A FURTHER REFINEMENT OF MORDELL S BOUND ON EXPONENTIAL SUMS

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1 A FURTHER REFINEMENT OF MORDELL S BOUND ON EXPONENTIAL SUMS TODD COCHRANE, JEREMY COFFELT, AND CHRISTOPHER PINNER 1. Introuction For a prime p, integer Laurent polynomial (1.1) f(x) = a 1 x k a r x kr, p a i, k i Z, where the k i are istinct an nonzero mo (p 1), an multiplicative character χ mo p we consier the mixe exponential sum S(χ, f) := χ(x)e p (f(x)), x=1 where e p ( ) is the aitive character e p ( ) = e 2πi /p on the finite fiel Z p. For such sums the classical Weil boun [5] (see [1] or [4] for Laurent f) yiels, (1.2) S(χ, f) p 1 2, where is the egree of f for a polynomial (egree of the numerator when f has both positive an negative exponents), nontrivial only if < p. Morell [3] gave a ifferent type of boun which epene rather on the prouct of all the exponents k i. In [2] we obtaine the following improvement in Morell s boun (1.3) S(χ, f) 4 1 r (l1 l 2 l r ) 1 r 2 p 1 1 2r, where (1.4) l i = k i, if k i > 0, r k i, if k i < 0, non-trivial as long as (l 1 l r ) 1 4 r p 1 2 r. We show here that some of the larger l i can in fact be omitte from the prouct (at the cost of a worse epenence on p) once r 3: Theorem 1.1. For any f an χ as above an positive integer m with 1 2 r < m r, S(χ, f) 4 1 m (l1 l m ) 1 m 2 p 1 1 m 2 (m 1 2 r), Date: September 6, Mathematics Subject Classification. 11L07;11L03. Key wors an phrases. exponential sums. 1

2 2 TODD COCHRANE, JEREMY COFFELT, AND CHRISTOPHER PINNER where l i = k i, if k i > 0, m k i, if k i < 0. The theorem thus implies a nontrivial boun on S(χ, f) as long as (l 1 l 2 l m ) < 4 m p m r/2 for some 1 2r < m r. Inequality (1.3) is just the case m = r. One on the state can in fact save an extra factor of ((k 1,..., k r, p 1)/(k 1,..., k m )) 1 m 2 boun, as we explain in Section 3 below. Theorem 1.1 is particularly useful when more than half of the exponents are small; in particular (for fixe r) if at least R = 1 2 r +1 of the k i are boune, l i B say, then one obtains a uniform boun S(χ, f) (4B) 1 R p 1 δ with δ = 1/R 2 or 1/2R 2 as r is even or o, irrespective of the size of the remaining l i. Notice one cannot expect a boun of orer p 1 δ with some δ > 0 if only 1 2 r of the k i are boune as can be seen by the sums S(χ, f) = 1 2 p + O(r p) when 1 r (1.5) f = εa 0 x () + a i (x i x i+ 1 2 () ), i=1 χ(x) = χ 0 (x) or with ε = 0 or 1 as r is even or o. For monomials an binomials we gain nothing new, but for trinomials f = ax k 1 + bx k 2 + cx k 3, we obtain the m = 2 Theorem 1.1 boun (1.6) S(χ, f) (k 1 k 2 ) 1 4 p 7 8, ( ) x, p avoiing entirely the nee to involve the largest exponent, in contrast to the Weil boun an our previous Morell type boun (m = 3): S(χ, f) maxk 1, k 2, k 3 }p , S(χ, f) 9 (k 1k 2 k 3 ) p 6. The proof of the theorem is very similar to that of (1.3) an involves bouning the number of solutions (x 1,..., x m, y 1,..., y m ) in Z p 2m to the system of simultaneous equations (1.7) x k i xk i m y k i yk i m mo p for i = 1,..., r. We enote the number of such solutions by M m. For m r we can merely use the first m equations (iscaring the remaining r m) an appeal to the boun of Morell [3] or Lemma 3.1 in [2] to obtain: (1.8) M m 4 m (l 1 l m )(p 1) m. The theorem is then immeiate from (1.8) by taking v = w = m in the following Lemma relating S(χ, f) to M m :

