ON SYSTEMS OF DIAGONAL FORMS

MICHAEL P. KNAPP

(March 11, 2005)

Abstract

In this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field ℚ_{p} provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in ℚ_{p}. In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.

1. Introduction

In this paper, we study conditions under which the system of homogeneous equations

\begin{aligned}F_{1}(x) & = & a_{11}x_{1}^{k_{1}} & + & ⋯ & + & a_{1s}x_{s}^{k_{1}} & = & 0 \\ ⋮ & & ⋮ & & & & ⋮ \\ F_{R}(x) & = & a_{R1}x_{1}^{k_{R}} & + & ⋯ & + & a_{Rs}x_{s}^{k_{R}} & = & 0 \end{aligned}  (1)

2000 Mathematics Subject Classification. 11D72 (primary), 11E76, 11E95 (secondary).

Work supported by NSF grant DMS-0344082.

\begin{matrix}c \\ ◯ \end{matrix} 0 Australian Mathematical Society 0263-6115/2000 $A2.00 + 0.00

with a_{ij} ∈ ℤ and k_{1}, …, k_{R} ∈ ℤ^{+} is guaranteed to have a nontrivial solution in p-adic integers. By nontrivial, we mean simply that at least one of the variables is not equal to zero. A conjecture commonly attributed to Artin suggests that regardless of the values of the coefficients, a nontrivial zero in ℤ_{p}^{s} should exist for each prime p provided only that

we have

s > ∑_{i=1}^{R} k_{i}^{2}.

If R = 1, then Davenport & Lewis [5] have shown that this bound is correct. Unfortunately, the following theorem of Lewis & Montgomery [8] shows that this conjecture is false, and that any such bound on s must in fact exhibit exponential growth.

Theorem 1 (Lewis-Montgomery). Suppose p is an odd prime. Let M be a positive integer, and let 𝔐 be a set of K integers in the range [M, 2M − 1]. Suppose that there are s integers x_{1}, …, x_{s}, not all divisible by p, such that

x_{1}^{(p−1)m} + ⋯ + x_{s}^{(p−1)m} ≡ 0 (mod p^{(p−1)M})

for all m in 𝔐. Then s ≥ p^{K}.

On the other hand, Ax & Kochen [1] have shown that if we ask only that a nontrivial solution exists in ℤ_{p}^{s} for p sufficiently large, then the Artin bound is sufficient. It is therefore an interesting problem to determine how small we can take the prime p to be before exponential growth is required. In particular, how small can we make p and still obtain polynomial bounds for s?

In this paper, we explore this problem in the situation where the degrees of the polynomials are all different. In order to write down our conclusions, we introduce the following notational convention. Let Γ_{p}^{∗}(k_{1}, …, k_{R}) be the smallest number such that any system of forms as in (1) has a nontrivial solution in ℤ_{p}^{s} whenever s ≥ Γ_{p}^{∗}(k_{1}, …, k_{R}). For example, if k_{1}, …, k_{R} are fixed, then the result of Ax & Kochen above states that one has

Γ_{p}^{∗}(k_{1}, …, k_{R}) ≤ 1 + ∑_{i=1}^{R} k_{i}^{2}

for sufficiently large p. The purpose of this paper is to prove the following theorem.

Theorem 2. Suppose that R ≥ 2 and let k_{1} > k_{2} > ⋯ > k_{R} be positive integers. For a fixed prime p, define numbers τ_{i} and k̃_{i}, (1 ≤ i ≤ R), so that k_{i} = p^{τ_{i}} k̃_{i} with (p, k̃_{i}) = 1.

i) Define

S_{1} = ∑_{i=1}^{R} k̃_{i} \frac{p^{τ_{i}+1} − 1}{p − 1}.

If p > k_{1} − k_{R} + 1, then we have

Γ_{p}^{∗}(k_{1}, …, k_{R}) ≤ (S_{1} + 1) ∑_{i=1}^{R} ik_{i} − ∑_{i=1}^{R} k_{i} + R.

This implies the bound

Γ_{p}^{∗}(k_{1}, …, k_{R}) ≤ \frac{3}{2}R (∑_{i=1}^{R} k_{i})^{2} + (R − 1) (∑_{i=1}^{R} k_{i}) + R

for such p.

ii) Define

S_{2} = ∑_{i=1}^{R} k̃_{i} \frac{p^{τ_{i}+3} − 1}{p − 1}.

If we have

p > 1 + max \{1, k_{1} − k_{R−1}, \frac{k_{1} − k_{R}}{2} \} ,

then we have

Γ_{p}^{∗}(k_{1}, …, k_{R}) ≤ (S_{2} + 1) ∑_{i=1}^{R} ik_{i} − ∑_{i=1}^{R} k_{i} + R.

This implies that for these p we have the bound

Γ_{p}^{∗}(k_{1}, …, k_{R}) ≤ \frac{3}{2}R(k_{1} − k_{R} + 1)^{2} (∑_{i=1}^{R} k_{i})^{2} + (R − 1) (∑_{i=1}^{R} k_{i}) + R.

This shows in particular that polynomial bounds are possible for primes smaller than the largest degree.

We prove this theorem through a method similar to the one used in [7]. First we apply a normalization process which shows that we need only consider systems which have certain desirable properties. Then we will consider the system (1), with each equation reduced modulo a power of p, and determine a number of variables which guarantees that this system of congruences will have a nonsingular solution. Finally, we will lift this solution of congruences to a nontrivial solution of (1) in ℤ_{p}^{s} through a version of Hensel’s Lemma.

