On Systems of Diagonal Forms II

Michael P. Knapp

1. Introduction

In a recent paper [8], we considered the system F of homogeneous additive forms

F_{1}(x) = a_{11}x_{1}^{k_{1}} + ⋯ + a_{1s}x_{s}^{k_{1}}
⋮ ⋮ ⋮
F_{R}(x) = a_{R1}x_{1}^{k_{R}} + ⋯ + a_{Rs}x_{s}^{k_{R}}

with coefficients in a p-adic field ℚ_{p}. If we had R ≥ 2, k_{1} > k_{2} > ⋯ > k_{R} (in particular, the degrees of the forms were all different) and p > k_{1} − k_{R} + 1, then we gave a bound on s, polynomial in terms of the degrees, which guarantees that the system F = 0 has a nontrivial solution in p-adic integers. One could view this as a “hybrid” between the results of Ax & Kochen [2] and Lewis & Montgomery [9].

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

Work supported by NSF grant DMS-0344082.

The Ax-Kochen theorem states that the (smaller) bound

s > k_{1}^{2} + ⋯ + k_{R}^{2}

suffices provided that p is sufficiently large, but gives no bound on how large p needs to be. The Lewis-Montgomery bound, on the other hand, implies that any bound on s which works for all primes must exhibit exponential growth. Thus one may think of the results in [8] as bounding how small the prime p can be before exponential growth on s is required.

The purpose of the present paper is to eliminate the restriction in our previous result that the degrees of the forms be distinct. In this situation, we are now able to prove that a polynomial bound on s suffices for any system, provided that p is mildly large. In particular, we prove the following theorem.

Theorem 1. Let p be a prime number. Consider the system F = 0 of diagonal forms with coefficients in ℤ_{p} given by

(1)  \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_{v}} + ⋯ + a_{Rs}x_{s}^{k_{v}} & = & 0. \end{aligned}

Suppose that k_{1} > k_{2} > ⋯ > k_{v} are the distinct degrees of the forms, and that for 1 ≤ i ≤ v there are R_{i} forms of degree k_{i}. Assume that p > max{k_{1} − k_{v} + 1, 2}. If we have

s ≥ \frac{3}{2}R (∑_{i=1}^{v} R_{i}k_{i})^{2} + R ∑_{i=1}^{v} R_{i}k_{i} − ∑_{i=1}^{v} k_{i} + v

then this system has a solution with each variable in ℤ_{p} and at least one variable not equal to 0.

Note that in this theorem, ∑_{i=1}^{v} k_{i} represents the sum of the distinct degrees in the system, and ∑_{i=1}^{v} R_{i}k_{i} represents the sum of the degrees with multiplicities taken into account.

If all of the coefficients are in ℤ, then Theorem 1 says that the system (1) has a nontrivial integral solution in every p-adic field ℚ_{p} for which p is at least the given bound. Note that if v ≥ 2, then Theorem 1 implies that the bound

s ≥ \frac{3}{2}R^{3} (k_{1} + k_{2} + ⋯ + k_{v})^{2}

suffices for these primes. One can compare this with the result given in [4] for systems of forms which all have the same degree (although this result is improved in [6]).

Theorem 1 is a trivial consequence of the following theorem, which is unfortunately somewhat more complicated to state.

Theorem 2. Let F(x_{1}, …, x_{s}) be a system as in Theorem 1, and assume again that p > max{k_{1} − k_{v} + 1, 2}. For 1 ≤ i ≤ v, define the numbers τ_{i} and k̃_{i} so that k_{i} = p^{τ_{i}} k̃_{i} for each i, with (p, k̃_{i}) = 1. Further, define

t = 1 + ∑_{j=1}^{v} R_{j}k̃_{j} \frac{p^{τ_{j}+1} − 1}{p − 1}.

If we have

s ≥ t ∑_{i=1}^{v}(1 + R_{1} + ⋯ + R_{i−1})R_{i}k_{i} − ∑_{i=1}^{v} k_{i} + v

then the system F = 0 has a solution with each variable in ℤ_{p} and at least one variable not equal to zero.

We note that Theorem 1 follows from Theorem 2 since each of the R_{i} is at least 1 and p/(p − 1) is at most 3/2. We also mention that we only require k_{1} > ⋯ > k_{v} as a convenience in the proof of Lemma 8 below. If we replace the expression k_{1} − k_{v} in these theorems by

(largest degree) − (smallest degree),

then we may relabel the degrees to possibly obtain a smaller bound on s.

Our proof of Theorem 2 is essentially a three-step process. First, we apply a normalization procedure which allows us to assume that the system F has certain desirable properties. Next we reduce the forms in F modulo various powers of p, obtaining a system of congruences, which we show must have a nonsingular solution. Finally, we are able to lift this nonsingular solution to a solution of (1) through an application of Hensel’s lemma.

To this end, Section 2 of this paper gives the notation that we will use throughout the proof and some preliminary results. Section 3 describes our normalization procedure and proves that normalized systems have the properties we will need. In Section 4, we prove an auxiliary result stating that under certain conditions, a particular matrix can be guaranteed to be nonsingular modulo a prime p. This lemma will eventually guarantee that our solution of congruences is a nonsingular solution. Finally, in Section 5 we use the results of the preceding sections to complete the proof of Theorem 2.

