SPARSE POLYNOMIAL EXPONENTIAL SUMS

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1 SPARSE POLNOMIAL EPONENTIAL SUMS TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE. Introduction In this paper we estimate the complete exponential sum (.) S(f; q) = q x= e q(f(x)); where e q( ) is the additive character e q( ) = e 2i =q, and f is a sparse integer polynomial, (.2) f(x) = a x k + + a rx kr with 0 < k < k 2 < < k r. We assume always that the content of f, (a ; a 2; : : : ; a r), is relatively prime to the modulus q. Let d = d(f) = k r denote the degree of f and for any prime p let d p(f) denote the degree of f read modulo p. A fundamental problem is to determine whether there exists an absolute constant C such that for an arbitrary positive integer q, (.3) js(f; q)j C q d ; if f is not a constant function modulo p for each prime pjq. It is well known that the exponent d is best possible. For the case of Gauss sums (r = ) Shparlinski [29], [30] showed that one may take C = +O(d =4+ ) and this was sharpened to C = +O(d + ) in his subsequent work with Konyagin [5, Theorem 6.7]. The best upper bounds available for general f are due to Steckin [33], and js(f; q)j e d+o( log d ) q d ; js(f; q)j e :74d q d ; due to Qi and Ding [27]; see also Chen [2], [3], Lu [9], [20], [2], Necaev [22], [23], Qi and Ding [25], [26] and Zhang and Hong [35]. These authors noted that in order to make any further improvement one must rst obtain a nontrivial upper bound on the prime modulus exponential sum js(f; p)j for p < (d ) 2, the interval where Weil's bound [34] js(f; p)j (d ) p p is worse than the trivial bound. In [5] we obtained a bound of this type in terms of the number of terms r of f(x). Using this bound we establish here Theorem.. For any positive integer r there exists a constant C(r) such that for any polynomial f of type (.2) and positive integer q relatively prime to the content of f, js(f; q)j C(r) q d : Although our proof yields C(r) e O(r4), no attempt was made to obtain the best possible value for C(r). For prime power moduli one can replace C(r) with an absolute constant as shown by Steckin [33] and Cochrane and Zheng [8], the latter result being Date: October 9, Mathematics Subject Classication. L07;L03. Key words and phrases. exponential sums. d

2 2 TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE Lemma.. [8, Theorem.] Let f be a polynomial over Z of degree d and p a prime with d p(f). Then for any m, (.4) js(f; p m )j 4:4 d ) : It is also well known (see [22], [3] or [8]) that for p (d ) 2d=(d 2) and m, (.5) js(f; p m )j d ) : The signicance of the constant one in (.5) lies in the fact that bounds Q for exponential k sums modulo prime powers lead to bounds for a general modulus q = i= pe i i via the multiplicative formula (.6) S(f; q) = k i= S( if; p e i i ); P k where the i are such that i= iq=pe i i =. Thus if (.5) holds for all prime power divisors of q then it follows that js(f; q)j q d. It is desirable to extend the inequality in (.5) to an interval of the type p > Cd for some constant C. In closing we note that for sums over reduced residue systems, (.7) S (f; q) = q x=; (x;q)= e q(f(x)); the exponent in the upper bound can be dramatically reduced. Shparlinski [3] showed that js (f; q)j C(d; ) q r + ; for any sparse polynomial in r terms with content relatively prime to q. Loh [7] obtained a related upper bound but an error in his Lemma 3 leaves his results in doubt. 