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Simultaneous diagonal equations over p-adic

ﬁelds

D. Brink, H. Godinho, P. H. A. Rodrigues

February 2008

Abstract. Consider a system of R diagonal forms of degree k in N

variables over a p-adic ﬁeld K. Write k = p

τ

m where p is the residue

ﬁeld characteristic and (m, p) = 1. We prove that this system has a

non-trivial p-adic solution if N exceeds (Rk)

2τ+5

, a bound independent

of the ﬁeld K. We prove the same if N exceeds 4nR

2

k

2

where

n = [K : Q

p

]. Both results improve previously known bounds.

Let K be a ﬁnite extension of the ﬁeld of p-adic numbers Q

p

. Let O be the

ring of integers in K and let p be O’s unique maximal ideal. We say that

K is a p-adic ﬁeld.

Consider R simultaneous diagonal equations

a

11

X

k

1

+ · · · + a

1N

X

k

N

= 0

.

.

.

.

.

.

.

.

.

a

R1

X

k

1

+ · · · + a

RN

X

k

N

= 0

(∗)

with coeﬃcients a

ij

in O. Write the degree as k = p

τ

m with p - m. A

solution x = (x

1

, . . . , x

N

) ∈ K

N

is called non-trivial if at least one x

j

is

non-zero. It is a special case of a conjecture of Emil Artin that (∗) has a

non-trivial solution whenever N > Rk

2

. This conjecture has been veriﬁed

by Davenport and Lewis for a single diagonal equation over Q

p

and for a

pair of equations of odd degree over Q

p

(see [3] and [4]), but the general

case remains open.

The main results of the present paper are the following two theorems.

Theorem 1. The system (∗) has a non-trivial solution if the number of

variables N exceeds (Rk)

2τ+5

.

Theorem 2. Let n be the degree of the ﬁeld extension K/Q

p

. Then (∗)

has a non-trivial solution if N exceeds 4nR

2

k

2

.

Theorem 1 has the virtue of being independent of K and can be compared

with Skinner [11] where the bound N > k

6τ+4

is given for a single diagonal

equation. Theorem 2 is a natural generalisation of Knapp [7, Theorem 1]

1

and improves Dodson [6, Theorem 1] and Knapp [7, Theorem 3]. See also

Skinner [11] for other references.

Deﬁne the integer Γ(R, k) as minimal with the property that any system

(∗) with N > Γ(R, k) has a non-trivial solution over K. Then Theorems

1 and 2 can be restated as Γ(R, k) ≤ (Rk)

2τ+5

and Γ(R, k) ≤ 4nR

2

k

2

,

respectively. The idea of the proof of the theorems is to ﬁrst solve (∗) in

the ﬁnite residue ring O/p

γ

(for a suitable exponent γ), and then lift this

solution to K via a version of Hensel’s lemma.

A solution x ∈ O

N

is called primitive if at least one coordinate x

j

is a

unit in O. Deﬁne the integer Φ(R, k, ν) as minimal with the property that

any system (∗) with N > Φ(R, k, ν) has a primitive solution modulo p

ν

.

The Chevalley-Warning theorem (see [2, Lemma 4]) states that any sys-

tem of homogeneous polynomials over a ﬁnite ﬁeld has a non-trivial zero

if the number of variables exceeds the sum of the polynomials’ degrees. In

the special case of systems of diagonal equations, the Chevalley-Warning

theorem gives

Φ(R, k, 1) ≤ Rk. (1)

For general moduli a, b ≥ 1 one has the relation

Φ(R, k, a + b) + 1 ≤ (Φ(R, k, a) + 1) (Φ(R, k, b) + 1) . (2)

This is shown using a well-known ”contraction” argument (see examples in

([4] and [11]). The idea is to construct a primitive solution modulo p

a+b

in

N = (Φ(R, k, a) + 1) · (Φ(R, k, b) + 1) variables as follows: First divide the

left hand side of (∗) into Φ(R, k, a) + 1 subsystems of diagonal forms, each

in Φ(R, k, b) + 1 variables, and solve each system primitively modulo p

b

.

Then multiply each of these solutions by a new variable to form a system

of diagonal forms in Φ(R, k, a) + 1 variables. Since every coeﬃcient is a

multiple of p

b

, to solve this new system primitively modulo p

a+b

is basically

to solve it modulo p

a

. This results in a primitive solution modulo p

a+b

to

(∗) which proves (2).