2. Notation and Preliminaries

Although much of the notation to be used throughout this paper is given in the statements of Theorems 1 and 2, we repeat it here to have it all available in one location. Let p be a prime number. We will consider systems of the shape

F_{1}(x) = a_{11}x_{1}^{k_{1}} + ⋯ + a_{1s}x_{s}^{k_{1}} = 0
⋮ ⋮ ⋮ ⋮
F_{R}(x) = a_{R1}x_{1}^{k_{v}} + ⋯ + a_{Rs}x_{s}^{k_{v}} = 0,

where the first R_{1} forms are of degree k_{1}, the next R_{2} forms are of degree k_{2}, and so on. Note that we have R_{1} + R_{2} + ⋯ + R_{v} = R. Here, k_{1} > k_{2} > ⋯ > k_{v} are the distinct degrees of the forms, and all of the coefficients are elements of ℤ_{p}. Given the degrees and the prime p, we define numbers τ_{i} and k̃_{i} so that k_{i} = p^{τ_{i}} k̃_{i} for each i and (p, k̃_{i}) = 1. Other notational conveniences that we will use require more context to define, and so we will defer those definitions until a more appropriate point.

Next, we state a few lemmata which will be needed as we complete the proof of Theorem 2. The first of these is a theorem of Alon [1, Theorem 1.2], which guarantees that a polynomial of a particular shape is not identically zero. This will be useful in Section 4 when we show that we can guarantee that certain matrices have a nonzero determinant modulo p.

Lemma 1. Let 𝔽 be an arbitrary field, and let f = f(x_{1}, …, x_{s}) be a polynomial in 𝔽[x_{1}, …, x_{s}]. Suppose the degree of f is ∑_{i=1}^{s} t_{i}, where each t_{i} is a nonnegative integer, and suppose the coefficient of ∏_{i=1}^{s} x_{i}^{t_{i}} in f is nonzero in 𝔽. Then, if Y_{1}, …, Y_{s} are subsets of 𝔽 with |Y_{i}| > t_{i}, then there exist y_{1} ∈ Y_{1}, y_{2} ∈ Y_{2}, …, y_{s} ∈ Y_{s} so that f(y_{1}, …, y_{s}) ≠ 0.

Our next lemma appears to have been independently discovered by Browkin [3] and Schanuel [11]. This is a version of Chevalley’s Theorem for congruences modulo powers of a prime. That is, this lemma provides a number of variables sufficient to guarantee that a system of congruences modulo powers of a fixed prime has a nontrivial solution.

Lemma 2. Suppose that f_{1}, …, f_{R} are (not necessarily homogeneous) polynomials without constant terms, and for each i, let d_{i} be the (total) degree of f_{i}. Here we make no restrictions on the degrees, and in particular they need not be distinct. Consider the system of congruences

f_{i}(x_{1}, …, x_{s}) ≡ 0 (mod p^{τ_{i}}), (1 ≤ i ≤ R).

If we have

s > ∑_{i=1}^{R} \frac{d_{i}(p^{τ_{i}} − 1)}{p − 1},

then this system has a nontrivial solution in which each variable lies in the Teichmüller set

T_{p} = {x ∈ ℤ_{p} : x^{p} = x}.

The final lemma in this section is a version of Hensel’s Lemma. This is Lemma 4 of [7], which shows that a nonsingular solution of a system of homogeneous additive congruences can be lifted to a p-adic solution of the system.

Lemma 3. Consider a system of additive equations

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

where we again make no assumptions about the degrees of the forms. Let p be a prime number, and for 1 ≤ i ≤ R we define numbers τ_{i} and d̃_{i} such that d_{i} = p^{τ_{i}} d̃_{i} with (p, d̃_{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)

such that the matrix

[ \begin{matrix}a_{11}z_{1}^{d_{1}−1} & … & a_{1s}z_{s}^{d_{1}−1} \\ ⋮ & & ⋮ \\ a_{R1}z_{1}^{d_{R}−1} & … & a_{Rs}z_{s}^{d_{R}−1} \end{matrix} ]

has an R × R submatrix M such that

det M ≢ 0 (mod p^{h}).

Then the system (2) has a solution y ∈ ℤ_{p}^{s} such that y ≡ z (mod p^{h}).

If a solution of (2) modulo various powers of p satisfies the determinant condition above, we will say that this solution is nonsingular modulo p^{h}.

3. Normalization

In this section we define our normalization procedure and show that if the system (1) is normalized, then it is explicit in a large number of variables when considered modulo p. That is, when a normalized system is reduced modulo p, a large number of variables will appear in at least one form with a nonzero coefficient. Since our normalization originates with the one given by Davenport & Lewis in [5] for systems of additive forms of equal degrees, we begin by recalling the function they used to normalize such systems. Suppose that we have a system F of R homogeneous additive forms

F_{i}(x) = a_{i1}x_{1}^{k} + ⋯ + a_{is}x_{s}^{k}, (1 ≤ i ≤ R)

all of degree k. Let [a_{j}]_{1≤j≤s} be the matrix of coefficients of the system, where a_{j} is the vector of coefficients of the variable x_{j}. Define the function

Θ(F) = ∏_{1≤j_{1}<j_{2}<···<j_{R}≤s} det([a_{j_{n}}]_{1≤n≤R}).