2. Normalization

In this section we describe the process by which we normalize the system of equations and derive a few properties of normalized systems. Our normalization process essentially combines the two used by Wooley in [10] and [11]. Suppose that F = (F_{1}, …, F_{R}) is a system of additive forms as in (1), and that the prime p is fixed. We define two fundamental operations on F. First, we may write

F′ = bF = (b_{1}F_{1}, …, b_{R}F_{R})

for some vector b of nonzero rational numbers. Second, we may make a change of variables of the form x_{i} ↦ p^{−v_{i}}x_{i}, where the v_{i} are rational integers, yielding a system of the form

F′′ = F(p^{v_{1}}x_{1}, …, p^{v_{s}}x_{s}).

Note that these operations commute. A system G with integer coefficients is said to be equivalent to F if G can be obtained from F through a combination of the above operations, i.e. if we can write

G = bF(p^{v_{1}}x_{1}, …, p^{v_{s}}x_{s}).

Now we wish to define a function ∂(F) whose value depends on the coefficients of F_{1}, …, F_{R} and which behaves nicely under the fundamental operations. Unfortunately, this will require a fair amount of notation. Define K = k_{1}k_{2} ⋯ k_{R} and for each i let k′_{i} = K/k_{i}. For any fixed integer r with R ≤ r ≤ s, we define

S_{r} = {(j_{1}, …, j_{R}) ∈ {1, …, r}^{R} : j_{i} ≠ j_{i′} when i ≠ i′}

and note that if we set L = |S_{r}|, then we have

L = r(r − 1) ⋯ (r − R + 1).

For each σ = (j_{1}, …, j_{R}) ∈ S_{r}, write

D_{σ}(F) = det ([a_{ij_{m}}^{k′_{i}}]_{1≤i,m≤R}) .

Further, for fixed numbers m_{1}, …, m_{R−1} with r + 1 ≤ m_{1} ≤ ⋯ ≤ m_{R−1} ≤ s, we define M_{1} = {r + 1, …, m_{1}}

M_{2} = {m_{1} + 1, …, m_{2}}
⋮
M_{R} = {m_{R−1} + 1, …, s},

taking M_{i} to be empty if m_{i} = m_{i−1}. Also, for i = 1, …, R, we set

N_{i} = \frac{RL}{r}k′_{i}.

Finally, we define

∂(F) = ∏_{σ∈S_{r}} D_{σ}(F) ∏_{j∈M_{1}} a_{1j}^{N_{1}} ∏_{j∈M_{2}} a_{2j}^{N_{2}} ⋯ ∏_{j∈M_{R}} a_{Rj}^{N_{R}}.

We now show that ∂(F) behaves nicely under the fundamental operations.

Lemma 3. Suppose that F is a system of forms as in (1) with integral coefficients. Then the following statements are true.

i) If we set F′ = bF = (b_{1}F_{1}, …, b_{R}F_{R}), then we have

∂(F′) = (∏_{i=1}^{R} b_{i}^{(L+|M_{i}|RL/r)k′_{i}}) ∂(F).

ii) If we set F′′ = F (p^{v_{1}}x_{1}, …, p^{v_{s}}x_{s}), then we have

∂(F′′) = p^{RLKv/r}∂(F),

where we have set v = v_{1} + ⋯ + v_{s}.

Proof. To prove the first statement, let F′ be the system

F′_{i} = a′_{i1}x_{1}^{k_{i}} + ⋯ + a′_{is}x_{s}^{k_{i}} (i = 1, …, R).

If F′ = bF, then we have a′_{ij} = b_{i}a_{ij} for each pair i, j. If σ = (j_{1}, …, j_{R}) ∈ S_{r}, then

D_{σ}(F′) = det ([b_{i}^{k′_{i}}a_{ij_{m}}^{k′_{i}}]_{i,m})
= b_{1}^{k′_{1}} ⋯ b_{R}^{k′_{R}} det ([a_{ij_{m}}^{k′_{i}}]_{i,m})
= b_{1}^{k′_{1}} ⋯ b_{R}^{k′_{R}}D_{σ}(F),

whence we obtain

∏_{σ∈S_{r}} D_{σ}(F′) = ∏_{σ∈S_{r}} b_{1}^{k′_{1}} ⋯ b_{R}^{k′_{R}}D_{σ}(F) = b_{1}^{k′_{1}L} ⋯ b_{R}^{k′_{R}L} ∏_{σ∈S_{r}} D_{σ}(F).  (2)

Moreover, for each i = 1, …, R, we have

∏_{j∈M_{i}} (a′_{ij})^{N_{i}} = ∏_{j∈M_{i}} b_{i}^{N_{i}}a_{ij}^{N_{i}} = b_{i}^{|M_{i}|N_{i}} ∏_{j∈M_{i}} a_{ij}^{N_{i}} = b_{i}^{|M_{i}|RLk′_{i}/r} ∏_{j∈M_{i}} a_{ij}^{N_{i}}.  (3)

Putting (2) and (3) together, we obtain

∂(F′) = (b_{1}^{k′_{1}L} ⋯ b_{R}^{k′_{R}L} ∏_{i=1}^{R} b_{i}^{|M_{i}|RLk′_{i}/r}) ∂(F)
= (∏_{i=1}^{R} b_{i}^{(L+|M_{i}|RL/r)k′_{i}}) ∂(F),

as desired.

In order to prove the second statement, we let F′′ be the system

F′′_{i} = a′′_{i1}x_{1}^{k_{i}} + ⋯ + a′′_{is}x_{s}^{k_{i}} (i = 1, …, R).