2. The method of recursion A standard method for bounding exponential sums modulo prime powers is the method of recursion, also known as the method of critical points. For any polynomial f let t = t p(f) = ord p(f 0 ) be the largest power of p dividing all of the coecients of f 0, d = d p(p t f 0 ), and let A = A(f; p) be the set of zeros of the congruence p t f 0 (x) 0 (mod p). A is called the set of critical points associated with the sum S(f; p m ), for any m 2. Write with S(f; p m ) = S (f; p m ) = p =0 x (mod p) S (f; p m ) e p m(f(x)): A fact of central importance is that if m is suciently large then S (f; p m ) = 0 unless is a critical point, Lemma 2.. [6, Proposition 4.] Suppose that p is an odd prime and m t + 2, or p = 2 and m t+3, or p = 2, t = 0 and m = 2. Then if is not a critical point, S (f; p m ) = 0. Consequently, For any 2 A dene S(f; p m ) = 2A S (f; p m ): (2.) = := ord p(f(px + ) f()); g (x) := p (f(px + ) f()):

3 EPONENTIAL SUMS 3 Lemma 2.2. [6, Proposition 4.] The Recursion Relationship. Suppose that p is an odd prime and m t + 2, or p = 2 and m t + 3, or p = 2, t = 0 and m = 2. Then if 2 A (2.2) S (f; p m ) = e p m(f())p S(g ; p m ); where (2.3) S(g ; p m ) = Under the hypotheses of the lemma we have (2.4) js(f; p m )j 2A (P p m x= e p m (g (x)) if m > ; p m if m : js (f; p m )j = 2A p js(g ; p m )j: In particular, since there are at most d critical points we have immediately the upper bound (2.5) js(f; p m )j d p m : In [8] we established the following bounds for S (f; p m ) and S(f; p m ), Lemma 2.3. [8, Theorem 2.] Let f be a polynomial over Z and p a prime with d p(f). Suppose that p is odd and m t + 2 or p = 2 and m t + 3. Set = (5=4) 5 3:05 and d = d p(p t f 0 ). Then (i) For any critical point of multiplicity we have (2.6) js (f; p m )j minf; g p t + + ) ; with equality if =. (ii) js(f; p m )j p For any critical point set d t + d (2.7) := ord p(g 0 (x)); g (x) := p g 0 (x): The following relations are well known (see eg. [6, Lemma 3.]) and play a central role in the proof of the preceding lemma. Lemma 2.4. (2.8) ( t + 2 if p is odd or > t + if p = 2 and =. (2.9) + + t : ( t + ordp(d p(g )) + + ordp(d p(g )) (2.0) d p(g ) + + t : (2.) d p(g ) + t : (2.2) p jd p(g ): An immediate consequence that we frequently make reference to is Lemma 2.5. Suppose that is a critical point of multiplicity with 2 and p > + 2. Then d p(g ) +. Proof. Let d p = d p(g ). Suppose that ord p(d p). If d p = p then by (2.0) we have p = d p + 2 contradicting our assumption. Otherwise d p 2p and we have p d p=2 d p ord p(d p) +, again a contradiction. Thus p - d p and we obtain (by (2.0)) d p +.

4 4 TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE 3. Preliminary Upper Bounds We begin with a couple of auxiliary lemmas. Lemma 3.. Dene i = i for i = ; 2; 3 and i = for i 4, where = (5=4) 5 3:05. Then for i d we have d i i d i+ i: Proof. For any xed i the function f i(x) := i i x i x i+ attains its maximum value at x = (i + )= log() < i +, and is decreasing for larger values of x. Thus for d i, the maximum value of f i(d) occurs at d = i or d = i +. Now, f i(i) = i and f i(i + ) = i( + i ) i+, as can be seen by considering the dierent cases i = ; 2; 3 and i 4. Lemma 3.2. If p > cd for some constant c then for i d we have i d 4p i+ i : cd d Proof. We rst note that i+ d d i 4; for i d. i This can be checked directly for i = ; 2; 3. For i 4 it follows from Lemma 3.. Then p > cd c i+ d(d=i) d i and the result follows. 