In [5, Lemma 16], Davenport & Lewis showed that Θ satisfies the following properties.

Lemma 4. 1) If a system F′ is defined by

F′(x) = F(p^{w_{1}}x_{1}, p^{w_{2}}x_{2}, …, p^{w_{s}}x_{s}),

then we have

Θ(F′) = p^{kRMw/s}Θ(F),

where M = s(s − 1) ⋯ (s − R + 1) and w = w_{1} + ⋯ + w_{s}.

2) If a system F′′ is defined by taking linear combinations of the forms in F, so that

F′′_{i} = ∑_{j=1}^{R} d_{ij}F_{i} (1 ≤ i ≤ R),

then we have

Θ(F′′) = D^{M}Θ(F),

where D = det([d_{ij}]_{1≤i,j≤R}).

We now turn to our situation, where the forms may have different degrees. Given a system F, we first define two fundamental operations which may be performed on the forms, similar to the ones mentioned in Lemma 4. First, we may multiply each variable by a power of p, yielding a new system

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

Second, we may take linear combinations of the forms, provided that for each linear combination, the forms involved all have the same degree and the total number of forms of each degree does not change. That is, we may form a new system F′′ defined by

F′′_{i}(x) = ∑_{\begin{matrix}n \\ deg F_{n}=deg F_{i} \end{matrix}} d_{in}F_{n}(x).

Note that these operations commute. If a system G with coefficients in ℤ_{p} can be obtained from F through a combination of these operations, we will say that G is equivalent to F. Note that when F and G are equivalent, F possesses a nontrivial zero if and only if G does.

Next, for positive integers j, a, b with 1 ≤ j ≤ v and b−a+1 ≥ R_{j}, let (F_{j}, a, b) be the subsystem of F consisting only of the forms of degree k_{j} and only the variables x_{i} with a ≤ i ≤ b. For example, if F is the system of forms

x_{1}^{k_{1}} + 2x_{2}^{k_{1}} + 3x_{3}^{k_{1}}
4x_{1}^{k_{2}} + 5x_{2}^{k_{2}} + 6x_{3}^{k_{2}}
7x_{1}^{k_{2}} + 8x_{2}^{k_{2}} + 9x_{3}^{k_{2}},

then (F_{2}, 2, 3) is the system

\begin{matrix}5x_{2}^{k_{2}} + 6x_{3}^{k_{2}} \\ 8x_{2}^{k_{2}} + 9x_{3}^{k_{2}}. \end{matrix}

We define a function ∂_{j}(F, a, b) by

∂_{j}(F, a, b) = Θ(F_{j}, a, b).

That is, we define ∂_{j} by applying the Davenport-Lewis function to the subsystem (F_{j}, a, b) of F. Note that performing a combination of the fundamental operations on F yields a combination of fundamental operations on (F_{j}, a, b). This is because if we form a linear combination of the forms of F, we must use forms of the same degree, and because we do not change the number of forms of degree k_{j}. Hence Lemma 4 applies to ∂_{j}, and so we immediately get the following lemma.

Lemma 5. 1) If the system F′ is defined by F′ = F(p^{w_{1}}x_{1}, …, p^{w_{s}}x_{s}), then we have

∂_{j}(F′, a, b) = p^{k_{j}R_{j}M_{j}w/(b−a+1)}∂_{j}(F),

where M_{j} = (b−a+1)(b−a) ⋯ (b−a−R_{j} +2) and w = w_{a} +⋯+w_{b}. 2) If the system F′′ is defined by

F′′_{i} = ∑_{\begin{matrix}n \\ deg F_{n}=deg F_{i} \end{matrix}} d_{in}F_{n}(x),

then we have

∂_{j}(F′′, a, b) = D_{j}^{M_{j}}∂_{j}(F, a, b),

where D_{j} = det([d_{in}]_{deg F_{n}=deg F_{i}=k_{j}}).

Now we can begin to define the function we will use to normalize F. Pick integers 1 < n_{0} < n_{1} < ⋯ < n_{v} = s such that n_{0} ≥ R and n_{i} − n_{i−1} ≥ R_{i} for each i. Define numbers M_{i} and L_{i} by

M_{i} = n_{0}(n_{0} − 1) ⋯ (n_{0} − R_{i} + 1)

and

L_{i} = (n_{i} − n_{i−1})(n_{i} − n_{i−1} − 1) ⋯ (n_{i} − n_{i−1} − R_{i} + 1).

Next, define the numbers K and M by

K = ∏_{i=1}^{v} k_{i} and M = ∏_{i=1}^{v} M_{i},

and for 1 ≤ i ≤ v, define k′_{i} = K/k_{i}, and M′_{i} = M/M_{i}. Finally, for 1 ≤ i ≤ v, we define

N_{i} = \frac{(n_{i} − n_{i−1})k′_{i}RM}{n_{0}R_{i}L_{i}} .