If F′′ = F(p^{v_{1}}x_{1}, …, p^{v_{s}}x_{s}), then we have a′′_{ij} = p^{k_{i}v_{j}}a_{ij} for each pair i, j. If σ = (j_{1}, …, j_{R}) ∈ S_{r}, then we have

D_{σ}(F′′) = det ([(a′′_{ij_{m}})^{k′_{i}}]_{i,m})
= det ([p^{Kv_{jm}} a_{ij_{m}}^{k′_{i}}]_{i,m})
= p^{Kv_{j_{1}}} ⋯ p^{Kv_{j_{R}}} det ([a_{ij_{m}}^{k′_{i}}]_{i,m})
= p^{Kv_{j_{1}}} ⋯ p^{Kv_{j_{R}}} D_{σ}(F).

Therefore we have

∏_{σ∈S_{r}} D_{σ}(F′′) = ∏_{σ=(j_{1},...,j_{R})∈S_{r}} p^{Kv_{j_{1}}} ⋯ p^{Kv_{j_{R}}} D_{σ}(F)
= ∏_{σ∈S_{r}} p^{K(v_{j_{1}}+···+v_{j_{R}})} ∏_{σ∈S_{r}} D_{σ}(F)
= p^{K ∑(v_{j_{1}}+···+v_{j_{R}})} ∏_{σ∈S_{r}} D_{σ}(F),

where the sum in the last line is over all σ ∈ S_{r}. Now there are L choices for σ, and each of j_{1}, …, j_{r} appears in RL/r of these choices. Hence we have

∑_{σ∈S_{r}}(v_{j_{1}} + ⋯ + v_{j_{R}}) = \frac{RL}{r}v_{1} + ⋯ + \frac{RL}{r}v_{r} = \frac{RL}{r} ∑^{∗} v_{j},

where we use the notation ∑^{∗} to represent a sum over all j ≤ r. Therefore we have

∏_{σ∈S_{r}} D_{σ}(F′′) = p^{(RLK ∑^{∗} v_{j})/r} ∏_{σ∈S_{r}} D_{σ}(F).  (4)

Additionally, for each i with 1 ≤ i ≤ R we have

\begin{aligned}∏_{j∈M_{i}} (a′′_{ij})^{N_{i}} & = & ∏_{j∈M_{i}} (p^{k_{i}v_{j}}a_{ij})^{N_{i}} & = & ∏_{j∈M_{i}} p^{k_{i}v_{j}N_{i}} ∏_{j∈M_{i}} a_{ij}^{N_{i}} \\ & = & p^{k_{i}N_{i} ∑^{i} v_{j}} ∏_{j∈M_{i}} a_{ij}^{N_{i}} & = & p^{(RLK ∑^{i} v_{j})/r} ∏_{j∈M_{i}} a_{ij}^{N_{i}}, \end{aligned}  (5)

where ∑^{i} represents a sum over all j ∈ M_{i}. Putting (4) and (5) together, we obtain

∂(F′′) = p^{RLK ∑^{∗} v_{j}/r}p^{RLK ∑^{1} v_{j}/r} ⋯ p^{RLK ∑^{R} v_{j}/r}∂(F)
= p^{RLKv/r}∂(F).

This completes the proof of the lemma. □

Suppose that F is a system of additive forms with integer coefficients. A standard argument (see for example page 572 of [6]) shows that in order to prove Theorem 2 for all systems of additive forms, it suffices to prove it for systems such that ∂(F) ≠ 0. We say that F is p-normalized if ∂(F) ≠ 0 and the power of p dividing ∂(F) is less than or equal to the power of p dividing ∂(G) for any system G of forms with integer coefficients which is equivalent to F. Note that since any system is equivalent to one which is p-normalized, it suffices to prove the theorem for p-normalized systems. We now prove a lemma showing that p-normalized systems are explicit in a relatively large number of variables when considered modulo p.

Lemma 4. Suppose that F is a p-normalized system of additive forms. Then the following statements are true.

i) If N is the number of variables in F which are explicit when F is considered modulo p, then one has

N ≥ ∑_{i=1}^{R} (|M_{i}| + \frac{r}{R}) \frac{1}{k_{i}} .

ii) If q_{i} is the number of variables explicit modulo p in the form F_{i} of degree k_{i}, then one has

q_{i} ≥ (|M_{i}| + \frac{r}{R}) \frac{1}{k_{i}} .

Proof. To prove the first statement, suppose (by relabeling if necessary) that the variables x_{1}, …, x_{N} are the variables which are explicit modulo p. Consider the system

F′ = p^{−1}F(px_{1}, …, px_{N}, x_{N+1}, …, x_{s}),

which has integer coefficients. The system F′ is obtained from F via a combination of the fundamental operations with b_{1} = ⋯ = b_{R} = p^{−1} and v = v_{1} + ⋯ + v_{s} = N. Then we have

∂(F′) = (p^{RLKN/r} ∏_{i=1}^{R}(p^{−1})^{(L+|M_{i}|RL/r)k′_{i}}) ∂(F)
= p^{A}∂(F),

where we have set

A = \frac{RLKN}{r} − ∑_{i=1}^{R} (|M_{i}|\frac{RL}{r} + L) k′_{i}.

Since the system F is p-normalized, we must have A ≥ 0, and the first part of the lemma follows.

For the second statement, fix i and suppose that the variables in the form F_{i} which are explicit modulo p are x_{1}, …, x_{q_{i}}. Consider the system

F′′ = bF(px_{1}, …, px_{q_{i}}, x_{q_{i}+1}, …, x_{s}),

where b_{i} = p^{−1} and b_{j} = 1 if j ≠ i. Note that F′′ is a system of forms with integer coefficients. Then we have v = q_{i} and hence ∂(F′′) = p^{B}∂(F), where

B = \frac{RLKq_{i}}{r} − \frac{|M_{i}|RLk′_{i}}{r} − Lk′_{i}.