Let A = (a

ij

) be the coeﬃcient matrix of (∗). A solution x ∈ O

N

is called

non-singular if the matrix (a

ij

x

k

j

) has rank R modulo p, or, equivalently, if

the columns of A corresponding to the indices j with x

j

6≡ 0 (mod p) have

rank R modulo p.

The following strong version of Hensel’s lemma is a natural generalisa-

tion of [5, Lemma 9], from p-adic to p-adic ﬁelds. The deﬁnition of γ here

is somewhat better than the value 2eτ + 1 often found in the literature (al-

though Alemu [1] has a result for one equation similar to the lemma below).

Lemma 1. Let e be the ramiﬁcation index of K over Q

p

and deﬁne

γ :=

1 for τ = 0,

e(τ + 1) for τ > 0 and p 6= 2,

e(τ + 2) for τ > 0 and p = 2.

The system (∗) then has a non-trivial solution in K if it has a non-singular

solution modulo p

γ

.

2

Proof. We ﬁrst show that a unit u ∈ O

∗

is a kth power if u ≡ ξ

k

(mod p

γ

)

for some ξ ∈ O

∗

. This is the standard Hensel’s lemma for τ = 0, so we may

assume τ > 0. Then multiplication x 7→ k·x maps p

e

onto k·p

e

= p

eτ+e

= p

γ

for p 6= 2, and p

2e

onto k · p

2e

= p

γ

for p = 2. For any n > e/(p − 1), the p-

adic exponential function and the p-adic logarithm are inverse isomorphisms

between the additive group p

n

and the the multiplicative group 1 + p

n

([9,

Kapitel II, Satz 5.5]). It follows that exponentiation x 7→ x

k

maps 1 + p

e

(for p 6= 2) and 1 + p

2e

(for p = 2) onto 1 + p

γ

. The diagram shows the

situation for p 6= 2:

1 + p

e

x7→x

k

// //

log

1 + p

γ

p

e

x7→k·x

// //

p

γ

exp

OO

Therefore, the elements of the set ξ

k

·(1+ p

γ

) = ξ

k

+p

γ

, to which u belongs,

are all kth powers.

Now let x = (x

1

, . . . , x

N

) be a non-singular solution to (∗) modulo p

γ

.

We may assume x

1

, . . . , x

R

6≡ 0 (mod p) and that the ﬁrst R columns of

A have rank R modulo p, i.e. form a non-singular matrix modulo p. Row

operations on A will not change the solution set, so we may assume

A =

a

11

0 a

1,R+1

. . . a

1N

.

.

.

.

.

.

.

.

.

0 a

RR

a

R,R+1

. . . a

RN

with a

11

, . . . , a

RR

6≡ 0 (mod p). For each i = 1, . . . , R we have x

k

i

≡

u

i

(mod p

γ

) with u

i

= −(a

i,R+1

x

k

R+1

+ · · · + a

iN

x

k

N

)/a

ii

. By the above,

the equation X

k

= u

i

has a solution x

0

i

because it has the solution x

i

mod-

ulo p

γ

. We conclude that (x

0

1

, . . . , x

0

R

, x

R+1

, . . . , x

N

) solves (∗).

The notion of a p-normalised system of diagonal equations over Q

p

was

introduced in [5]. It is shown there that any system of the form (∗) over

Q

p

has a non-trivial solution provided that any p-normalised system has a

non-trivial solution. All of this is easily generalised to π-normalised systems

with p-adic coeﬃcients (see [7] for details).

Let µ(d) be the maximal number of columns of the coeﬃcient matrix A

which, when considered modulo p, lie in a d-dimensional subspace of F

N

q

.

The key property of π-normalised systems is the inequality

µ(d) ≤ N − (R − d)N/Rk for d = 0, . . . , R − 1. (3)

This is [5, Lemma 11] combined with [2, eq. (9)]. An equivalent statement

of this inequality is that any matrix having (R − d) rows which are linear

combinations of the rows of A, independent modulo p, contains at least

(R − d)N/Rk columns which are nonzero modulo p.

The following slight strengthening of [2, Lemma 2] essentially gives one

extra non-singular submatrix.