At last, for any system F, we can define our normalization function ∂(F) by

∂(F) = [∏_{i=1}^{v} ∂_{i}(F, 1, n_{0})^{k′_{i}M′_{i}}] [∏_{i=1}^{v} ∂_{i}(F, n_{i−1} + 1, n_{i})^{N_{i}}] .

We will say that the system F with coefficients in ℤ_{p} 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 with coefficients in ℤ_{p} which is equivalent to F.

We now prove a lemma showing that the function ∂ behaves nicely under the fundamental operations. This is the analogue of Lemma 4 for our normalization function.

Lemma 6. Suppose that F is a system of additive forms.

1) If the system F′ is defined by F′ = F(p^{w_{1}}x_{1}, …, p^{w_{s}}x_{s}), then we have

∂(F′) = p^{KRMw/n_{0}}∂(F),

where w = w_{1} + ⋯ + w_{s}.

2) If the system F′′ is defined by

F′′_{j} = ∑_{\begin{matrix}n \\ deg F_{n}=deg F_{j} \end{matrix}} d_{jn}F_{n},

then we have

∂(F′′) = (∏_{i=1}^{v} D_{i}^{A_{i}}) ∂(F),

where

D_{i} = det([d_{jn}]_{deg F_{n}=deg F_{j}=k_{i}})

and

A_{i} = Mk′_{i} + \frac{(n_{i} − n_{i−1})k′_{i}RM}{n_{0}R_{i}} .

Proof. For part 1) of the lemma, we have

∂(F′) = [∏_{i=1}^{v} ∂_{i}(F′, 1, n_{0})^{k′_{i}M′_{i}}] [∏_{i=1}^{v} ∂_{i}(F′, n_{i−1} + 1, n_{i})^{N_{i}}]

= [∏_{i=1}^{v} (p^{B_{i}}∂_{i}(F, 1, n_{0}))^{k′_{i}M′_{i}}] [∏_{i=1}^{v} (p^{C_{i}}∂_{i}(F, n_{i−1} + 1, n_{i}))^{N_{i}}]

where

B_{i} = \frac{k_{i}R_{i}M_{i}(w_{1} + ⋯ + w_{n_{0}})}{n_{0}}

and

C_{i} = \frac{k_{i}R_{i}L_{i}(w_{n_{i−1}+1} + ⋯ + w_{n_{i}})}{(n_{i} − n_{i−1})}.

Combining the powers of p and simplifying, we find that ∂(F′) = p^{A}∂(F), where

A = ∑_{i=1}^{v} B_{i}k′_{i}M′_{i} + ∑_{i=1}^{v} C_{i}N_{i}
= ∑_{i=1}^{v} \frac{R_{i}(w_{1} + ⋯ + w_{n_{0}})KM}{n_{0}} + ∑_{i=1}^{v} \frac{(w_{n_{i−1}+1} + ⋯ + w_{n_{i}})KRM}{n_{0}}
= \frac{(w_{1} + ⋯ + w_{n_{0}})KM}{n_{0}} ∑_{i=1}^{v} R_{i} + \frac{KRM}{n_{0}} ∑_{i=1}^{v}(w_{n_{i−1}+1} + ⋯ + w_{n_{i}})
= \frac{KRM}{n_{0}} (w_{1} + w_{2} + ⋯ + w_{s}) .

To prove part 2) of the lemma, note that we have

∂(F′′) = [∏_{i=1}^{v} ∂_{i}(F′′, 1, n_{0})^{k′_{i}M′_{i}}] [∏_{i=1}^{v} ∂_{i}(F′′, n_{i−1} + 1, n_{i})^{N_{i}}]
= [∏_{i=1}^{v} (D_{i}^{M_{i}}∂_{i}(F, 1, n_{0}))^{k′_{i}M′_{i}}] [∏_{i=1}^{v} (D_{i}^{L_{i}}∂_{i}(F, n_{i−1} + 1, n_{i}))^{N_{i}}]
= ∂(F) · ∏_{i=1}^{v} D_{i}^{M_{i}k′_{i}M′_{i}+L_{i}N_{i}}
= ∂(F) · ∏_{i=1}^{v} D_{i}^{Mk′_{i}+(n_{i}−n_{i−1})k′_{i}RM/n_{0}R_{i}},

as desired.□

Now we can prove that p-normalized forms have certain desirable properties. These properties essentially state that if the forms in F, or linear combinations of these forms, are reduced modulo p, then many variables will appear in the reduced forms with a nonzero coefficient.

While we develop here only a few results along these lines, we note that it is possible to go further and develop results analogous to Lemmata 2.2 and 2.3 of [12].

Lemma 7. Let the system F of additive forms be p-normalized.

1) Suppose that when this system is reduced modulo p, there are m variables which have a nonzero coefficient in at least one of the reduced forms. Then we have

m ≥ ∑_{i=1}^{v} \frac{1}{k_{i}} (\frac{n_{0}R_{i}}{R} + n_{i} − n_{i−1}) .