Since the system F is p-normalized and F′′ is equivalent to F, we must have B ≥ 0, and part ii) of the lemma follows. This completes the proof of the lemma. □

3. Preliminary Lemmata

In this section we establish some lemmata which will be needed in the proof of Theorem 2. Our first lemma, due to Schanuel [9], provides a bound on the number of variables necessary to solve a system of congruences modulo various powers of a prime p.

Lemma 5. For 1 ≤ i ≤ R, let F_{i} be a (not necessarily homogeneous) polynomial of degree k_{i} in N variables with coefficients in ℤ_{p} and no constant term. Also let T_{p} = {x ∈ ℤ_{p} : x^{p} = x} be the set of Teichmüller representatives of {0, 1, 2, …, p − 1}. Then the system of equations

F_{i}(x_{1}, …, x_{N}) ≡ 0 (mod p^{v_{i}}) (1 ≤ i ≤ R)

has a nontrivial solution in T_{p}^{N} provided that

N > ∑_{i=1}^{R} k_{i} \frac{p^{v_{i}} − 1}{p − 1}.

Our next lemma is a version of Hensel’s Lemma, which allows us to lift a nonsingular solution of a system of congruences to a p-adic solution. This is Lemma 4 of [7].

Lemma 6. Consider the system (1). Let p be a prime number, and for 1 ≤ i ≤ R we define numbers τ_{i} and k̃_{i} such that k_{i} = p^{τ_{i}} k̃_{i} with (p, k̃_{i}) = 1. Further, for 1 ≤ i ≤ R, we define

γ_{i} = \{ \begin{matrix}τ_{i} & if p is odd \\ τ_{i} + 1 & if p = 2. \end{matrix}

Let h be a positive integer and suppose that z is a nontrivial solution of the system of congruences

F_{i}(x) ≡ 0 (mod p^{2h+γ_{i}−1})  (1 ≤ i ≤ R)  (6)

such that the matrix

[ \begin{matrix}a_{11}z_{1}^{k_{1}−1} & ⋯ & a_{1s}z_{s}^{k_{1}−1} \\ ⋮ & & ⋮ \\ a_{R1}z_{1}^{k_{R}−1} & ⋯ & a_{Rs}z_{s}^{k_{R}−1} \end{matrix} ]  (7)

has an R × R submatrix M such that

det M ≢ 0 (mod p^{h}).  (8)

Then the system (1) has a solution y ∈ ℤ_{p}^{s} such that y ≡ z (mod p^{h}).

Our final goal for this section is to prove Lemma 10, a result stating that under certain conditions the determinant of a matrix similar to (7) can be made to be nonzero modulo a power of a prime p. This will be needed later to ensure that our solutions of congruences are nonsingular. In order to prove this lemma, we will need some properties of Bhargava’s generalized factorial function (see [2, 3, 4]), and refer the reader to [4] for the definition of a p-ordering and the definitions of the functions v_{k}(S, p), w_{p}(a) and k!_{S}. In order to prove Lemma 10, we will need the following preliminary lemmata.

Lemma 7. The sequence 0, 1, 2, … of nonnegative integers is a p-ordering for ℤ for any prime p.

This is Proposition 6 of [4].

Lemma 8. Let p be a prime number, and let S be the set S = ℤ − pℤ. Then the sequence

(a_{0}, a_{1}, a_{2}, …) = (1, 2, 3, …, p − 1, p + 1, …, 2p − 1, 2p + 1, …)

is a p-ordering for S.

Proof. In the definition of a p-ordering, we may take a_{0} to be any element of S. Hence it is permissible to set a_{0} = 1. Now suppose that a_{0}, …, a_{k} are the first k + 1 terms of a p-ordering for S. We wish to show that a_{k+1} is allowable for the next term. We divide the proof into two cases. First, if p ∤ (a_{k} + 1), then a_{k+1} = a_{k} + 1. Let mp be the largest multiple of p such that mp < a_{k}. Suppose by way of contradiction that we cannot use a_{k+1} as the next term. Then there is some number y ∈ S − {a_{0}, …, a_{k}} such that

w_{p}((y − a_{0}) ⋯ (y − a_{k})) < w_{p}((a_{k+1} − a_{0}) … (a_{k+1} − a_{k})).

Since any element of S is relatively prime to p, we have

w_{p} (∏_{i=0}^{a_{k}}(y − i)) = w_{p} (∏_{j=0}^{k}(y − a_{j})) · w_{p} (∏_{j=0}^{m}(y − jp))
< w_{p} (∏_{j=0}^{k}(a_{k+1} − a_{j})) · w_{p} (∏_{j=0}^{m}(a_{k+1} − jp))
= w_{p} (∏_{i=0}^{a_{k}}(a_{k+1} − i)) ,

where the inequality holds because

w_{p} (∏_{j=0}^{m}(y − jp)) = w_{p} (∏_{j=0}^{m}(a_{k+1} − jp)) = 1

since a_{k+1} and y are both relatively prime to p. However we cannot have an element y ∈ S − {a_{0}, …, a_{k}} such that

w_{p} (∏_{i=0}^{a_{k}}(y − i)) < w_{p} (∏_{i=0}^{a_{k}}(a_{k+1} − i))

since then the sequence 0, 1, 2, … would not be a valid p-ordering of ℤ, contradicting Lemma 7.

