On Waring s problem in finite fields
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1 ACTA ARITHMETICA LXXXVII.2 (1998 On Waring problem in finite field by Arne Winterhof (Braunchweig 1. Introduction. Let g( p n be the mallet uch that every element of F p n i a um of th power in F p n. In Section 2 we ummarize the baic reult on g( p n. In Section 3 we generalize Dodon upper bound for mall ([5] Lemma 2.5.4: g( p < 8 ln p + 1; p 1 p/2 < 2 < p and deduce g( p n 32 ln + 1 for p n > 2. The object of Section 4 i to invetigate to what extent Waring problem for F p n can be reduced to the problem for F p. It i proven that if g( p n exit then g( p n ng(d p; d = ( (p n 1/(p 1 pn 1. It i well nown (ee [3] that g( p /2 + 1; < (p 1/2. [15] Theorem 1 implie that if g( p n exit and p i odd then g( p n /2 + 1 for < (p n 1/2. Whether p ha to be odd ha not been nown yet. In Section 5 we how that p need not be odd. 2. Baic reult on g( p n. Every ( p n 1th power i at the ame time a th power. Hence (1 g( p n = g(( p n 1 p n. It i ufficient to retrict ourelve to the cae (2 p n 1. Remember that the multiplicative group F p n (3 g( p n = 1 = Mathematic Subject Claification: 11P05 11T99. i cyclic. Hence [171]
2 172 A. Winterhof Since L := {x x x 1... x F p n N} i a field ([16] Lemma 1 g( p n exit if and only if L i not a proper ubfield of F p n and thu (4 g( p n exit if and only if pn 1 p d for all n d n. 1 Thi reult i eentially that of [1] Theorem G. We hall uppoe that from now on g( p n exit. Let A i = {z zi z 1... z i F p n}. If A i A i+1 then y A i+1 \A i implie xy A i+1 \A i for each 0 x A 1 o that A i+1 A i + A 1 1 = A i + pn 1. Hence in the chain A 1 A 2... A = F p n there are at mot 1 trict incluion and therefore (5 g( p n which i a pecialization of [10] Théorème Equality hold for the following example: ( p 1 g(1 p n = 1 g(2 p n = 2 g 2 p = p 1 g(p 1 p = p 1. 2 Since A ( p n we get a trivial lower bound for g( p n : (6 g( p n ln p n ln ( p n (7 (8 (9 (10 (11 (12 (13 For n = 1 the following reult are well nown: g( p max(3 32 ln + 1; p > 2 [6] g( p 68(ln 2 1/2 ; p > [7] g( p /2 + 1; p > [3] g( p ( (1 + 2 log p 1 2 p ; p > 3/2 [2] 7/3 g( p 170 (p 1 ln p; p 4/3 7/4 + 1 [8] g( p c ε (ln 2+ε ; 2 p g( p c ε ; < p 2/3 ε ε > 0 [9]. ln ε > 0 [11] (ln(ln ε 3. Extenion of Dodon bound for mall. Now we conider the cae 0 < ( 1 2 < p n. In thi cae g( p n exit.
3 Waring problem in finite field 173 The number N (b; b F pn of olution of the equation x x = b; x 1... x F p n can be expreed in term of Jacobi um ([12] Theorem 6.34 N (b = p n( j 1...j =1 λ j j (bj(λ j 1... λ j where λ i a multiplicative character of F p n of order. Uing the fact that { J(λ j 1... λ j p n( 1/2 if λ = j j i non-trivial p n( 2/2 if λ j j i trivial ([12] Theorem 5.22 we obtain and in particular Hence N (b p n( 1 ( 1 p n( 1/2 N (b p n( 1 ( 1 p n( 1/2. (14 g( p n for p n( 1 > ( 1 2. For = 2 thi i Small [14] reult. If 0 < θ( 1 2 p n for θ > 1 then > ln θ( 12 ln θ ln p n ln(p n /( 1 2 implie p n( 1 > ( 1 2 and thu ln θ( 1 (15 g( p n for 0 < θ( 1 2 p n ; θ > 1. ln θ We define S(b = ψ(bx where ψ(x = e 2πi p Tr(x denote the additive canonical character. We denote by b a ummation in which b 0 run through a et of repreentative one from each of the 1 non-power clae and one from the th power cla. Lemma 1. b S(b 2 = ( 1p n.
