A FURTHER REFINEMENT OF MORDELL’S BOUND ON EXPONENTIAL SUMS TODD COCHRANE, JEREMY COFFELT, AND CHRISTOPHER PINNER 1. Introduction For a prime p, integer Laurent polynomial (1.1) f (x)= a 1 x k 1 + ··· + a r x kr , p ∤ a i , k i ∈ Z, where the k i are distinct and nonzero mod (p − 1), and multiplicative character χ mod p we consider the mixed exponential sum S (χ, f ) := p−1  x=1 χ(x)e p (f (x)), where e p (·) is the additive character e p (·)= e 2πi·/p on the finite field Z p . For such sums the classical Weil bound [5] (see [1] or [4] for Laurent f ) yields, (1.2) |S (χ, f )|≤ dp 1 2 , where d is the degree of f for a polynomial (degree of the numerator when f has both positive and negative exponents), nontrivial only if d< √ p. Mordell [3] gave a different type of bound which depended rather on the product of all the exponents k i . In [2] we obtained the following improvement in Mordell’s bound (1.3) |S (χ, f )|≤ 4 1 r (ℓ 1 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 omitted from the product (at the cost of a worse dependence on p) once r ≥ 3: Theorem 1.1. For any f and χ as above and positive integer m with 1 2 r<m ≤ r, |S (χ, f )|≤ 4 1 m (ℓ 1 ··· ℓ m ) 1 m 2 p 1− 1 m 2 (m− 1 2 r) , Date : September 6, 2009. 1991 Mathematics Subject Classification. 11L07;11L03. Key words and phrases. exponential sums. 1