3

Lemma 2. Suppose (∗) is π-normalised and has more than k(tR − 1)

variables, where t is arbitrary. Then the coeﬃcient matrix A contains t

disjoint R × R submatrices which are non-singular modulo p.

Proof. For every d = 0, . . . , R − 1, the assumption N > k(tR − 1) combined

with (3) implies µ(d) ≤ N − (R − d)t since µ(d) is integral. Now the

conclusion follows by a combinatorial result of Aigner (see [8, Lemma 1] or

the comment before [2, Lemma 2]).

Next, we extend and improve [2, Lemma 5] using the same idea of proof.

Lemma 3. Suppose (∗) is π-normalised and has more than Rk ·Φ(R, k, ν)−

k(R − 1)

2

variables, where ν is arbitrary. Then (∗) has a non-singular

solution modulo p

ν

.

Proof. Suppose ﬁrst (∗) has N = k(tR − 1) + 1 variables for some t to be

deﬁned later. Then, by Lemma 2, A has t disjoint R×R submatrices which

are non-singular modulo p. Discard all variables not belonging to one of

these t submatrices. Then we have tR variables left. In each of all but one

of the t submatrices, replace all R variables by one new variable. Then we

have a new system with t − 1 + R variables. This system, by deﬁnition,

has a primitive solution modulo p

ν

if t − 1 + R > Φ(R, k, ν), hence if

t = Φ(R, k, ν) − R + 2. Not all the new variables of this solution can be

zero modulo p since the columns corresponding to the old variables form a

non-singular submatrix modulo p and so are linearly independent modulo p.

Therefore, ”inﬂating” the new variables again gives a non-singular solution

to our original system (∗) in N = Rk · Φ(R, k, ν) − k(R − 1)

2

+ 1 variables,

and the lemma is proved.

Recall that Γ(R, k) is the minimal integer such that any system (∗) with

N > Γ(R, k) has a non-trivial solution. From Lemmas 1 and 3 it follows

that

Γ(R, k) ≤ Rk · Φ(R, k, γ) − k(R − 1)

2

(4)

since any bound on Γ(R, k) may be proved under the assumption that (∗)

is π-normalised. For degree k not divisible by p, (4) and (1) give

Γ(R, k) ≤ (Rk)

2

− k(R − 1)

2

, (5)

which extends [2, Theorem 3].

Now, Theorem 2 follows from (4) and the following lemma.

Lemma 4. With γ deﬁned as in Lemma 1, we have

Φ(R, k, γ) ≤

(

p(p − 1)

−1

nRk for p > 2,

4nRk for p = 2.

Proof. To bound Φ(R, k, γ), we must ﬁnd a primitive solution modulo p

γ

4

to (∗). The additive group of the ﬁnite residue ring O/p

γ

is equal to the

direct sum of n cyclic subgroups of order p

γ/e

,

O/p

γ

= Zλ

1

⊕ · · · ⊕ Zλ

n

.

This can be seen for example by counting the number of elements of any

given order in both groups and noting that these numbers are the same (see

also [1] for a diﬀerent proof and a more general statement). Writing each

coeﬃcient a

ij

of (∗) as a Z-linear combination of the λ

i

’s, we see that it

suﬃces to solve nR congruences

c

i1

X

k

1

+ · · · + c

iN

X

k

N

≡ 0 (mod p

γ/e

), i = 1, . . . , nR (6)

with coeﬃcients c

ij

∈ Z. We shall only look for solutions x ∈ T

N

where

T = {x ∈ Q

p

| x

p

= x} is the set of Teichm¨uller representatives. Since

{x

k

| x ∈ T} = {x

(k,p−1)

| x ∈ T}, we may in (6) replace the exponent

k by (k, p − 1). Now, by a theorem of Schanuel [10], the system (6) has a

non-trivial solution x ∈ T

N

if N > nR(k, p−1)(p

γ/e

−1)(p−1)

−1

. Recalling

k = p

τ

m, we see that (k, p − 1) divides m and conclude that Φ(R, k, γ) is

bounded by nR(k, p − 1)p

τ+1

(p − 1)

−1

≤ p(p − 1)

−1

nRk for p 6= 2, and by

4nRk for p = 2.

The next two lemmas and the ﬁnal proof of Theorem 1 are much inspired

by the ideas presented in Skinner [11].