Now suppose that p | (a_{k} + 1), and write a_{k} + 1 = mp. Then a_{k+1} = a_{k} + 2. As before, suppose by way of contradiction that there is an element y ∈ S − {a_{0}, …, a_{k}} with

w_{p}((y − a_{0}) ⋯ (y − a_{k})) < w_{p}((a_{k+1} − a_{0}) ⋯ (a_{k+1} − a_{k})).

Again noting that both y and a_{k+1} are prime to p, we have

w_{p} ( ∏_{i=0}^{a_{k}+1}(y − i)) = w_{p} (∏_{i=0}^{k}(y − a_{i})) · w_{p} (∏_{j=0}^{m}(y − jp))
< w_{p} (∏_{i=0}^{k}(a_{k+1} − a_{i})) · w_{p} (∏_{j=0}^{m}(a_{k+1} − jp))
= w_{p} ( ∏_{i=0}^{a_{k}+1}(a_{k+1} − i)) ,

and the existence of such an element y again violates Lemma 7. Hence, after having chosen a_{0}, …, a_{k}, the element a_{k+1} is allowable for the next term of a p-ordering on S. So the lemma is true by induction.□

Lemma 9. Suppose that the sequence a_{0}, a_{1}, … is a p-ordering for the set S. A polynomial F of degree k, written in the form

F (x) = ∑_{n=0}^{k} e_{n}(x − a_{0})(x − a_{1}) ⋯ (x − a_{n−1}),

vanishes on S modulo p^{r} if and only if e_{n} is a multiple of p^{r}/(p^{r}, n!_{S}) for 0 ≤ n ≤ k.

This is Lemma 14 of [4].

Lemma 10. Consider the matrix

B = [ \begin{matrix}a_{11}x_{1}^{k_{1}−1} & ⋯ & a_{1R}x_{R}^{k_{1}−1} \\ ⋮ & & ⋮ \\ a_{R1}x_{1}^{k_{R}−1} & ⋯ & a_{RR}x_{R}^{k_{R}−1} \end{matrix} ] ,  (9)

and assume that a_{11}a_{22} ⋯ a_{RR} ≢ 0 (mod p). Then the following statements hold.

i) If p > k_{1} − k_{R} + 1, then there exist integers t_{2}, …, t_{R}, all relatively prime to p, such that if we set x_{2} = t_{2}x_{1}, …, x_{R} = t_{R}x_{1} and let x_{1} be any integer relatively prime to p, then det B ≢ 0 (mod p). ii) If we have

p > 1 + max \{k_{1} − k_{R−1}, \frac{k_{1} − k_{R}}{2} \} ,

then there exist integers t_{2}, …, t_{R}, all relatively prime to p, such that if we set x_{2} = t_{2}x_{1}, …, x_{R} = t_{R}x_{1} and let x_{1} be any integer relatively prime to p, then det B ≢ 0 (mod p^{2}).

Note that if R = 1 we interpret the condition in part ii) of the lemma as p > 1.

Proof. First note that if we set x_{2} = t_{2}x_{1}, …, x_{R} = t_{R}x_{1}, then we have

det B = x_{1}^{k_{1}+···+k_{R}−R}(t_{2} ⋯ t_{R})^{k_{R}−1} det C,

where C is the matrix

C = [ \begin{matrix}a_{1,1} & a_{1,2}t_{2}^{k_{1}−k_{R}} & ⋯ & a_{1,R}t_{R}^{k_{1}−k_{R}} \\ ⋮ & ⋮ & & ⋮ \\ a_{R−1,1} & a_{R−1,2}t_{2}^{k_{R−1}−k_{R}} & ⋯ & a_{R−1,R}t_{R}^{k_{R−1}−k_{R}} \\ a_{R,1} & a_{R,2} & ⋯ & a_{R,R} \end{matrix} ] .

Since we require x_{1}, t_{2}, …, t_{R} to all be nonzero modulo p, the matrix B has the desired property if and only if det C is nonzero modulo the appropriate power of p.

We prove part i) of the lemma by induction on R. If R = 1, then det B = a_{11}x_{1}^{k_{1}−1}. If a_{11} and x_{1} are both relatively prime to p, then so is det B. Now suppose that the statement is true for R = M − 1. We wish to prove that it holds for R = M. In this situation, we have

B = [ \begin{matrix}a_{11}x_{1}^{k_{1}−1} & ⋯ & a_{1M}x_{M}^{k_{1}−1} \\ ⋮ & & ⋮ \\ a_{M1}x_{1}^{k_{M} −1} & ⋯ & a_{MM}x_{M}^{k_{M} −1} \end{matrix} ] ,

and the matrix C becomes

C = [ \begin{matrix}a_{1,1} & a_{1,2}t_{2}^{k_{1}−k_{M}} & ⋯ & a_{1,M}t_{M}^{k_{1}−k_{M}} \\ ⋮ & ⋮ & & ⋮ \\ a_{M−1,1} & a_{M−1,2}t_{2}^{k_{M−1}−k_{M}} & ⋯ & a_{M−1,M}t_{M}^{k_{M−1}−k_{M}} \\ a_{M,1} & a_{M,2} & ⋯ & a_{M,M} \end{matrix} ] .

Now consider the upper left-hand (M − 1) × (M − 1) submatrix of B. By the inductive hypothesis, choose integers t_{2}, …, t_{M−1} all nonzero modulo p such that the determinant of this matrix is nonzero modulo p whenever x_{1} is relatively prime to p. Hence the determinant of the upper left-hand (M − 1) × (M − 1) submatrix D of C is also nonzero modulo p. Then we have

C = [ D  a_{1,M}t_{M}^{k_{1}−k_{M}} ; ⋮ ; a_{M−1,M}t_{M}^{k_{M−1}−k_{M}}_{a_{M,1} a_{M,2} ⋯ a_{M,M−1}  a_{M,M}} ]

and by expanding along the rightmost column we get

det C = a_{MM} det D + p(t_{M}),

where

p(t_{M}) = c_{1}t_{M}^{k_{1}−k_{M}} + ⋯ + c_{M−1}t_{M}^{k_{M−1}−k_{M}}

is a polynomial with no constant term. If c_{1}, …, c_{M−1} are all divisible by p, then we can set t_{M} = 1 and obtain

det C ≡ a_{MM} det D ≢ 0 (mod p).