2) Suppose that F is one of the forms in F, having degree k_{l}. If q_{l} denotes the number of variables which appear with a nonzero coefficient when F is reduced modulo p, then we have

q_{l} ≥ \frac{1}{k_{l}} (\frac{n_{0}}{R} + \frac{n_{l} − n_{l−1}}{R_{l}} ) .

3) Consider the forms in F of degree k_{l}, and suppose that we make any H ≤ R_{l} linear combinations of these forms which are linearly independent over ℤ/pℤ. Suppose that when these linear combinations are reduced modulo p, there are Q = Q(k_{l}, H) variables which appear in at least one of these reduced forms with a nonzero coefficient. Then we have

Q ≥ \frac{H}{k_{l}} (\frac{n_{0}}{R} + \frac{n_{l} − n_{l−1}}{R_{l}} ) .

Proof. Since part 2 is a special case of part 3 of this lemma, we will only give explicit proofs of part 1 and part 3. To prove part 1, suppose that the variables which are explicit modulo p are x_{i}, i ∈ I. Define a system F′ by F′ = p^{−1}F(y), where

y_{i} = \{ \begin{matrix}px_{i} & if i ∈ I \\ x_{i} & if i ∉ I, \end{matrix}

and note that all the coefficients of this system are in ℤ_{p}. By Lemma 6, we have ∂(F′) = p^{A}∂(F), where

A = \frac{KRMm}{n_{0}} − ∑_{i=1}^{v} (R_{i}Mk′_{i} + \frac{(n_{i} − n_{i−1})k′_{i}RM}{n_{0}} ) .

Now, since the system F is p-normalized, we must have A ≥ 0, whence we obtain

m ≥ \frac{n_{0}}{KRM} ∑_{i=1}^{v} (R_{i}Mk′_{i} + \frac{(n_{i} − n_{i−1})k′_{i}RM}{n_{0}} )
= ∑_{i=1}^{v} \frac{1}{k_{i}} (\frac{n_{0}R_{i}}{R} + n_{i} − n_{i−1}) ,

as desired.

In order to prove part 3, we first complete the original H linear combinations to a set of R_{l} linear combinations of the forms of degree k_{l} which are linearly independent over ℤ/pℤ. Let F′ be the system consisting of these R_{l} forms and the R − R_{l} forms in F of degrees other than k_{l}. Let F′_{j}, j ∈ J, be the forms in F′ produced by our original H linear combinations, and suppose that x_{i}, i ∈ I are the variables which have a nonzero coefficient when at least one of these H forms are reduced modulo p. Finally, consider the system F′′ defined by

F′′_{j} = \{ \begin{matrix}p^{−1}F′_{j}(y) & if j ∈ J \\ F′_{j}(y) & if j ∉ J, \end{matrix}

where

y_{i} = \{ \begin{matrix}px_{i} & if i ∈ I \\ x_{i} & if i ∉ I. \end{matrix}

Note that all of the coefficients in both F′ and F′′ are elements of ℤ_{p}. By Lemma 6, we have

∂(F′′) = p^{MRKQ/n_{0}}p^{−H(Mk′_{l}+(n_{l}−n_{l−1})k′_{l}RM/n_{0}R_{l})}∂(F′)

and

∂(F′) = D^{Mk′_{l}+(n_{l}−n_{l−1})k′_{l}RM/n_{0}R_{l}}∂(F),

where D is the determinant of the matrix corresponding to the linear combinations taken to produce the R_{l} forms of degree k_{l} in F′. Hence we may write ∂(F′′) = p^{A}D^{B}∂(F), where

A = \frac{MRKQ}{n_{0}} − H (Mk′_{l} + \frac{(n_{l} − n_{l−1})k′_{l}RM}{n_{0}R_{l}} )

and B is the exponent on D above. Since these combinations are linearly independent over ℤ/pℤ, we have p |̸ D. Now, since the system F is p-normalized, we must have A ≥ 0, or in other words

\frac{MRKQ}{n_{0}} − H (Mk′_{l} + \frac{(n_{l} − n_{l−1})k′_{l}RM}{n_{0}R_{l}} ) ≥ 0.

Solving this for Q yields

Q ≥ \frac{H}{k_{l}} (\frac{n_{0}}{R} + \frac{n_{l} − n_{l−1}}{R_{l}} ) ,

as desired. This completes the proof of the lemma. □

4. Nonsingular Matrices

This section is devoted to the proof of the following lemma, which shows that we can guarantee that certain matrices are nonsingular modulo a prime p.