4 Lemma 3.3. Let p be a prime and f be any integer polynomial with t = 0 and either d = 0; or p > d 2+4=(d ) where d = d p(p t f 0 ). Then for m 2 (3.) js(f; p m )j d Proof. If d = 0 then there are no critical points and the sum is zero. If d = then there is a single critical point of multiplicity one and the result follows from Lemma 2.3(i). Suppose that d 2. Let A = A(f; p) F p be the set of critical points. We prove by induction on m that under the hypotheses of the theorem (3.2) js j + ) ; for any critical point 2 A. We rst note that (3.) is an immediate consequence of (3.2). Indeed, if p m P (p=d ) d+ then using the trivial upper bound js (f; p m )j p m we have js(f; p m )j 2A js(f; pm )j d p m d + ). Next, if there is a critical point of multiplicity d then it is the only critical point and we have js(f; p m )j = js (f; p m )j d + ). Finally, suppose that p m > (p=d ) d+ and that every critical point is of multiplicity less than d. Letting n i denote the number of critical points of multiplicity i we obtain from (3.2) (3.3) js(f; p m )j 2A js (f; p m )j d i= n i i+ ) = d + ) d Then from p m > (p=d ) d +, p > 4d and Lemma 3.2 with c = 4 we obtain js(f; p m )j d + ) d i= n i(p=d ) i d i+! i=! n m(i d ) ip (i+)(d +) : d ( n ii=d ) d + ) d i=

5 EPONENTIAL SUMS 5 We proceed now to establish (3.2). If = then by Lemma 2.3 we have equality in (3.2). So we may assume that 2. When m = 2 the bound is trivial, js j p p 2( + ). Suppose m 3. If m then the result follows trivially, js j p m + ) p + + ) ; the latter inequality following from (2.9). Suppose next that = m. Put d p = d p(g ). Since p > d it follows from Lemma 2.5 that d p + d +. If d p 3 then p (d p ) 2+4=(dp 2), so by the Weil bound, js(g ; p)j (d p ) p p p dp p +. If d p = or 2 the same bound is elementary. It follows from the recursion formula of Lemma 2.2 that js j = p js(g ; p)j p m + = p + + ) Suppose nally that m + 2. We note that = 0 since by (2.2), p d p(g ) + d + < p, and so we can apply the induction assumption to S(g ; p m ). Putting d 2 = d p(g) 0 d and noting that either d 2 = 0; or p d 2+ d we obtain js j = p js(g ; p m )j p p (m )( d 2 + ) p p (m )( + ) 4. Multiplicity Estimates Next, we obtain an upper bound on the multiplicity of a nonzero zero of a sparse polynomial f(x) = a x k + + a rx kr (mod p): Let a 6 0 (mod p) be a zero of multiplicity mod p, that is, For i r let and set (4.) (4.2) (x a) jjf(x) (mod p): S(i; ) = fk j : k j k i (mod p )g; i = maxf : js(i; )j 2g; r i = js(i; i)j: Lemma 4.. The multiplicity of any nonzero zero of f(x) (mod p) satises < min i r ip i : In particular, if p does not divide any k i k j with i 6= j then < r. Lemma 4. follows from the more precise Lemma 4.2. Suppose that k ; :::; k t are the smallest distinct exponents mod p so that where f(x) = x k f (x) p + + x kt f t(x) p f i(x) = k j =k i +l j p a jx lj : (mod p); Then if f(x) has a nonzero zero a of multiplicity mod p, we have = kp + u where u < t and (x a) k is the highest power dividing all the f ; :::; f t.