If some of the c_{i} are nonzero modulo p, then we note that det C is a polynomial of degree at most k_{1} − k_{M}. If p − 1 > k_{1} − k_{M}, then det C cannot be divisible (as a polynomial) by

t_{M}^{p−1} − 1 = (t_{M} − 1)(t_{M} − 2) ⋯ (t_{M} − (p − 1)).

Since the ring (ℤ/pℤ)[t_{M}] has unique factorization, there must be a value for t_{M} which is nonzero modulo p and for which det C ≢ 0 (mod p). Therefore the values we have chosen for t_{2}, …, t_{M} ensure that det B ≢ 0 (mod p) whenever (x_{1}, p) = 1. This completes the proof of part i) of the lemma.

To prove part ii), first note that if R = 1, then the same argument as above shows that the statement is true whenever p > 1. Now consider the matrix B given in (9). Since p > k_{1} − k_{R−1} + 1, we can choose values of t_{2}, …, t_{R−1} such that the upper lefthand (R−1)×(R−1) submatrix of B will be nonsingular modulo p whenever (x_{1}, p) = 1. As in the proof of part i), this implies that the upper left-hand (R −1)×(R −1) submatrix D of C will also be nonsingular modulo p. Thus we just need to choose a value for t_{R}.

We can now write

det C = p(t_{R}) = c_{k_{1}−k_{R}}t_{R}^{k_{1}−k_{R}} + ⋯ + c_{k_{R−1}−k_{R}}t_{R}^{k_{R−1}−k_{R}} + a_{RR} det D.

Note that this polynomial is slightly different than the one we called p(t_{M}) earlier. We wish to show that this polynomial does not vanish modulo p^{2} on the set S = ℤ − pℤ. Since a_{RR} det D ≢ 0 (mod p), this is certainly true if c_{k_{i}−k_{R}} ≡ 0 (mod p) for 1 ≤ i ≤ R − 1. If at least one of these coefficients is nonzero modulo p, let d be the smallest number such that c_{k_{d}−k_{R}} ≢ 0 (mod p).

Let a_{0}, a_{1}, … be the p-ordering for S given in Lemma 8, and write p(t_{R}) in the form

p(t_{R}) = ∑_{n=0}^{k_{1}−k_{R}} e_{n}(t_{R} − a_{0}) ⋯ (t_{R} − a_{n−1}),  (10)

as in Lemma 9. Note that because of the way we chose d, we have p | e_{n} whenever n > k_{d} − k_{R}. It is then not too hard to see that we must have

e_{k_{d}−k_{R}} ≡ c_{k_{d}−k_{R}} ≢ 0 (mod p).

We now show that if p > 1 + (k_{1} − k_{R})/2, then e_{k_{d}−k_{R}} is not a multiple of

\frac{p^{2}}{(p^{2}, (k_{d} − k_{R})!_{S})}.

Once this is done, our proof will be complete by Lemma 9.

In order to prove this divisibility criterion, we examine the values of p^{2}/(p^{2}, n!_{S}). First note that we have

n!_{S} = ∏_{q prime} v_{n}(S, q)

and that v_{n}(S, q) is a power of q. (Again, see [4] for an elementary explanation of Bhargava’s factorial function and this notation.) Since p is prime, the terms v_{n}(S, q) with q ≠ p do not contribute anything to (p^{2}, n!_{S}) and so we have

(p^{2}, n!_{S}) = (p^{2}, v_{n}(S, p)).

By simply writing out the terms a_{0}, a_{1}, … for the p-ordering for S given in Lemma 8, it is easy to see that

v_{n}(S, p) = \{ \begin{matrix}1 & if & n ≤ p − 2 \\ p & if & p − 1 ≤ n ≤ 2p − 3 \\ p^{2}L_{n}, L_{n} ∈ ℤ & if & n ≥ 2p − 2. \end{matrix}

Hence we see that

\frac{p^{2}}{(p^{2}, n!_{S})} = \frac{p^{2}}{(p^{2}, v_{n}(S, p))} = \{ \begin{matrix}p^{2} & if & n ≤ p − 2 \\ p & if & p − 1 ≤ n ≤ 2p − 3 \\ 1 & if & n ≥ 2p − 2. \end{matrix}

Note that if p > 1 + (k_{1} − k_{R})/2, then k_{1} − k_{R} ≤ 2p − 3. Since k_{d} − k_{R} ≤ k_{1} − k_{R}, this implies that

\frac{p^{2}}{(p^{2}, (k_{d} − k_{R})!_{S})} = p or p^{2}.

But this number cannot divide e_{k_{d}−k_{R}} since e_{k_{d}−k_{R}} is nonzero modulo p. This completes the proof of the lemma.□

4. The Proof of Theorem 2

Since the proofs of the first bound in both parts of Theorem 2 are essentially identical, we prove them together. In what follows, setting m = 1 proves the first bound in part i) of the theorem and setting m = 2 proves the first bound in part ii). We remark first that if there is some number N such that any system like (1) in N variables has a nontrivial p-adic solution, then any such system in s > N variables also has one. This can be seen by simply setting s − N of the variables equal to zero, leaving a system in N variables. It therefore suffices to assume that we have

s = (S_{m} + 1) ∑_{i=1}^{R} ik_{i} − ∑_{i=1}^{R} k_{i} + R

variables and show that the system (1) has a nontrivial p-adic solution.