Lemma 8. Suppose that R is a positive integer, that R_{1}, …, R_{v} are positive integers such that R_{1}+⋯+R_{v} = R, that k_{1}, …, k_{v} are positive integers with k_{1} > k_{2} > ⋯ > k_{v}, and that p is a prime satisfying p > k_{1} − k_{v} + 1. Suppose that A is the matrix

A = [ \begin{matrix}a_{11}x_{1}^{k_{1}−1} & ⋯ & a_{1R}x_{R}^{k_{1}−1} \\ ⋮ & & ⋮ \\ a_{R1}x_{1}^{k_{v}−1} & ⋯ & a_{RR}x_{R}^{k_{v}−1} \end{matrix} ] ,

where the exponent on the x_{i} is k_{1} −1 in the first R_{1} rows, the exponent is k_{2} − 1 in the next R_{2} rows, and so on. Suppose further that for 1 ≤ n ≤ v, the R_{n} × R_{n} submatrix

A_{n} = [a_{ij}]_{R_{1}+···+R_{n−1}+1≤i,j≤R_{1}+···+R_{n}}

is nonsingular modulo p (i.e. det A_{n} ≢ 0 (mod p)). Then there exist numbers b_{2}, …, b_{R}, all nonzero modulo p, such that if we set

x_{2} = b_{2}x_{1}, …, x_{R} = b_{R}x_{1}

and let x_{1} be any number which is nonzero modulo p, then the matrix A will be nonsingular modulo p.

To begin the proof, notice that if we make the substitution

x_{j} = b_{j}x_{1}, 2 ≤ j ≤ R,

and then pull as many factors of x_{1} and b_{j} as possible out of the determinant, we obtain

det A = x_{1}^{R_{1}k_{1}+···+R_{v}k_{v}−R}(b_{2} ⋯ b_{R})^{k_{v}−1} det B,

where B is the matrix

B = [ \begin{matrix}a_{11} & a_{12}b_{2}^{k_{1}−k_{v}} & ⋯ & a_{1R}b_{R}^{k_{1}−k_{v}} \\ ⋮ & ⋮ & & ⋮ \\ a_{R1} & a_{R2} & ⋯ & a_{RR} \end{matrix} ] .

Therefore it suffices to show that we can choose b_{2}, …, b_{R} all nonzero modulo p such that B is nonsingular modulo p. We prove this by induction on v, the number of distinct degrees. If v = 1, then the matrix

[ \begin{matrix}a_{11} & ⋯ & a_{1R} \\ ⋮ & & ⋮ \\ a_{R1} & ⋯ & a_{RR} \end{matrix} ]

is nonsingular modulo p by hypothesis, and so it suffices to set b_{2} = ⋯ = b_{R} = 1.

Suppose now that the lemma is true when there are v − 1 distinct degrees. Set m = R_{1} + ⋯ + R_{v−1}, and choose values for b_{2}, …, b_{m}, all nonzero modulo p, which make the upper left-hand m × m submatrix C of B nonsingular modulo p. Now we just need to show that we can choose values for b_{m+1}, …, b_{R} which work. Since the submatrix C of B is nonsingular modulo p, we can perform elementary row operations on B to obtain a matrix

[ \begin{matrix}C & ∗ \\ 0_{p} & M \end{matrix} ] ,

where all the entries of 0_{p} are divisible by p, the asterisk could represent any matrix, and M has the form

M = [a_{ij} − p_{ij}(b_{j})]_{m+1≤i,j≤R} ,

where each p_{ij}(b_{j}) is a polynomial of degree at most k_{1} − k_{v}. Since C is nonsingular modulo p, it suffices to show that we can choose b_{m+1}, …, b_{R} to make M nonsingular modulo p. Note that if it happens that each p_{ij}(b_{j}) is identically zero, then setting b_{m+1} = ⋯ = b_{R} = 1 yields the desired conclusion.

Suppose then that at least one of the p_{ij}(b_{j}) is not identically zero. Then expanding det M yields a polynomial det M = q(b_{m+1}, …, b_{R}) in which deg_{b_{j}} q ≤ k_{1} − k_{v} for each variable b_{j}. Select a term in this polynomial with highest total degree d which has a nonzero coefficient modulo p (if several such terms exist, any one of them can be selected). Without loss of generality, suppose that the variables involved in this term are b_{m+1}, b_{m+2}, …, b_{r}. Set b_{r+1} = ⋯ = b_{R} = 1. Then det M becomes a polynomial of total degree d in b_{m+1}, …, b_{r} and contains a term

b_{m+1}^{d_{m+1}}b_{m+2}^{d_{m+2}} ⋯ b_{r}^{d_{r}}

whose coefficient is nonzero modulo p, and

d_{m+1} + ⋯ + d_{r} = d.

Also, for each j, we have d_{j} ≤ k_{1} − k_{v}. Since p > k_{1} − k_{v} + 1, we have p − 1 > d_{j} for each j. Then by Lemma 1, there exist values of b_{m+1}, …, b_{r} ∈ {1, 2, …, p − 1} such that

q(b_{m+1}, …, b_{r}, 1, …, 1) ≢ 0 (mod p).

The lemma follows upon noting that the value of each of b_{1}, …, b_{R} in our construction is nonzero modulo p.

5. Proof of Theorem 2

In this section, we will prove Theorem 2 using the tools developed in Sections 2, 3, and 4. Suppose that F is a system as in (1) defined over ℤ_{p} and that p > max{k_{1} − k_{v} + 1, 2}. Let s_{0} be the bound on the number of variables given in Theorem 2. Assume briefly that we can prove that the theorem is true for systems containing exactly s_{0} variables. If F contains more than s_{0} variables, then setting s − s_{0} of them equal to zero yields a new system F′ of diagonal forms in exactly s_{0} variables. Since the theorem applies to F′, we know that F′ has a nontrivial zero, and hence so does F. Thus, in order to prove Theorem 2, it suffices to prove it for systems containing exactly s_{0} variables.