4 174 A. Winterhof P r o o f. The deduction i the ame a for Dodon Lemma We have S(b 2 = ψ(b(x y = p n M b F p n xy F p n b F p n where M denote the number of olution of x = y in F p n. Since M = 1 + (p n 1 and S(0 = p n we obtain b F p n S(b 2 = ( 1p n (p n 1. The lemma follow ince S(b ha the ame value for each element of the ame cla. Lemma 2. Suppoe that x x doe not repreent every element of F p n. Then there exit ome c F p uch that n ( S(mc > p n 2 ln pn 1 m ; m = 1... p 1. P r o o f. The proof i a direct extenion of Dodon proof for Lemma Verify that N (b = p n ψ(t(x x b = p n S(t ψ( tb x 1...x F p n t F p n t F p n and uppoe that there exit a b F p n uch that N (b = 0. Hence we get S(t ψ( tb = p n. t F p n It follow that there exit an element c F p n whence S(c S(c > p n exp ( which i the reult for m = 1. For ome real ϑ we have S(c = and thu uch that pn p n 1 > pn( 1 ln pn > p (1 n ln pn ( 2πi exp p (Tr(cx ϑ ( 2π co p (Tr(cx ϑ > p (1 n ln pn
5 whence Waring problem in finite field 175 ( π in 2 p (Tr(cx ϑ < pn ln p n. 2 Since in mϕ m in ϕ and Tr(mx = mtr(x for m = 1... p 1 we deduce that ( π in 2 p (Tr(mcx mϑ < m2 p n ln p n 2 whence and thu ( co 2 π p (Tr(mcx mϑ > p n (1 m2 ln p n ( S(mc > p n 1 m2 ln p n. Lemma 3. Suppoe that 2 i a th power in F p n and g( p n exit. Then ( ln p g( p n < n + 1. ln 2 P r o o f. If g( p n exit then there exit a bai {b 1... b n } of th power. Let x = a 1 b a n b n be any element of F p n; 0 a i < p i = 1... n. For i = 1... n we can expre a i a a i = a i0 + a i a ihi 2 h i ; a ij {0 1} j = 0... h i 1 a ihi = 1. Since 2 h i a i < p x i a um of at mot (h (h n +1 < n ( ln p ln 2 +1 th power. Lemma 4. If p n > 2 then g( p n < 8 ln p n + 1. P r o o f. We uppoe that for = 8 ln p n + 1 there exit an element b F p n that i not of the form b = x x and obtain a contradiction. By Lemma 2 there exit c F p uch that ( n S(c > p n ln pn 1 > 78 pn and S(2c > p (1 n 4 ln pn > 1 2 pn. If 2 i not a th power then c and 2c are repreentative of two different clae in the um of Lemma 1. Since 2 < p n thi give ( 2 ( p 2n < p 2n + p 2n ( 1p n < p 2n. 8 2 Hence 2 mut be a th power and Lemma 3 implie that b i a um of n ( ln p ln th power. Corollary 1. If p n /θ 2 < p n for ome θ > 1 then g( p n 8 ln θ
6 176 A. Winterhof From Corollary 1 with θ = 2 and (14 with = 2 we get: Theorem 1. g( p n 32 ln + 1 for p n > 2. Thi generalize [6] p. 151 (6. 4. A relation between g( p n and g(d p Theorem 2. If g( p n exit then g( p n ng(d p; d = ( p n 1 = p 1 p 1 ( p n 1 p 1. P r o o f. If g( p n exit then there exit a bai {b 1... b n } of F p n over F p coniting of th power. The th power are exactly the pn 1 th root of unity. Thu the th power of element of F p in n F p are exactly the ( p n 1 p 1 th root of unity which are the dth power of element of F p. Hence all element of F p are um of g(d p th power of element of F p n o that all element of the form b i a; a F p i = 1... n are um of g(d p th power. Thu an arbitrary element x = a 1 b a n b n F p n; a i F p i = 1... n i a um of ng(d p th power. 