Since it is enough to prove each part of the theorem for p-normalized systems of forms, we will assume throughout this section that all systems are p-normalized. However, we must define the quantities r and |M_{1}|, …, |M_{R}| used in the normalization process. To do this, we set r = R and

|M_{i}| = ik_{i}(S_{m} + 1) − k_{i} (1 ≤ i ≤ R).

For each i, Lemma 4 yields

q_{i} ≥ (|M_{i}| + \frac{r}{R}) \frac{1}{k_{i}}
= i(S_{m} + 1) − 1 + \frac{1}{k_{i}} .

But since q_{i} must be an integer, this implies that we have q_{i} ≥ i(S_{m} + 1). In other words, for each i the form F_{i} of degree k_{i} contains at least i(S_{m} + 1) variables which are explicit when F_{i} is reduced modulo p.

We now relabel the variables in our system using the following procedure. Since q_{1} ≥ S_{m} + 1, we can choose S_{m} + 1 variables which are explicit when F_{1} is reduced modulo p. Let 𝔘_{1} be the set containing these variables. Since q_{2} ≥ 2(S_{m} + 1), we can choose a set 𝔘_{2} containing S_{m}+1 variables which are explicit when F_{2} is reduced modulo p and which are not in 𝔘_{1}. We can continue this procedure to define sets 𝔘_{3}, …, 𝔘_{R}, where each 𝔘_{i} contains S_{m} + 1 variables, all of which are explicit when F_{i} is considered modulo p, such that 𝔘_{1}, …, 𝔘_{R} are pairwise disjoint. We now relabel the variables in such a manner that for each i, the variables in the set 𝔘_{i} are labeled

x_{i}, x_{R+i}, …, x_{RS_{m}+i}.

If i > R(S_{m} + 1), then we set x_{i} = 0. This leaves us with a system

\begin{aligned}F_{1}(x) & = & a_{1,1}x_{1}^{k_{1}} & + ⋯ + & a_{1,R(S_{m}+1)}x_{R(S_{m}+1)}^{k_{1}} & = & 0 \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ F_{R}(x) & = & a_{R,1}x_{1}^{k_{R}} & + ⋯ + & a_{R,R(S_{m}+1)}x_{R(S_{m}+1)}^{k_{R}} & = & 0, \end{aligned}  (11)

which has the property that for 0 ≤ j ≤ S_{m}, we have

a_{1,jR+1}a_{2,jR+2} ⋯ a_{R,jR+R} ≢ 0 (mod p).

In other words, if we let A_{1} be the matrix of coefficients of the first R variables, A_{2} be the matrix of coefficients of the second R variables, and so on, then each diagonal element of each of these matrices is nonzero modulo p.

To find a ℚ_{p}-integral solution to (11), we will first find a solution of the system

\begin{matrix}a_{1,1}x_{1}^{k_{1}} + ⋯ + a_{1,R(S_{m}+1)}x_{R(S_{m}+1)}^{k_{1}} ≡ 0 (mod p^{2m+τ_{1}−1}) \\ ⋮ & ⋮ & ⋮ \\ a_{R,1}x_{1}^{k_{R}} + ⋯ + a_{R,R(S_{m}+1)}x_{R(S_{m}+1)}^{k_{R}} ≡ 0 (mod p^{2m+τ_{R}−1}) \end{matrix}  (12)

which is nonsingular modulo p^{m}, where we recall that each τ_{i} is defined so that k_{i} = p^{τ_{i}} k̃_{i} with (p, k̃_{i}) = 1. Note that since both parts of the theorem require p to be odd, the powers of p in (12) are the powers required in Lemma 6 when h = m.

If the bounds on p given in the statement of the theorem hold, then Lemma 10 tells us that for each j with 0 ≤ j ≤ S_{m}, we can find integers t_{jR+2}, …, t_{jR+R} such that if we set

x_{jR+i} = t_{jR+i}x_{jR+1}, (2 ≤ i ≤ R)

and let B_{j} be the matrix

B_{j} = [ \begin{matrix}a_{1,jR+1}x_{jR+1}^{k_{1}−1} & ⋯ & a_{1,jR+R}x_{jR+R}^{k_{1}−1} \\ ⋮ & & ⋮ \\ a_{R,jR+1}x_{jR+1}^{k_{R}−1} & ⋯ & a_{R,jR+R}x_{jR+R}^{k_{R}−1} \end{matrix} ] ,

then we have det B_{j} ≢ 0 (mod p^{m}) whenever x_{jR+1} ≢ 0 (mod p).

After making the identifications above, we obtain a new system

\begin{matrix}c_{1,1}x_{1}^{k_{1}} + c_{1,R+1}x_{R+1}^{k_{1}} + ⋯ + c_{1,RS_{m}+1}x_{RS_{m}+1}^{k_{1}} ≡ 0 (mod p^{2m+τ_{1}−1}) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ c_{R,1}x_{1}^{k_{R}} + c_{R,R+1}x_{R+1}^{k_{R}} + ⋯ + c_{R,RS_{m}+1}x_{RS_{m}+1}^{k_{R}} ≡ 0 (mod p^{2m+τ_{R}−1}). \end{matrix}  (13)

Suppose that we can find a solution to this system with at least one of the variables, say x_{jR+1}, not divisible by p. This would lead to a solution of the system (12) in which the matrix B_{j} satisfies det B_{j} ≢ 0 (mod p^{m}). Then the solution of (12) lifts to a nontrivial solution of (11) by Lemma 6, and this gives us a nontrivial solution of (1). Thus it suffices to show that the system (13) has a nontrivial solution.

