SPARSE POLYNOMIAL EXPONENTIAL SUMS TODD COCHRANE, CHRISTOPHER PINNER, AND JASON ROSENHOUSE 1. Introduction In this paper we estimate the complete exponential sum (1.1) S (f,q)= q  x=1 e q (f (x)), where e q (·) is the additive character e q (·)= e 2πi·/q , and f is a sparse integer polynomial, (1.2) f (x)= a 1 x k 1 + ··· + a r x kr with 0 <k 1 <k 2 < ··· <k r . We assume always that the content of f , (a 1 ,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, (1.3) |S (f,q)|≤ Cq 1− 1 d , Date : November 18, 2002. 1991 Mathematics Subject Classification. 11L07;11L03. Key words and phrases. exponential sums. The research of the second author was supported in part by the National Science Foundation under grant EPS-9874732 and matching support from the State of Kansas. 1