6 6 TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE Proof. Suppose that (x a) k jf ; :::; f t with (x a) k+ - f, and write f i(x) = (x a) k g i(x) mod p, = kp + u, so that (x a) u jjg(x) = x k g (x) p + + x kt g t(x) p : Writing r = x d dx we must have ri g(a) 0 mod p for j = 0; :::; u. That is k j a k g (a) p + + k j t a kt g t(a) p 0 (mod p) Q for j = 0; :::; u. Since j det(ki j ) j=0;:::;t i=;:::;t j = jk i k jj 6 0 (mod p) and a k g (a) p 6 0 (mod p) we must therefore have u < t. Proof of Lemma 4.. Pick an arbitrary k i, i = ; :::; t, and use the preceding lemma and induction on i: If i = 0 then plainly k = 0 and = u < t r = r i. If i then since (x a) k jf i(x) we have (by induction) k < r ip i and u < p giving = pk + u (r ip i )p + (p ) < r ip i : In practice we apply the multiplicity estimate to the polynomial p t f 0 (x) and so we let r = r (f; p) be the number of nonzero terms mod p of the polynomial p t f 0 (x). For critical points having multiplicity less than r we have the following upper bound. Lemma 4.3. Let f be a sparse polynomial as in (.2) and suppose that either r = ; 2 or that p > (r ) 2r =(r 2). Then if m t + 2 and is a critical point of multiplicity < r we have (4.3) js j p t + Proof. If = the result follows from Lemma 2.3, and so we may assume 2 and r 3. Let d p = d p(g ). Since p > 2+4=( ), d p + by Lemma 2.5, and thus p > (d p ) 2dp=(dp 2). Also, since p d p(g ) + r + < p we must have = 0. If m the result follows trivially, Suppose next that = m js j p m = p m + + ) p t +. Then applying the bound in (.5) to S(g ; p) we obtain, js j = p js(g ; p)j p p + = p + + ) p t + Finally, if m 2 then we can apply Lemma 3.3 to S(g ; p m ), since d 2 := d p(g 0 ) < r and so p d 2+4=(d 2 ) 2. We obtain js j p js(g ; p m )j p p (m )( d 2 + ) = p + 5. Bounds for exponential sums with p small relative to d First we consider sums modulo p. From the bound of Weil, one deduces (see [8, Lemma 3.]) the upper bound (5.) js(f; p)j :75 p d ; for any polynomial f with d p(f). Moreover the constant.75 may be replaced by provided p d 2. For our purposes here we need constant for p d. We obtain this from the following result established in the author's work [5, Corollary.]. Lemma 5.. Let f be an integer polynomial of degree d as in (.2). Then for any > 0 9 if p > d and p > C(), :06 then p (5.2) e p(f(x)) rp : x= p

7 EPONENTIAL SUMS 7 Lemma 5.2. Let f be a polynomial as in (.2) of degree d = d p(f) suppose that p > C 2, (an absolute constant), p > 50d and p > r 4. Then (mod p) and js(f; p)j p d : Proof. The result is elementary for d = ; 2 and so we assume d > 2. If p > 6d 2 then the result follows from the Weil bound js(f; p)j (d ) p p. Suppose that p 6d 2. Applying Lemma 5. with = we obtain that if p > 50d and p > C( ) then js(f; p)j 5 5 p( =rp =5 ). Since, p > r 4 it follows that js(f; p)j p( =p 9=20 ), and since p 6d 2 the latter is p d for p > Lemma 5.3. Let f be a sparse polynomial as in (.2) with p 50(d + ), p > C 2 (the constant in Lemma 5.2), p > r 4 and m t + 2. Then for any critical point of multiplicity we have (5.3) js j p t + + ) ; and (5.4) js(f; p m )j p t d + d Proof. We rst observe that (5.4) is always an immediate consequence of (5.3). Indeed, if p m t (p=d ) d+ then using the trivial upper bound js (f; p m )j p m we have d t + P js(f; p m )j 2A js(f; pm )j d p m p d + ). Next, if there is a critical point of multiplicity d then it is the only critical point and we have js(f; p m )j = d t + d + ). js (f; p m )j p Finally, suppose that p m t > (p=d ) d+ and that every critical point is of multiplicity less than d. Letting n i denote the number of critical points of multiplicity i we obtain from (5.3) js(f; p m )j p d i= n ip t i+ i+ ) p t d + d + ) i d t + d + ) i n ip (m t)(i d ) (i+)(d +i) n i(p=d ) i d i+ p t d + d + ) ; P the last inequality following from Lemma 3.2 (with c = 4) and i nii d. We now establish (5.3) by induction on m. The result is trivial if m = 2. Suppose that m > 2. If m then from (2.9), js j p m p t + If = m and 6= 0 then since p > d it follows from Lemma 4. that < r. Also, since p 50d 50 we have by Lemma 2.5 that d p(g) + r < p =4 ; and so by (.5), js(g ; p)j p dp(g) p +. It then follows from the recursion relation that (5.5) js j p js(g ; p)j p + p t + + ) ; by (2.9). If = 0 then we have to argue dierently since the multiplicity may be larger than r. In this case g (x) = f(px) is a sparse polynomial with the same number of terms as f. Since p > 50(d + ) 50( + ) 50d p(g ) we can apply Lemma 5.2 to obtain js(g ; p)j p dp(g), and the result follows as before.