3 EXPONENTIAL SUMS 3 Lemma 1.1. For any f an χ as above, an positive integers v, w, S(χ, f) (p 1) 1 1 v 1 w p r w (Mv M w ) 1 w. 2. Slight improvements in the boun for M m Although it seems wasteful to simply iscar the remaining (r m) equations in (1.7) there are certainly cases where these equations are reunant. For instance, if the first m exponents take the form k i = il, i = 1,..., m with l k i for the remaining k i then the x l i are merely a permutation of the yi l whatever those remaining exponents. Moreover when m = 2 our [2] boun for the first two equations M 2 k 1 k 2 (p 1) 2 if k 1 k 2 > 0, 3 k 1 k 2 (p 1) 2 if k 1 k 2 < 0, can be asymptotically sharp; for example for exponents k 1 = l, k 2 = 2l, with l k i, i = 3,..., r an l (p 1) or k 1 = l, k 2 = l or 3l an l k i, i = 3,..., r with the k i /l o an 2l (p 1), it is not har to see that M 2 = 2l 2 (p 1) 2 l 3 (p 1) M 2 = 3l 2 (p 1) 2 3l 3 (p 1), respectively. In certain cases though we can utilize the remaining equations for a slight saving: Lemma 2.1. If r 2 an then for m = 2 we have L ij = M 2 (k 1, k 2,..., k r, p 1) k i k j if k i k j > 0, 3 k i k j if k i k j < 0, min 1 i<j r L ij (k i, k j ) (p 1)2. Thus for example in the trinomial case (1.6) can be slightly refine to ( ) 1 (k1, k 2, k 3, p 1) 4 S(χ, f) (k1 k 2 ) p 8, (k 1, k 2 ) of use if k 1 an k 2 share a common factor not share with k 3. More generally a slight moification of the proof of Lemma 3.1 in [2] allows a similar saving of a factor (k 1, k 2,..., k r, p 1)/(k 1, k 2,..., k m ) on the previous boun (1.8): Lemma 2.2. If r 3, then for any 3 m r an choice of m exponents k 1,..., k m, M m 4e ( ) 2m (k1, k 2,..., k r, p 1) m 2 (l 1 l m )(p 1) m. m (k 1, k 2,..., k m )

4 4 TODD COCHRANE, JEREMY COFFELT, AND CHRISTOPHER PINNER 3. Proof of Lemma 1.1 For u = (u 1,..., u r ) Z r p an positive integer m, we efine an observe that (3.1) N m ( u) = # (x 1,..., x m ) Z p m : N m ( u) = (p 1) m, m i=1 x k j i } = u j, j = 1,..., r, Nm( u) 2 = M m. For any multiplicative character χ an positive integer m, the simple observation that u Z p e p (au) = p if a 0 (mo p) an zero otherwise, gives (3.2) 2m χ(x)e p (a 1 u 1 x k a r u r x kr ) x=1 = χ(x 1 x m y1 1 ym 1 ) r e p a j u j (x k j xk j m y k j 1 yk j x 1,...,xm, y 1,..,y m Z p = p r χ(x1 x m y 1 1 y 1 m ) p r M m, where enotes a sum over the x 1,..., x m, y 1,..., y m in Z p satisfying m m yk i j (mo p) for 1 i r. Writing S = S(χ, f), we have (p 1)S w = = = ( χ(mx)e p (a 1 (mx) k a r (mx) kr ) x=1 χ w (m) x 1,...,x w Z p x 1,...,x w Z p χ(x 1 x w ) ) w xk i j r χ(x 1 x w )e p a j m k j (x k j xk j r χ w (m)e p w ) a j m k j (x k j xk j an so (3.3) (p 1) S w r N w ( u) χ w (m)e p a j u j m k j. Applying Höler s inequality twice, the secon time splitting (3.4) N w ( u) = Nw ( u) 2 Nw ( u) 2, w ), m )

5 an using (3.1) an (3.2) gives (p 1) S w (3.5) ( u N w ( u) ( N w ( u) u EXPONENTIAL SUMS 5 ) ) 2 u χ w (m)e p (a 1 u 1 m k a r u r m kr ) ( Nw( u) 2 = ((p 1) w ) v 1 v (Mw ) 1 (Mv p r ) 1 u ) 1 (M v p r ) 1 = (p 1) w(1 1 v ) p r (Mv M w ) 1. 1 Hence S < (p 1) 1 1 v 1 w p r w (Mv M w ) 1 w. Write M 2 = C( u)2 where 4. Proof of Lemma 2.1 C(u 1, u 2,..., u r ) = #(x, y) Z 2 p : x k i y k i = u i for i = 1, 2,..., r} = #x Z p : y Z p satisfying x k i y k i = u i for i = 1, 2,..., r}, an = (k 1, k 2,..., k r, p 1) (since for each x with a solution y 0 there will be solutions y satisfying y (k 1,k 2,...,k r) = y (k 1,k 2,...,k r) 0 ). Note the trivial boun C( u) (p 1). If 0 < k 1 < k 2 an (u 1, u 2 ) (0, 0) then any x in the latter set must be a root of the non-zero polynomial f = (x k 1 u 1 ) k 2/(k 1,k 2 ) (x k 2 u 2 ) k 1/(k 1,k 2 ) which has egree at most k 1 (k 2 /(k 1, k 2 ) 1), an so C( u) k 1k 2 (k 1, k 2 ) k 1. On the other han, if k 1 < 0 < k 2 an (u 1, u 2 ) (0, 0) then x will be a root of the non-zero polynomial f = (x k 2 u 2 ) k 1 /(k 1,k 2 ) (1 u 1 x k 1 ) k 2/(k 1,k 2 ) x k 1 k 2 /(k 1,k 2 ) of egree at most 2 k 1 k 2 /(k 1, k 2 ), an so C( u) 2 (k 1, k 2 ) k 1 k 2.

