On Waring’s problem in finite fields by

LXXXVII.2 (1998)

On Waring’s problem in finite fields

by

Arne Winterhof (Braunschweig)

1. Introduction. Let g(k, p

) be the smallest s such that every element of F

is a sum of s kth powers in F

.

In Section 2 we summarize the basic results on g(k, p

). In Section 3 we generalize Dodson’s upper bound for small k ([5], Lemma 2.5.4):

g(k, p) < b8 ln pc + 1; k | p − 1, p/2 < k

< p, and deduce

g(k, p

) ≤ b32 ln kc + 1 for p

> k

.

The object of Section 4 is to investigate to what extent Waring’s problem for F

can be reduced to the problem for F

. It is proven that if g(k, p

) exists, then

g(k, p

) ≤ ng(d, p); d = k

(k, (p

− 1)/(p − 1)) , k | p

− 1.

It is well known (see [3]) that

g(k, p) ≤ bk/2c + 1; k < (p − 1)/2.

[15], Theorem 1, implies that if g(k, p

) exists and p is odd, then g(k, p

) ≤ bk/2c + 1 for k < (p

− 1)/2. Whether p has to be odd has not been known yet. In Section 5 we show that p need not be odd.

2. Basic results on g(k, p

). Every (k, p

− 1)th power is at the same time a kth power. Hence,

(1) g(k, p

) = g((k, p

− 1), p

).

It is sufficient to restrict ourselves to the case

(2) k | p

− 1.

Remember that the multiplicative group F

is cyclic. Hence

(3) g(k, p

) = 1 ⇔ k = 1.

1991 Mathematics Subject Classification: 11P05, 11T99.

[171]

Since L := {x

+ . . . + x

| x

, . . . , x

∈ F

, s ∈ N} is a field ([16], Lemma 1), g(k, p

) exists if and only if L is not a proper subfield of F

, and thus

(4) g(k, p

) exists if and only if p

− 1

p

− 1 - k for all n 6= d | n.

This result is essentially that of [1], Theorem G.

We shall suppose that from now on g(k, p

) exists.

Let A

= {z

+. . .+z

| z

, . . . , z

∈ F

}. If A

$ A

then y ∈ A

\A

implies xy ∈ A

\A

for each 0 6= x ∈ A

, so that

|A

| ≥ |A

| + |A

| − 1 = |A

| + p

− 1

k .

Hence in the chain A

⊂ A

⊂ . . . ⊂ A

= F

there are at most k − 1 strict inclusions and therefore

(5) g(k, p

) ≤ k,

which is a specialization of [10], Th´eor`eme 7.14.

Equality holds for the following examples:

g(1, p

) = 1, g(2, p

) = 2, g

 p − 1 2 , p



= p − 1

2 , g(p − 1, p) = p − 1.

Since |A

| ≤

+ 1

, we get a trivial lower bound for g(k, p

):

(6) g(k, p

) ≥

 ln p

ln

+ 1

 . For n = 1 the following results are well known:

g(k, p) ≤ max(3, b32 ln kc + 1); p > k

[6], (7)

g(k, p) ≤ 68(ln k)

k

; p > 2k + 1 [7], (8)

g(k, p) ≤ bk/2c + 1; p > 2k + 1 [3], (9)

g(k, p) ≤



1 + 2k

p − 1



(1 + b2 log

pc); p > k

[2], (10)

g(k, p) ≤ 170 k

(p − 1)

ln p; p ≤ k

+ 1 [8], (11)

g(k, p) ≤ c

(ln k)

; k ≥ 2, p ≥ k ln k

(ln(ln k + 1))

, ε > 0 [11], (12)

g(k, p) ≤ c

; k < p

, ε > 0 [9].

(13)

3. Extension of Dodson’s bound for small k. Now we consider the

case 0 < (k − 1)

< p

. In this case g(k, p

) exists.

The number N

(b); b ∈ F

, of solutions of the equation x

+ . . . + x

= b; x

, . . . , x

∈ F

, can be expressed in terms of Jacobi sums ([12], Theorem 6.34)

N

(b) = p

+

k−1

X

j1,...,js=1

λ

(b)J(λ

, . . . , λ

), where λ is a multiplicative character of F

of order k.

