Exponential Sums over Finite Fields and Differential Equations over the Complex Numbers: Some Interactions

A fundamental question (perhaps the fundamental question) we can ask about such an ƒ is: Does ƒ = 0 have a solution in integers? Clearly, if there exists an integer solution, then for any prime p there exists a solution in the finite field Z/pZ ; simply reduce mod p any integer solution. (For any ring R, ƒ maps R to R. When we say "a solution of ƒ = 0 in R," we mean an «-tuple of elements (a{, . . . , an) of R such that f (a) = 0 in R.) Thus, for example, equations of the form