which is not a

zero of y_{o}...y_{m} and replace y_{j} = y_{j}(t) ∈ F [t] by z_{j} = t^{M}y(1/t + β). Note that the family z_{0}, z_{1}, ..., z_{m} ∈ F [t] is irreducible, gcd(z_{1}, ..., z_{m}) = 1, deg(z_{j}) = M for all j, ν(y_{j}) = ν(z_{j}) for at least two js, and ν(y_{j}) ≤ ν(z_{j}) ≤ ν(y_{j}) + 1 for all j.

Now we can use Case 2 with y_{j} replaced by z_{j} and obtain that

deg(y_{0}) ≤ M = deg(z_{0}) ≤ m − 2 − d′(d′ − 1)/2 + (d′ − 1) ∑_{j=0}^{m} ν(z_{j})

= m − 2 − d′(d′ − 1)/2 + (d′ − 1) ∑_{j=0}^{m} ν(y_{j}).

If m − 2 − d′(d′ − 1)/2 ≤ 0, we are done. Otherwise we replace z_{j} = z_{j}(t) by z_{j}(t^{N}) with a large integer N and obtain that MN ≤ m−2−d′(d′ −1)/2+M(d′ −1) ∑_{j=0}^{m} ν(y_{j}).

Dividing both sides by N and sending N to ∞, we obtain the conclusion of the theorem. Corollary 4.7. Under the conditions of Theorem 4.1, deg(y_{0}) ≤ (d − 1) ∑_{j=0}^{m} ν(y_{j}).

If not all y_{j} are constant, then deg(y_{0}) < (m − 1) ∑_{j=0}^{m} ν(y_{j}).

If not all y_{j} are constant and they are pairwise coprime, then deg(y_{0}) < (d′ − 1) ∑_{j=0}^{m} ν(y_{j}). Proof. This follows from Theorem 4.1 using the following observations. If all y_{j} are constant, i.e., d = 1, then the conclusion is that 0 ≤ 0. When y_{j} are constant and they are pairwise coprime we can apply Theorem 3.1 or Theorem 4.1 to a minimal set containing the index 0.

Our next result is similar to Theorem B of [BM] (in the case g = 0). For any nonzero polynomials y_{0}, ..., y_{m} ∈ F [t] and any α ∈ P^{1}(F̄) let μ(α) + 1 be the number of js with minimal value of ord_{α}(y_{j}) (over 0 ≤ j ≤ m). Note that 1 ≤ μ(α) ≤ m. When m ≥ 1 and either y_{0} = y_{1} + ⋯ + y_{m} or a family y_{0}, ..., y_{m} is irreducible, we have 1 ≤ μ(α) for all α.

When gcd(y_{0}, ..., y_{m}) = 1, the minimal value is 0 for all α ∈ F̄. For α = ∞, μ(∞) is the number of js with maximal deg(y_{j}). Theorem 4.8. Under the conditions of Theorem 4.1,

deg(y_{0}) ≤ 2d′ − d^{′2} − d + ∑_{α∈P^{1}(F̄)}(d(d − 1)/2 − μ′(α)(μ′(α) − 1)/2), where μ′(α) = max(d − d′, d − m + μ(α)). Proof. The proof is similar to those of Theorems 4.1 and 4.4 and we leave it to the reader.

5 NONZERO CHARACTERISTIC CASE

Now let p = char(F) ≠ 0. It was shown in [P, p.54] that the equation

x_{1}^{n} + x_{2}^{n} = x_{3}^{n} + x_{4}^{n}  (5.1)

has nontrivial solutions for every n. When n is odd or F is algebraically closed, (5.1) is essentially the same equation as (1.1) with m = 3.

In [V2] it is shown that for every n, every polynomial in F [t] is the sum of at most n_{(p)} n-th powers provided that F is an algebraically closed field with char(F) = p ≠ 0 and n is coprime with p. Here n_{(p)} = ∏(a_{i} + 1) − 1 ≤ n, where the a_{i} are the digits of n in base p. Moreover [V2] gives solutions with pairwise coprime x_{i}. It follows that (1.1) has nontrivial solutions (with an arbitrary x_{0}) when m ≥ S(n, p), where S(n, p) is the minimum of (nk)_{(p)} over all natural numbers k. Obviously, S(n, p) could be small for very large n. For example, S(n, p) = 3 if n > 2 divides p^{k} + 1 for some k. So for all such n and any x_{0}, the equation (1.1) with m = 3 has nontrivial solutions in F̄[t], where F̄ is an algebraic closure of F.

Thus, when p ≠ 0, no upper bound on m prevents the existence of indecomposable primitive nonconstant solutions of (1.1) in F [t], and one cannot place any bound of the form m < f(n) with f(n) → ∞ as n → ∞ to prevent pairwise coprime nonconstant solutions. One can hope to obtain a bound of the form n < f(S(n, p)) (under additional conditions on solutions).