Using the fact that

|J(λ

, . . . , λ

)| =

 p

if λ

is non-trivial, p

if λ

is trivial ([12], Theorem 5.22), we obtain

|N

(b) − p

| ≤ (k − 1)

p

and in particular

N

(b) ≥ p

− (k − 1)

p

. Hence,

(14) g(k, p

) ≤ s for p

> (k − 1)

. For s = 2 this is Small’s [14] result.

If 0 < θ(k − 1)

≤ p

for θ > 1, then s > ln θ(k − 1)

ln θ ≥ ln p

ln(p

/(k − 1)

) implies p

> (k − 1)

, and thus

(15) g(k, p

) ≤

 ln θ(k − 1)

ln θ



+ 1 for 0 < θ(k − 1)

≤ p

; θ > 1.

We define

S(b) = X

x∈F_pn

ψ(bx

),

where ψ(x) = e

denotes the additive canonical character. We denote by P

b

a summation in which b 6= 0 runs through a set of representatives, one from each of the k − 1 non-power classes and one from the kth power class.

Lemma 1.

X

b

|S(b)|

= k(k − 1)p

.

P r o o f. The deduction is the same as for Dodson’s Lemma 2.5.1. We

have X

b∈F_pn

|S(b)|

= X

x,y∈F_pn

X

b∈F_pn

ψ(b(x

− y

)) = p

M,

where M denotes the number of solutions of x

= y

in F

. Since M = 1 + (p

− 1)k and S(0) = p

we obtain

X

b∈F^∗_pn

|S(b)|

= (k − 1)p

(p

− 1).

The lemma follows since S(b) has the same value for each element of the same class.

Lemma 2. Suppose that x

+ . . . + x

does not represent every element of F

. Then there exist some c ∈ F

such that

|S(mc)| > p



1 − m

ln p

s



; m = 1, . . . , p − 1.

P r o o f. The proof is a direct extension of Dodson’s proof for Lemma 2.5.2. Verify that

N

(b) = p

X

x₁,...,x_s∈F_pn

X

t∈F_pn

ψ(t(x

+. . .+x

−b)) = p

X

t∈F_pn

S(t)

ψ(−tb) and suppose that there exists a b ∈ F

such that N

(b) = 0. Hence we get

X

t∈F^∗_pn

S(t)

ψ(−tb) = −p

.

It follows that there exists an element c ∈ F

such that

|S(c)|

≥ p

p

− 1 > p

, whence

|S(c)| > p

exp



− ln p

s



> p



1 − ln p

s

 , which is the result for m = 1.

For some real ϑ we have

|S(c)| = X

x∈Fpn

exp

 2πi

p (Tr(cx

) − ϑ)



and thus

X

x∈F_pn

cos

 2π

p (Tr(cx

) − ϑ)



> p



1 − ln p

s



,

whence

X

x∈Fpn

sin

 π

p (Tr(cx

) − ϑ)



< p

ln p

2s .

Since |sin mϕ| ≤ |m sin ϕ| and Tr(mx) = mTr(x) for m = 1, . . . , p − 1, we deduce that

X

x∈Fpn

sin

 π

p (Tr(mcx

) − mϑ)



< m

p

ln p

2s ,

whence

X

x∈F_pn

cos

 2 π

p (Tr(mcx

) − mϑ)



> p



1 − m

ln p

s

 , and thus

|S(mc)| > p



1 − m

ln p

s

 .

Lemma 3. Suppose that 2 is a kth power in F

and g(k, p

) exists. Then g(k, p

) < n

 ln p ln 2

 + 1

 .

P r o o f. If g(k, p

) exists, then there exists a basis {b

, . . . , b

} of kth powers. Let x = a

b

+ . . . + a

b

be any element of F

; 0 ≤ a

< p, i = 1, . . . , n. For i = 1, . . . , n we can express a

as

a

= a

+ a

2 + . . . + a

2

; a

∈ {0, 1}, j = 0, . . . , h

− 1, a

= 1.