8 8 TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE Suppose now that m 2. We rst note that by (2.2), = 0 since p > d p(g ). Set d 2 = d p(g 0 ). If 6= 0 then we have by (2.) and Lemma 4., d 2 < r < p =4. Thus by Lemma 3.3 js(g ; p m )j p (m )( d 2 If = 0 then we can apply the induction assumption to the polynomial g = p f(px) and obtain the same bound. From the recursion relationship we then obtain js(f; p m )j p p (m )( d 2 + ) p ) p t + Next we obtain a bound valid for even smaller values of p. Again, let d and r = r (f; p) be the degree and number of nonzero terms of the polynomial p t f 0 (x) read mod p. Lemma 5.4. Let f be a sparse polynomial in r terms and p a prime with p > r 4, p > C 3 and such that p - (k j k i) for all k i < k j d. Then for m t + 2 and any critical point of multiplicity we have (i) If 6= 0 then js (f; p m )j p t + (ii) (iii) For = 0, js 0(f; p m )j p 2r+t + + ). js(f; p m )j p 2r+t d + d Proof. We take C 3 = maxfc 2; 200g where C 2 is the constant in Lemma 5.3. The condition p - (k i k j) implies (by Lemma 4.) that < r for any nonzero critical point. So (i) is implied by Lemma 4.3. If p 50(d + ) then the lemma is implied by Lemma 5.3 and so we may assume p < 50(d + ). In particular, it follows that r d (if r 4 then r 4 < p < 50(d + ) implies r < r r3 < d + ; if r 3 then p > 200 implies d > 3 r). 50 The proof of (ii) is by induction on m, but rst we show that (i) and (ii) together imply (iii). If zero is the only critical point then (ii) immediately implies (iii) and so we assume henceforth that r 2 and that (0) < d. If m t 2r then the upper bound in (iii) follows from the trivial bound js(f; p m )j p m. Next write m t = 2r + + j with j 0 and set the desired bound. We have = p t+2r d + d + ) ; js(f; p m )j js 0(f; p m )j + 6=0 For the rst term we have the trivial bound js (f; p m )j: (5.6) js 0(f; p m )j p m = p j d d + : Now there are at most p nonzero critical points, each of multiplicity r r and so by (i) (5.7) 6=0 js (f; p m )j p p t=r r ) = p j(r d ) rd d (d +)r = p j d d + +j r : Combining (5.6) and (5.7) we have for j d =2, and for d > j > d =2, d js(f; p m )j p 2(d +) ( + p =r ) < 2 p =4 < ; js(f; p m )j (p d =2r + )p =(d +) (r 2d =r + )r 4=(d +) < :

9 EPONENTIAL SUMS 9 If j d then by the bound in (ii) (replacing with d ) we obtain j js 0(f; p m )j p d (d +) p =d : For the remaining critical points we use the upper bound of (5.7) replacing j with d. Thus js(f; p m )j p d + p d r (r 4=d + r 4(d+)=r ) < : We return to the task of proving (ii) by induction on m. The bound follows trivially from js 0(f; p m )j p m if m + + t + 2r, and so we assume m t + 2r. By (2.9) we have m t + 2r ( + + t ) = + 2r + + 2; and by the recursion formula of Lemma 2.2, js 0(f; p m )j = p js(g 0; p m )j, where g 0(x) = p f(px). Since g 0 has the same degree monomials as f we can apply the induction assumption to g 0 and obtain, js 0(f; p m )j p p +2r d 2 + p (m )( d 2 + ) ; where d 2 := d p(p g 0) d. Now by (2.) d 2 and so replacing d 2 by in the previous inequality and using the upper bound in (2.9) we deduce the inequality in (ii). 