The main trouble is that the determinant D in Section 3 may vanish even when y_{1}, ..., y_{m} are linearly independent over F. In fact [K] it vanishes if and only if y_{1}, ..., y_{m} are linearly dependent over F (t^{p}). If D ≠ 0, then it is easy to obtain much sharper conclusions (cf. the case k = 1 of the next proposition).

For any integer k ≥ 1, we set F_{k} = F (t^{p^{k}}). For any natural number n = ∑ s_{i}p^{i} written in base p, we set mod(m, p^{k}) = ∑_{i=0}^{k−1} s_{i}p^{i}. For any nonzero polynomial z = z(t) = c ∏(t − α)^{n(α)} ∈ F [t], where 0 ≠ c ∈ F, α ∈ F̄, n(α) = ord_{α}(z), and any integer k ≥ 1, we define deg_{k}(z) = ∑_{α} mod(n(α), p^{k}). It is a nondecreasing function of k and deg_{k}(z) = deg(z) when p^{k} ≥ deg(z). We denote by ν_{k}(z) the number of exponents n(α) which are not divisible by p^{k}. In particular, ν(z) = ν_{1}(z). Proposition 5.2. Let m ≥ 2 and y_{0} = y_{1} + ⋯ + y_{m} with y_{j} ∈ F [t]. Suppose that gcd(y_{0}, ..., y_{m}) = 1. For some integer k ≥ 1 assume that y_{1}, ..., y_{m} are linearly independent over F_{k}. Then

deg(y_{0}) ≤ −m(m − 1)/2 + ∑_{j=0}^{m} deg_{k}(y_{j}) ≤ −m(m − 1)/2 + (p^{k} − 1) ∑_{j=0}^{m} ν_{k}(y_{j}). Moreover, in the case k = 1,

deg(y_{0}) ≤ −m(m − 1)/2 + (m − 1) ∑_{j=0}^{m} ν(y_{j}).

Proof. The condition of linear independence implies that p^{k} ≥ m. We write y_{j} = y_{j}(t) = ∑_{s=0}^{p^{k}−1} y_{j,s}(t)^{p^{k}}t^{s} with y_{j,s}(t) = y_{j,s} ∈ F [t]. Such a representation is unique which is easy to see by induction on k. Also by induction on k it is easy to see that

ord_{α}(y_{j,s}) ≥ (ord_{α}(y_{j})−mod(ord_{α}(y_{j}), p^{k}))/p^{k} for all α, j, s. Now we consider the matrix (y_{j,s})_{1≤j≤m,0≤s≤p^{k}−1}. Its rank is m. We choose a square m by m submatrix with nonzero determinant D_{0} ∈ F [t]. Since p^{k} deg(y_{j,s}) + s ≤ deg(y_{j}), we have deg(y_{j,s}) ≤ (deg(y_{j}) − s)/p^{k} and

deg(D_{0}) ≤ (−m(m − 1)/2 + ∑_{j=1}^{m} deg(y_{j}))/p^{k}.

On the other hand,

ord_{α}(D_{0}) ≥ ∑_{j=0}^{m}(ord_{α}(y_{j})−mod(ord_{α}(y_{j}), p^{k}))/p^{k} for every α ∈ F̄ which is not a zero of y_{0}. Moreover this inequality also holds for every α ∈ F̄ which is a zero of y_{0}, because D_{0} = D_{j}, where D_{j} is the determinant of the matrix obtained by replacing y_{j,s} by y_{0,s}, and 0 ≠ c_{j} ∈ F [t^{p^{l}}] ⊂ F [t^{p^{k}}] for all j.

Thus,

∑_{j=0}^{m} (deg(y_{j})− deg_{k}(y_{j}))/p^{k} = ∑_{j=0}^{m} ∑_{α∈F̄}(ord_{α}(y_{j})−mod(ord_{α}(y_{j}), p^{k}))/p^{k} ≤ ∑_{α∈F̄} ord_{α}D_{0} = deg(D_{0}) ≤ (−m(m − 1)/2 + ∑_{j=1}^{m} deg(y_{j}))/p^{k},

hence we obtain the first conclusion.

Now we consider the case k = 1. Note that the linear independence implies that m ≤ p and that the Wronskian D_{0} = det(y_{j}^{(i)})_{0≤j≤m,1≤i≤m−1} ≠ 0. It is clear that the row j is divisible by (t − α)^{n} with

n = ord_{α}(y_{j})− (the last digit of ord_{α}(y_{j}) in base p) ≤ ord_{α}(y_{j}) − p + 1

≤ ord_{α}(y_{j}) − m + 1.