Suppose then that F has exactly s_{0} variables. Note that since our normalization function ∂ is a polynomial in the coefficients of F, a standard argument (see for example pages 572-573 of [5]) shows that if the theorem is true for all systems with ∂(F) ≠ 0, then it is true for all systems with ∂(F) = 0 as well. Hence we may assume that ∂(F) ≠ 0. Additionally, since every system with ∂(F) ≠ 0 is equivalent to a pnormalized system, we may assume without loss of generality that F is p-normalized. When we normalize, we must specify the values of n_{0}, n_{1}, …, n_{v} used in the normalization procedure. To this end, we set n_{0} = R and for 1 ≤ i ≤ v, we define n_{i} by the recursion

n_{i} = n_{i−1} + tR_{i}k_{i} − k_{i} − R_{i} + t(R_{1} + ⋯ + R_{i−1})R_{i}k_{i} + 1,

noting that we have n_{v} = s_{0}.

Let A be the matrix of coefficients of F. We wish to find a collection of submatrices of A which satisfy the following properties.

(1) Let t be as defined in the theorem. Then for 1 ≤ j ≤ v, there are t submatrices of size R_{j} × R_{j} in the collection.

(2) Each submatrix of size R_{j} × R_{j} contains only coefficients of the forms of degree k_{j}.

(3) Each submatrix in the collection is nonsingular modulo p (i.e. if B is one of the submatrices, then det B ≢ 0 (mod p)).

(4) Each column of A contributes elements to at most one subma-

trix in the collection. To accomplish this, we follow the method used by Brüdern & Godinho in [4], which adapts the work of Low, Pitman & Wolff in [10]. In the rest of this section, when we use the term “nonsingular,” we will really mean “nonsingular modulo p.”

First, we will find t disjoint R_{1} × R_{1} nonsingular submatrices of A. Let A_{1} be the submatrix of A containing the coefficients of the forms of degree k_{1}. Then A_{1} is an R_{1} × s_{0} matrix. Let s_{1,m} be the minimal number of nonzero columns in any m linearly independent linear combinations of the rows of A_{1}. Then Brüdern & Godinho show in [4] that A_{1} contains t disjoint nonsingular R_{1} × R_{1} submatrices if and only if

s_{1,m} ≥ tm for all 0 ≤ m ≤ R_{1}.

Clearly this condition holds for m = 0. Now, from Lemma 7, we have

s_{1,m} = Q(k_{1}, m)
≥ \frac{m}{k_{1}} (1 + \frac{tk_{1}R_{1} − k_{1} − R_{1} + 1}{R_{1}})
= tm − \frac{m}{R_{1}} (1 − \frac{1}{k_{1}} ) .

Since \frac{m}{R_{1}} ≤ 1 and 1 − \frac{1}{k_{1}} < 1, we see that s_{1,m} > tm − 1, and since s_{1,m} is an integer we have s_{1,m} ≥ tm as desired. So we can form t disjoint nonsingular submatrices of A_{1} (and hence also of A) of size R_{1} × R_{1}.

These submatrices involve a total of tR_{1} columns of A. When we find our submatrices corresponding to the degree k_{2} forms, we do not want these columns involved. Hence we let A_{2} be the submatrix of coefficients of the degree k_{2} forms, excluding the coefficients from any columns of A which were involved in our previous nonsingular submatrices. Then A_{2} has R_{2} rows and s_{0} − tR_{1} columns. Again, we find from [4] that if s_{2,m} is the minimal number of nonzero columns in any m linearly independent linear combinations of the rows of A_{2}, then the submatrices we want exist provided that

s_{2,m} ≥ tm for all 0 ≤ m ≤ R_{2}.

Again, this is clearly true if m = 0. From the normalization process, we know that any m linearly independent linear combinations of the forms of degree k_{2} are explicit in at least

Q(k_{2}, m) ≥ \frac{m}{k_{2}} (\frac{n_{0}}{R} + \frac{n_{2} − n_{1}}{R_{2}} )

variables. However, up to tR_{1} of these variables might not contribute to the columns of A_{2}. Hence, we know that the number of nonzero columns in any m linearly independent linear combinations of the rows of A_{2} is at least

s_{2,m} ≥ \frac{m}{k_{2}} (1 + \frac{tR_{2}k_{2} − k_{2} − R_{2} + tR_{1}R_{2}k_{2} + 1}{R_{2}}) − tR_{1}
= tm + (m − 1)tR_{1} − \frac{m}{R_{2}} (1 − \frac{1}{k_{2}} )

whenever m ≥ 1. Again, we can see that this bound is strictly larger than tm − 1, and since s_{2,m} is an integer, we obtain s_{2,m} ≥ tm when m ≥ 1. Hence we have the bound that we need, and can get the required R_{2} × R_{2} nonsingular submatrices of A.