Lemma 5. Any a ∈ O can be written as

a ≡ c

p

τ

0

+ πc

p

τ

1

+ π

2

c

p

τ

2

+ · · · + π

p

τ

−1

c

p

τ

p

τ

−1

(mod p)

with c

j

∈ O and π being a prime element of O.

Proof. If R ⊂ O is a set of representatives for O/p, then so is {r

p

τ

| r ∈ R},

because the map x 7→ x

p

τ

is a bijection F

q

→ F

q

. Hence, with suitable

r

n

∈ R, we can write

a =

∞

X

n=0

r

p

τ

n

π

n

=

p

τ

−1

X

j=0

π

j

∞

X

i=0

r

p

τ

j+ip

τ

π

ip

τ

≡

p

τ

−1

X

j=0

π

j

∞

X

i=0

r

j+ip

τ

π

i

!

p

τ

(mod p),

which proves the lemma.

Lemma 6. Φ(R, k, e) ≤ Φ(Rp

τ

, m, e).

Proof. We have to ﬁnd a primitive solution x ∈ O

N

to the R congruences

a

i1

X

k

1

+ · · · + a

iN

X

k

N

≡ 0 (mod p), i = 1, . . . , R.

Write each polynomial in this system as a sum of p

τ

polynomials using the

above lemma on each coeﬃcient a = a

ij

. Thus it suﬃces to ﬁnd a primitive

solution to Rp

τ

congruences

c

p

τ

i1

X

k

1

+ · · · + c

p

τ

iN

X

k

N

≡ 0 (mod p), i = 1, . . . , Rp

τ

.

5

Since

c

p

τ

i1

X

k

1

+ · · · + c

p

τ

iN

X

k

N

≡ (c

i1

X

m

1

+ · · · + c

iN

X

m

N

)

p

τ

(mod p),

it suﬃces to ﬁnd a primitive solution to the Rp

τ

congruences

c

i1

X

m

1

+ · · · + c

iN

X

m

N

≡ 0 (mod p), i = 1, . . . , Rp

τ

.

Such a solution exists by deﬁnition for N > Φ(Rp

τ

, m, e).

We can ﬁnally prove Theorem 1. Clearly, Φ(Rp

τ

, m, e) is bounded by

Γ(Rp

τ

, m) which is in turn bounded by (Rk)

2

− m(Rp

τ

− 1)

2

by (5) since

m is not divisible by p. For τ = 0 we already have the bound (5) which is

superior to the one given in Theorem 1. So assume τ > 0. Then Lemma 6

implies

Φ(R, k, e) < (Rk)

2

. (7)

From (4), (2), and (7) it now follows that

Γ(R, k) ≤ Rk · Φ(R, k, γ)

≤ Rk · (Φ(R, k, e) + 1)

γ/e

≤ (Rk)

2γ/e+1

≤ (Rk)

2τ+5

.

This concludes the proof of Theorem 1.

References

[1] Y. Alemu, On zeros of diagonal forms over p-adic ﬁelds, Acta Arith.

48 (1987), no. 3, 261–273.

[2] J. Br¨udern & H. Godinho, On Artin’s Conjecture, I: Systems of diag-

onal forms, Bull. London Math. Soc. 31 (1998), 305–313.

[3] H. Davenport & D. J. Lewis, Homogeneous additive equations, Proc.

Roy. Soc. London Ser. A 267 (1963), 443–460.

[4] H. Davenport & D. J. Lewis, Two additive equations, Number Theory

(ed. W. J. LeVeque and E. G. Strauss), Proc. Symp. Pure Math. 12

(1969), 74–98.

[5] H. Davenport & D. J. Lewis, Simultaneous equations of additive type,

Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 557–595.

[6] M. Dodson, Some estimates for diagonal equations over p-adic ﬁelds,

Acta Arith. 40 (1982), 117–124.

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don Math. Soc. (2) 63 (2001), no. 2, 257–267.

6

[8] L. Low, J. Pitman & A. Wolﬀ, Simultaneous diagonal congruences, J.

Number Theory 29 (1988), 31–59.

[9] J. Neukirch, Algebraische Zahlentheorie, Springer, Berlin, 1992.

[10] S. H. Schanuel, An extension of Chevalley’s theorem to congruences

modulo prime powers, J. Number Theory 6 (1974), 284–290.

[11] C. Skinner, Local solvability of diagonal equations (again), Acta Arith.

124 (2006), no. 1, 73–77.

7