5. Extenion of the Chowla/Mann/Strau bound Theorem 3. If g( 2 n exit then g( 2 n ( + 1/2. P r o o f. By Theorem 2 we have g( 2 n n which implie the reult for (16 n ( + 1/2. Moreover (14 with = 2 implie the reult for (17 2 n > ( 1 4. Hence it i ufficient to conider 2 n 21. By (4 (16 and (17 we have 12 pair ( 2 n to invetigate: g(3 2 4 g(7 2 6 g(5 2 8 g(7 2 9 g( g( g( g( g( g( g( and g( For 5 and 2 n > ( 1 3 or 7 and 2 3n > ( 1 8 we get the reult by (14. Hence only g(3 2 4 and g(7 2 6 are undecided. It i well nown that for p n 4 and 7 every element of F p n i a um of two cube (ee [13] which implie g(3 2 4 = 2. A in the proof of Theorem 2 we get g( g(1 2 2 which complete the proof. Remar. For mall it i hown in [4] that g( p n /2 + 1 for < min(p (p n 1/2. For arbitrary but p 2 [15] Theorem 1 implie g( p n /2 + 1 for < (p n 1/2.
7 Waring problem in finite field 177 Reference [1] M. Bhaaran Sum of mth power in algebraic and abelian number field Arch. Math. (Bael 17 ( ; Correction ibid. 22 ( [2] J. D. Bovey A new upper bound for Waring problem mod p Acta Arith. 32 ( [3] S. Chowla H. B. Mann and E. G. Strau Some application of the Cauchy Davenport theorem Nore Vid. Sel. Forh. Trondheim 32 ( [4] G. T. Diderrich and H. B. Mann Repreentation by -th power in GF (q J. Number Theory 4 ( [5] M. M. Dodon Homogeneou additive congruence Philo. Tran. Roy. Soc. London Ser. A 261 ( [6] On Waring problem in GF[p] Acta Arith. 19 ( [7] M. M. Dodon and A. Tietäväinen A note on Waring problem in GF[p] ibid. 30 ( [8] A. Garcia and J. F. Voloch Fermat curve over finite field J. Number Theory 30 ( [9] D. R. Heath-Brown and S. Konyagin New bound for Gau um derived from th power and for Heilbronn exponential um ubmitted to Quart. J. Math. Oxford. [10] J. R. J o l y Somme de puiance d-ième dan un anneau commutatif Acta Arith. 17 ( [11] S. V. K o n y a g i n On etimate of Gauian um and Waring problem for a prime modulu Trudy Mat. Int. Stelov. 198 ( (in Ruian; Englih tranl.: Proc. Stelov Int. Math no [12] R. Lidl and H. Niederreiter Finite Field Encyclopedia Math. Appl. 20 Addion-Weley [13] S. Singh Analyi of each integer a um of two cube in a finite integral domain Indian J. Pure Appl. Math. 6 ( [14] C. Small Sum of power in large finite field Proc. Amer. Math. Soc. 65 ( [15] A. Tietäväinen On diagonal form over finite field Ann. Univ. Turu Ser. A I 118 ( pp. [16] L. Tornheim Sum of n-th power in field of prime characteritic Due Math. J. 4 ( Intitut für Algebra und Zahlentheorie TU Braunchweig Poceltr Braunchweig Germany A.Winterhof@tu-b.de Received on and in revied form on (3351
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