6 6 TODD COCHRANE, JEREMY COFFELT, AND CHRISTOPHER PINNER Now for (u 1, u 2 ) = (0, 0), we will evaluate the sum C( u). Since x k 1 = y k 1 an x k 2 = y k 2 imply x (k 1,k 2 ) = y (k 1,k 2 ), we have } C( u) = # (x, y) Z 2 p : x (k 1,k 2 ) = y (k 1,k 2 ), x k l y k l = u l for l 1, 2 = # (x, y) Z 2 p : x (k 1,k 2 ) = y (k 1,k 2 ) } = (k 1, k 2, p 1)(p 1) Finally, since u Z r C( u) = (p 1)2, we have for 0 < k p 1 < k 2, M 2 = C( u) 2 + C( u) 2 ( ) k1 k 2 (k 1, k 2 ) k 1 C( u) + (p 1) C( u) ( k1 k 2 = (k 1, k 2 ) ( k 1 (k 1, k 2, p 1) ) ) ( ) (p 1) 2 k1 k 2 (k 1, k 2, p 1) (k 1, k 2 ) k 1 (p 1) < k 1k 2 (k 1, k 2 ) (p 1)2, an for k 1 < 0 < k 2, M 2 = C( u) 2 + C( u) 2 2 (k 1, k 2 ) k 1 k 2 C( u) + (p 1) C( u) ( ) = 2 (k 1, k 2 ) k 1 k 2 + (k 1, k 2, p 1) (p 1) 2 2 (k 1, k 2 ) (k 1, k 2, p 1) k 1 k 2 (p 1) < 3 (k 1, k 2 ) k 1 k 2 (p 1) 2. Since the proof hols when the k i s are interchange, we have the esire result. 5. Proof of Lemma 2.2 The proof is almost ientical to that of Lemma 3.1 in [2]. Simply ignore the (r m) remaining equations for all of the proof except for the instance where Wooley s result [6] was applie to boun the number of solutions to u k j 1 + uk j uk j t = α j for j = 1,..., t, for some 1 t m with D t ( u) 0. Instea of bouning the number of solutions to the above system, boun instea the number of solutions to X k j/ 1 + X k j/ X k j/ t = α j for j = 1,..., t

7 EXPONENTIAL SUMS 7 where = (k 1, k 2,..., k m ) an X i = u i. By the previously mentione result of Wooley, we know that the number of solutions to the secon system is no more than (k 1 /)(k 2 /) (k t /). However, for a given value of X i there are at most (, p 1) values for u i such that u i = X i. After fixing values for all but one of the u i, say u 1, the values u k 1 1,..., ukr 1 are all etermine, so that the number of choices for u 1 is at most (k 1, k 2,..., k r, p 1). This gives no more than (k 1, k 2,..., k r, )(, ) t 1 (k 1 /)(k 2 /) (k t /) (k 1, k 2,..., k r, p 1) k 1 k 2 k t solutions, improving on the previous boun of k 1 k 2 k t (given by the irect application of Wooley s result on only the first t equations) by the esire factor. References [1] F. N. Castro & C. J. Moreno, Mixe exponential sums over finite fiels, Proc. Amer. Math. Soc. 128, no. 9 (2000), [2] T. Cochrane & C. Pinner, An improve Morell type boun for exponential sums, Proc. Amer. Math. Soc. to appear. [3] L. J. Morell, On a sum analagous to a Gauss s sum, Quart. J. Math. 3 (1932), [4] G. I. Perel muter, Estimate of a sum along an algebraic curve, Mat. Zametki 5 (1969), [5] A. Weil, On some exponential sums, Proc. Nat. Aca. Sci. U.S.A. 34 (1948), [6] T. Wooley, A note on simultaneous congruences, J. Number Theory 58 (1996), no. 2, Department of Mathematics, Kansas State University, Manhattan, KS aress: cochrane@math.ksu.eu Department of Mathematics, Kansas State University, Manhattan, KS aress: jcoffelt@math.ksu.eu Department of Mathematics, Kansas State University, Manhattan, KS aress: pinner@math.ksu.eu