6. Dealing with the primes that divide k i k j for some i 6= j. If pj(k j k i) for some k i < k j d then there may be nonzero critical points of multiplicity exceeding r and so we have to argue more carefully. Let f(x) be a sparse polynomial as in (.2) of degree d and set d = d p(p t f 0 (x)). For any pair (i; j) with i < j r let p ij be the maximal prime divisor of k j k i (taking p ij = in case k j k i = ) and put (6.) P = fp ij : i < j rg: Assume now that p > 4r, pj(k j k i) for some k i < k j d but that p =2 P. Let and dene p`s = minfp ij : pj(k j k i); k i < k j d g; M := rd =p`s : Then if p e k(k j k i) is the maximum power of p dividing any of the dierences k j k i that actually occur in the critical point congruence for S(f; p m ), it follows from Lemma 4. that the multiplicity of any nonzero critical point satises (6.2) < rp e r(k j k i)=p ij M: Let S (f; p m ) denote the sum over a reduced residue system (mod p m ) as in (.7). For j 0 dene j, t j by (6.3) p j k(a p jk ; : : : ; a rp jkr ); p j +t j k(a k p jk ; : : : ; a rk rp jkr ): Then we can write (6.4) S(f; p m ) = where for 0 j m m j=0 S (f(p j x); p m j ) = m j=0 (6.5) S j = S (p j f(p j x); p m j ): p j j S j ; The critical point congruence associated with the sum S j is just g j(x) := p j t j (a k p jk x k + + a rk rp jkr x kr ) 0 (mod p); Viewing g j(x) as a polynomial over F p we observe that for any j < m the largest degree term of g j+(x) is at most the smallest degree term of g j(x). Indeed, if p t j + j ka Ik Ip jk I

10 0 TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE then p t j + j +k I ka Ik Ip (j+)k I and p t j + j +k I + ka`k`p (j+)k` for ` > I. It follows that the degrees of the g j are nonincreasing (with j) and that at most r of the g j(x) can have more than one nonzero term. The rest of the g j(x) are monomials and therefore the associated sums S j are zero, provided m j 2. Thus there are at most r values of j m for which m j 2 and S j is nonzero. Moreover, for these nonzero sums the multiplicity of any nonzero critical point is bounded above by M. Say d = k I for some I. Then since p t ka Ik I it is easily seen that for 0 j m (6.6) j + t j t + j(d + ): We split the sum in (6.4) into two parts according as m t j j 8M or not. If this inequality holds then since S j has at most p critical points, each of multiplicity M, it follows from Lemma 2.3 that p j j js j j p j j 4p p t j M p (m j )( M ) = 4p p (m j t j )=(2M) p m j p (m j t j )=(2M) Now (m j t j)=(2m) 