Since 2

≤ a

< p, x is a sum of at most (h

+1)+. . .+(h

+1) < n 

ln 2

 +1
kth powers.

Lemma 4. If p

> k

, then g(k, p

) < b8 ln p

c + 1.

P r o o f. We suppose that for s = b8 ln p

c + 1 there exists an element b ∈ F

that is not of the form b = x

+ . . . + x

and obtain a contradiction.

By Lemma 2 there exists c ∈ F

such that

|S(c)| > p



1 − ln p

s



> 7

8 p

and |S(2c)| > p



1 − 4 ln p

s



> 1 2 p

. If 2 is not a kth power then c and 2c are representatives of two different classes in the sum of Lemma 1. Since k

< p

this gives

p

<

 7 8



p

+

 1 2



p

≤ k(k − 1)p

< p

.

Hence 2 must be a kth power and Lemma 3 implies that b is a sum of n 

ln 2

 + 1

≤ s kth powers.

Corollary 1. If p

/θ ≤ k

< p

for some θ > 1, then

g(k, p

) ≤ b8 ln θk

c + 1.

From Corollary 1 with θ = k

and (14) with s = 2 we get:

Theorem 1. g(k, p

) ≤ b32 ln kc + 1 for p

> k

. This generalizes [6], p. 151, (6).

4. A relation between g(k, p

) and g(d, p) Theorem 2. If g(k, p

) exists, then

g(k, p

) ≤ ng(d, p); d = k

k,

= p − 1

pⁿ−1

k

, p − 1
.

P r o o f. If g(k, p

) exists, then there exists a basis {b

, . . . , b

} of F

over F

consisting of kth powers.

The kth powers are exactly the

th roots of unity. Thus, the kth powers of elements of F

in F

are exactly the

, p − 1

th roots of unity which are the dth powers of elements of F

. Hence, all elements of F

are sums of g(d, p) kth powers of elements of F

, so that all elements of the form b

a; a ∈ F

, i = 1, . . . , n, are sums of g(d, p) kth powers. Thus an arbitrary element x = a

b

+ . . . + a

b

∈ F

; a

∈ F

, i = 1, . . . , n, is a sum of ng(d, p) kth powers.

5. Extension of the Chowla/Mann/Straus bound Theorem 3. If g(k, 2

) exists, then g(k, 2

) ≤ (k + 1)/2.

P r o o f. By Theorem 2 we have g(k, 2

) ≤ n, which implies the result for

(16) n ≤ (k + 1)/2.

Moreover, (14) with s = 2 implies the result for

(17) 2

> (k − 1)

.

Hence it is sufficient to consider 2 ≤ n ≤ 21. By (4), (16) and (17) we have 12 pairs (k, 2

) to investigate: g(3, 2

), g(7, 2

), g(5, 2

), g(7, 2

), g(11, 2

), g(9, 2

), g(13, 2

), g(15, 2

), g(21, 2

), g(17, 2

), g(27, 2

), and g(33, 2

).

For k ≥ 5 and 2

> (k − 1)

or k ≥ 7 and 2

> (k − 1)

we get the result by (14). Hence only g(3, 2

) and g(7, 2

) are undecided. It is well known that for p

6= 4 and 7 every element of F

is a sum of two cubes (see [13]), which implies g(3, 2

) = 2. As in the proof of Theorem 2 we get g(7, 2

) ≤ 3g(1, 2

), which completes the proof.

Remark. For small k it is shown in [4] that g(k, p

) ≤ bk/2c + 1 for k < min(p, (p

− 1)/2).

For arbitrary k but p 6= 2, [15], Theorem 1, implies g(k, p

) ≤ bk/2c + 1

for k < (p

− 1)/2.

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Institut f¨ ur Algebra und Zahlentheorie TU Braunschweig

Pockelsstr. 14

38106 Braunschweig, Germany E-mail: A.Winterhof@tu-bs.de

Received on 13.3.1998

and in revised form on 13.8.1998 (3351)