Since c_{0}D_{0} = c_{j}D_{j}, where D_{j} is the determinant obtained when we replace y_{j} by y_{0}, it follows that

deg(D_{0}) ≥ −(m − 1)ν(y_{j}) + ∑_{j=0}^{m} deg(y_{j}).

On the other hand,

deg(D_{0}) ≤ −m(m − 1)/2 + ∑_{j=1}^{m} deg(y_{j}). Comparing this upper bound for deg(D_{0}) with the above lower bound and taking into account that deg(y_{j}) = deg_{l}(y_{j}), we obtain the conclusion.

Remarks. We do not pursue in this paper improvements of the proposition similar to Theorem 3.1.

We could prove the proposition using divided derivatives (cf. [GV], [Wa]).

If y_{1}, ...., y_{m} are linearly independent over F then they are linearly independent over F_{k} for all sufficiently large k. However the proposition gives a meaningful upper bound on deg(y_{0}) only when k is small.

The proposition can be easily generalized to the situation when every linearly independent over F subsequence in y_{1}, ..., y_{m} stays linearly independent over F_{k}. This is always true for sufficiently large k, but the larger the k the weaker the conclusion.

REFERENCES

[B] Beukers, F. The Diophantine Equation Ax^{p} + By^{q} = Cz^{r}. Duke Math. J. 1998, 91,

61–88.

[BB] Browkin, J.; Brzeziński J. Some Remarks on the abc-Conjecture. Mathematics of

Computation 1994, 62, 931–939.

[BM] Brownawell, D.; Masser, D. Vanishing Sums in Function Fields. Math. Proc.

Cambbridge Philos. Soc. 1986, 100, 427–434.

[BS] Beukers, F.; Schlickewei, H.P. The Equation x + y = 1 in Finitely Generated

Groups. Acta Arith. 1966, 78, 189–199.

[DG] Darmon, H.; Granville, A. On the Equations z^{m} = F (x, y) and Ax^{p} + By^{q} = Cz^{r}.

Bull. London Math. Soc. 1995, 27, 513–543.

[E] Elkies, N. D. On A^{4} + B^{4} + C^{4} = D^{4}. Mathematics of Computation 1988, 51,

825–836.

[GV] García, A.; Voloch, J. F. Wronskians and Linear Independence in Fields of Prime

Characteristic. Manuscripta Math. 59 (1987), 457–469.

[K] Kaplansky, I. An Introduction to Differential Algebra. Hermann: Paris, 1957.
[LP] Lander, L. J.; Parkin, T. R. Counterexamples to Euler’s Conjecture on Sums of

Like Powers. Bull. Amer. Math. Soc. 1996, 72, 1079.

[L] Lang, S. Math Talks for Undergraduates. Springer-Verlag: New York, 1999.
[Ma] Mason, R. C. Norm Form Equations. I. Journal of Number Theory 1986, 22,

190–207.

[Mu1] Mueller J. The abc-Inequality and the Generalized Fermat Equation in Function

Fields. Acta Arithmetica 1993, LXIV.1, 7–18.

[Mu2] Mueller J. S-Unit Equations in Function Fields via the abc-Theorem. Bull.

London Math. Soc. 2000, 32, 163–170.

[NS] Newman, D. J.; Slater, M. Waring’s Problem for the Ring of Polynomials.

Journal of Number Theory 1979, 11, 477-487.

[P] Paley, R. E. A. C. Theorems on Polynomials in a Galois Field. Quarterly Journal

of Math. 1993, 4, 52–63.

[S] Stepanov, S.A. Diophantine Equations over Function Fields. (in Russian) Mat.

Zametki 1982, 32, 753–764, 909.

[V1] Vaserstein, L. Waring’s Problems for Algebras over Fields. Journal of Number

Theory 1987, 26, 286–298.

[V2] Vaserstein, L. Ramsey’s Theorem and Waring’s Problem for Algebras over Fields.

In Proceedings of the Workshop at The Ohio State University, June 17-26, 1991, Walter de Gruyter: Berlin, New York 1992, 435–441.

[Vo1] Voloch, J. F. Diagonal Equations over Function Fields. Bol. Soc. Brasil. Mat.

1985, 16, 29–39.

[Vo2] Voloch, J. F. The Equation ax + by = 1 in Characteristic p. Journal of Number

Theory 1998, 73, 195–200.

[Wa] Wang, J. T.-Y. The Truncated Second Main Theorem of Function Fields. Journal

of Number Theory 1996, 58, 139–157.

[Wo] Wooley, T. D. Large Improvement in Waring’s Problem. Ann. Math. 1992,

135:1, 131–164.

[Z] Zannier, U. Some Remarks on the S-Unit Equation in Function Fields. Acta Arith.

1993, 64, 87–98.