4. Also, since p > 2r, 2M < d and so by (6.6) It follows that m j t j 2M m t j(d + ) d + = m t d + (6.7) p j js j j 4 p 3 p t d + d We consider next the set of j for which m t j j < 8M and let j o denote the least such j. Then p j j js j j p m jo = p d m t + jo p d t + d + ) Now by (6.6). Thus (6.8) jjo (m t) j o(d + ) 8M + t j o + j o t j o(d + ) 8M = 8rd =p`s ; jjo p j j js j j p 8r From (6.7) and (6.8) we conclude that js(f; p m )j This establishes t p`s p d + d 4r p 3 + p 8r p`s p t d + d + ) p 8r p`s j: + p p 2 d t + p m d + Lemma 6.. Suppose that pj(k j k i) for some k i < k j d, p > 4r and p =2 P. Then js(f; p m )j + p 8r t p`s p d p 2 + p m d + ; for some p`s 2 P with pj(k` k s), p < p`s. 7. Proof of Theorem. For any prime power p m and polynomial f let (7.) R(f; p m ) = js(f; pm )j d ) : Let f be a sparse polynomial with r terms and let q be a positive integer such that d p(f) for all prime divisors p of q. Write p m kq R(f; p m ) = P P 2 P 3 P 4 P 5 P 6 :

11 EPONENTIAL SUMS where the P i are products over the prime power divisors of q satisfying the following constraints (counting prime powers only once if they happen to satisfy more than one constraint): (7.2) (7.3) (7.4) (7.5) (7.6) (7.7) P = m= R(f; p m ); P 2 = P 3 = P 4 = P 5 = P 6 = <mt+ pc 3 or pr 4 or p2p R(f; p m ); R(f; p m ); p>r 4 ; pj(k j k i ); for some k i < k j d, p=2p mt+2; 50d>p>r 4 ; p>c 3 and p-(k j k i ) for all k i < k j d, mt+2; p>r 4 ; p>c 2 and p50d R(f; p m ); R(f; p m ); R(f; p m ); where C 2; C 3 are the constants in Lemmas 5.3 and 5.4 respectively, and P is the set of exceptional primes (6.). By (.6) the theorem follows if we show that each of the products P i is bounded by a constant depending only on r. By Lemma 5.2 the Weil bound (5.) and the trivial bound R(f; p) p =d we have P R(f; p) R(f; p) R(f; p) (:75) C 2+r 4 p =d (:75) r4 : p<c 2 pr 4 p<50d p<50d For the next few products we need the following Lemma 7.. Let f be a sparse polynomial with r terms of degree d. For any prime p let t p = ordp(f 0 (x)). Then letting p run through the set of all primes for which d p(f) we have p;dp(f) p tp d r : Proof. Let f(x) = a x k + + a rx kr and p be a prime with d p(f). Then for some i, p - a i, and so for this value of i, p tp jk i. Thus the product over all such p tp is a divisor of k k 2 : : : k r. (We continue to write t for t p.) For P 2 the condition < m t + implies that t and so m 2t. Thus we have trivially, P 2 p p m=d p p 2t=d d 2r=d (2:) r : The number of primes in the product P 3 is less than r 4 =2 + r 2 + C 3 < r 4 + C 3 and so by Lemma. P 3 5 r4 +C 3. For P 4 we apply Lemma 6. to obtain P 4 p t=d 8r p ) ij d r=d C 4r 5 :5 r C5 2r3 ; p + p 2 ( p p i<jr pp i;j i<jr P for some absolute constant C 5. We may take C 5 = sup x e (x)=x ; where (x) = px log p.