We find a nontrivial solution of (13) with the variables restricted to the Teichmüller set T_{p} = {x ∈ ℤ_{p} : x^{p} = x}. Note that when x ∈ T_{p}, we have

x^{k_{i}} = x^{p^{τ_{i}} k̃_{i}} = xk̃_{i}.

Therefore any solution of the system

\begin{matrix}c_{1,1}xk̃_{1} ; 1 + c_{1,R+1}xk̃_{1} ; R+1 + ⋯ + c_{1,RS_{m}+1}xk̃_{1} ; RS_{m}+1 ≡ 0 (mod p^{2m+τ_{1}−1}) \\ ⋮ & ⋮ & ⋮ \\ c_{R,1}xk̃_{R} ; 1 + c_{R,R+1}xk̃_{R} ; R+1 + ⋯ + c_{R,RS_{m}+1}xk̃_{R} ; RS_{m}+1 ≡ 0 (mod p^{2m+τ_{R}−1}) \end{matrix}  (14)

with all the variables in T_{p} is also a solution of (13). By Lemma 5, we can solve (14) nontrivially whenever the number of variables is greater than

∑_{i=1}^{R} k̃_{i} \frac{p^{2m+τ_{i}−1} − 1}{p − 1} = S_{m}.

Since we have S_{m} + 1 variables, there exists a nontrivial solution to (14) with each variable in T_{p}. As mentioned above, this is also a nontrivial solution of (13), and this leads to a nontrivial solution of (1). The first bound in each part of the theorem follows.

To finish the proof, we need to show that the second bound in each part of the theorem holds. For part i), note that we have

S_{1} = ∑_{i=1}^{R} k̃_{i} \frac{p^{τ_{i}+1} − 1}{p − 1} = \frac{p}{p − 1} ∑_{i=1}^{R} k_{i} − ∑_{i=1}^{R} \frac{k̃_{i}}{p − 1}.

Since p ≥ 3, we obtain

S_{1} + 1 < 1 + \frac{p}{p − 1} ∑_{i=1}^{R} k_{i}
≤ 1 + \frac{3}{2} ∑_{i=1}^{R} k_{i},

and since i ≤ R, we have

∑_{i=1}^{R} ik_{i} ≤ R ∑_{i=1}^{R} k_{i}.

From the first bound in part i), we then find that

Γ_{p}^{∗}(k_{1}, …, k_{R}) ≤ \frac{3}{2}R (∑_{i=1}^{R} k_{i})^{2} + (R − 1) (∑_{i=1}^{R} k_{i}) + R,

as desired. For part ii), note that if we have

p > k_{1} − k_{R} + 1,

then part i) of the theorem applies and yields a smaller bound than given in part ii). Hence we may assume that

p ≤ k_{1} − k_{R} + 1.

Then since we are assuming that p ≥ 3, we have

S_{2} + 1 = 1 + ∑_{i=1}^{R} k̃_{i} \frac{p^{τ_{i}+3} − 1}{p − 1}
< 1 + ∑_{i=1}^{R} \frac{k̃_{i}p^{τ_{i}+3}}{p − 1}
= 1 + \frac{p}{p − 1}p^{2} ∑_{i=1}^{R} k_{i}
≤ 1 + \frac{3}{2}(k_{1} − k_{R} + 1)^{2} ∑_{i=1}^{R} k_{i}.

Therefore we obtain

(S_{2} + 1) ∑_{i=1}^{R} ik_{i} < (1 + \frac{3}{2}(k_{1} − k_{R} + 1)^{2} ∑_{i=1}^{R} k_{i}) ∑_{i=1}^{R} Rk_{i}
= \frac{3}{2}R(k_{1} − k_{R} + 1)^{2} (∑_{i=1}^{R} k_{i})^{2} + R (∑_{i=1}^{R} k_{i}) ,

and the second bound in part ii) of the theorem follows easily. This completes the proof of the theorem.□

References

[1] J. Ax and S. Kochen, “‘Diophantine problems over local fields I”’, Amer. J. Math. 87 (1965), 605–630.

[2] M. Bhargava, “‘P -orderings and polynomial functions on arbitrary subsets of Dedekind rings”’, J. reine angew. Math. 490 (1997), 101–127.

[3] , “‘Generalized factorials and fixed divisors over subsets of a Dedekind domain”’, J. Number Theory 72 (1998), 67–75.

[4] , “‘The factorial function and generalizations”’, Amer. Math. Monthly 107 (2000), 783–799.

[5] H. Davenport and D. J. Lewis, “‘Homogeneous additive equations”’, Proc. Royal Soc. London Ser. A 274 (1963), 443–460.

[6] , “‘Simultaneous equations of additive type”’, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 264 (1969), 557–595.

[7] M. Knapp, “‘Diagonal equations of different degrees over p-adic fields”’, (submitted) .

[8] D. J. Lewis and H. L. Montgomery, “‘On zeros of p-adic forms”’, Michigan Math J. 30 (1983), 83–87.

[9] S. H. Schanuel, “‘An extension of Chevalley’s theorem to congruences modulo prime powers”’, J. Number Theory 6 (1974), 284–290.

[10] T. D. Wooley, “‘On simultaneous additive equations III”’, Mathematika 37 (1990), 85–96.

[11] , “‘On simultaneous additive equations I”’, Proc. London Math. Soc. (3) 63 (1991), 1–34.

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