We can now proceed inductively to show that we can find all of the necessary nonsingular submatrices. Assume that we can find the submatrices associated with the forms of degrees k_{1}, k_{2}, …, k_{l−1}. These submatrices involve t(R_{1} + ⋯ + R_{l−1}) columns of A which must not contribute to the submatrices for the forms of degree k_{l}. Let A_{l} be the matrix containing the coefficients of the forms of degree k_{l}, but excluding any coefficients coming from a column of A which was involved in any of the submatrices we have found previously. Let s_{l,m} be the minimal number of nonzero columns in any m linearly independent linear combinations of the rows of A_{l}. As before, we can find t disjoint nonsingular R_{l} × R_{l} submatrices of A_{l} if and only if we have

(3)  s_{l,m} ≥ tm for all 0 ≤ m ≤ R_{l}.

Once again, this is trivially true for m = 0. If m ≥ 1, then we have

s_{l,m} ≥ Q(k_{l}, m) − t(R_{1} + R_{2} + ⋯ + R_{l−1})
≥ \frac{m}{k_{l}} (1 + \frac{tR_{l}k_{l} − k_{l} − R_{l} + t(R_{1} + ⋯ + R_{l−1})R_{l}k_{l} + 1}{R_{l}})
− t(R_{1} + ⋯ + R_{l−1})
= mt + (m − 1)t(R_{1} + ⋯ + R_{l−1}) − \frac{m}{R_{l}} (1 − \frac{1}{k_{l}} )
> mt − 1.

Again, since s_{l,m} is an integer, this implies that the inequality in (3) holds for 1 ≤ m ≤ R_{l}. Hence this inequality holds for all the necessary values of m, and so the matrices we are looking for do indeed exist.

Next, by relabeling the variables if necessary, we may assume that the t(R_{1} + ⋯ + R_{v}) = tR columns of A involved in our nonsingular submatrices are the columns corresponding to the first tR variables. We may also assume that the coefficients of x_{1}, …, x_{R} correspond (in order) first to the columns of one of the R_{1}×R_{1} nonsingular submatrices, then the columns of one of the nonsingular R_{2} × R_{2} submatrices, and so on, ending with the columns of one of the R_{v} ×R_{v} nonsingular submatrices. Further, we may assume that this is true for each set of R variables

x_{jR+1}, …, x_{(j+1)R}, (0 ≤ j ≤ t − 1).

Now, for each degree k_{j}, define the numbers τ_{j} and k̃_{j} as in the statement of Theorem 2. Consider the system

(4)  \begin{aligned}a_{11}x_{1}^{k_{1}} & + ⋯ + & a_{1s}x_{s}^{k_{1}} & ≡ 0 (mod p^{τ_{1}+1}) \\ ⋮ & & ⋮ & ⋮ \\ a_{R1}x_{1}^{k_{v}} & + ⋯ + & a_{Rs}x_{s}^{k_{v}} & ≡ 0 (mod p^{τ_{v}+1}). \end{aligned}

Note that we have arranged the variables so that the coefficients of each set of R variables

x_{jR+1}, …, x_{(j+1)R}, (0 ≤ j ≤ t − 1)

satisfy the nonsingularity hypothesis of Lemma 8. Thus for each j we can find numbers

b_{jR+2}, …, b_{(j+1)R},

all nonzero modulo p, such that if we set

x_{jR+2} = b_{jR+2}x_{jR+1}, …, x_{(j+1)R} = b_{(j+1)R}x_{jR+1},

then the matrix

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

is nonsingular modulo p whenever x_{jR+1} ≢ 0 (mod p). If we make these identifications and set x_{i} = 0 whenever i > tR, we obtain a new system of congruences

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

Note that any nontrivial solution of (5) yields a nontrivial solution of (4). In fact, because of the way we chose the multipliers b_{j}, this solution of (4) will be nonsingular modulo p.

We wish to solve (5) with all the variables lying in the Teichmüller set

T_{p} = {x ∈ ℤ_{p} : x^{p} = x}.

Note that if we write k = p^{τ} k̃ with (p, k̃) = 1, then we have

x^{k} = (x^{p^{τ}})k̃ = xk̃

whenever x ∈ T_{p}. Hence any nontrivial solution of the system

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

with all variables in T_{p} will also be a solution of (5).

Now, since we have

t > ∑_{j=1}^{v} R_{j}k̃_{j} \frac{p^{τ_{j}+1} − 1}{p − 1},

Lemma 2 tells us that we can solve the system (6) nontrivially with each variable in T_{p}. As noted above, this leads to a solution of (4) which is nonsingular modulo p. Then Lemma 3 shows that we can lift this solution of (4) to a solution of (1). Finally, since the solution of (4) contains at least one variable which is not divisible by p and our solutions of (1) and (4) are congruent modulo p, the solution of (1) is a nontrivial solution. This completes the proof of the theorem.

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Michael P. Knapp Mathematical Sciences Department Loyola College 4501 North Charles Street Baltimore, MD 21210-2699 Email: mpknapp@loyola.edu