12 2 TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE For P 5, we apply Lemma 5.4 (iii) to obtain, P 5 p<50d p 2r+t d p p t=d p<50d d r=d e 2r(50d)=d :5 r C 00r 5 : Finally, we apply Lemma 5.3 to P 6 to obtain p 2r=d P 6 p t=d d r=d :5 r : p Thus the product P P 2P 3P 4P 5P 6 is bounded above by a constant depending only on r. References [] J.H.H. Chalk, On Hua's estimate for exponential sums, Mathematika 34 (987), [2] J.R. Chen, On the representation of natural numbers as a sum of terms of the form x(x + ) : : : (x + k )=k!, Acta Math. Sin. 8 (958), [3] On Professor Hua's estimate of exponential sums, Sci. Sinica 20 (977), [4] T. Cochrane, Mixed exponential sums modulo prime powers, preprint. [5] T. Cochrane, C. Pinner and J. Rosenhouse, Bounds on exponential sums and the polynomial Waring's problem mod p, preprint. [6] T. Cochrane and Z. Zheng, Pure and mixed exponential sums, Acta Arithmetica 9, no. 3, (999), [7] T. Cochrane and Z. Zheng, Exponential sums with rational function entries, to appear in Acta Arithmetica. [8] T. Cochrane and Z. Zheng, On upper bounds of Chalk and Hua for exponential sums, preprint. [9] P. Ding, An improvement to Chalk's estimation of exponential sums, Acta Arith. 59 no. 2 (99), [0] On a conjecture of Chalk, J. Number Theory 65 no. 2 (997),6-29. [] L.K. Hua, On exponential sums, J. Chinese Math. Soc. 20 (940), [2] On exponential sums, Sci. Record (Peking) (N.S.) (957), -4. [3] Additiv Primzahltheorie, Teubner, Leipzig (959), 2-7. [4] S.V. Konyagin and I.E. Shparlinski, On the distribution of residues of nitely generated multiplicative groups and their applications, Macquarie Mathematics Reports, Macquarie University, 995. [5] Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge, 999. [6] W.K.A. Loh, Hua's Lemma, Bull. Australian Math. Soc. (3) 50 (994), [7] Exponential sums on reduced residue systems, Canad. Math. Bull. Vol. 4 (2) (997), [8] J.H. Loxton and R.C. Vaughan, The estimation of complete exponential sums, Canad. Math. Bull. 28 no. 4 (985), [9] M. Lu, A note on the estimate of a complete rational trigonometric sum, Acta Math. Sin. 27 (984), [20] The estimate of complete trigonometric sums, Sci. Sin. 28, no. 6, (985), [2] A note on complete trigonometric sums for prime powers, Sichuan Daxue uebao 26 (989), [22] V.I. Necaev, An estimate of a complete rational trigonometric sum, Mat. Zametki 7 (975), ; English translation in Math. Notes 7 (975). [23] On the least upper bound on the modulus of complete trigonometric sums of degrees three and four, Investigations in number theory (Russian),Saratov. Gos. Univ., Saratov, (988), [24] V.I. Necaev and V.L. Topunov, Estimation of the modulus of complete rational trigonometric sums of degree three and four, Trudy Mat. Inst. Steklov, 58 (98), 25-29; English translation in Proceedings of the Steklov Institute of Mathematics 983, no. 4, Analytic number theory, mathematical analysis and their applications, Amer. Math. Soc., [25] M. Qi and P. Ding, Estimate of complete trigonometric sums, Kexue Tongbao 29 (984), [26] On Estimates of complete trigonometric sums, China Ann. Math. B6 (985), 0-20.

13 EPONENTIAL SUMS 3 [27] Further estimates of complete trigonometric sums, J. Tsinghua Univ. 29, no. 6, (989), [28] W.M. Schmidt, Equations over nite elds, L.N.M. 536, Springer-Verlag, Berlin, (976). [29] I.E. Shparlinski, On bounds of Gaussian sums, Matem. Zametki, 50 (99), (in Russian). [30] On Gaussian sums for nite elds and elliptic curves, Proc. st French-Soviet Workshop on Algebraic Coding, Paris, 99, Lect. Notes in Computer Sci., 537 (992), 5-5. [3] On exponential sums with sparse polynomials and rational functions, J. Number Theory 60 (996), [32] S.A. Stepanov, Arithmetic of Algebraic Curves, English translation, Monographs in Contemporary Mathematics, Consultants Bureau, New ork, (994). [33] S.B. Steckin, Estimate of a complete rational trigonometric sum, Proc. Steklov Inst. 43 (977), , English translation, A.M.S. Issue (980), [34] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (948), [35] M. Zhang and. Hong, On the maximum modulus of complete trigonometric sums, Acta Math. Sinica, New Series 3, no. 4 (987), Department of Mathematics, Kansas State University, Manhattan, KS address: cochrane@math.ksu.edu Department of Mathematics, Kansas State University, Manhattan, KS address: pinner@math.ksu.edu Department of Mathematics, Kansas State University, Manhattan, KS address: jasonr@math.ksu.edu